(Part 2) Best mathematics books according to redditors
We found 7,687 Reddit comments discussing the best mathematics books. We ranked the 2,734 resulting products by number of redditors who mentioned them. Here are the products ranked 21-40. You can also go back to the previous section.
The answer is "virtually all of mathematics." :D
Although lots of math degrees are fairly linear, calculus is really the first big branch point for your learning. Broadly speaking, the three main pillars of contemporary mathematics are:
You might also think of these as the three main "mathematical mindsets" — mathematicians often talk about "thinking like an algebraist" and so on.
Calculus is the first tiny sliver of analysis and Spivak's Calculus is IMO the best introduction to calculus-as-analysis out there. If you thought Spivak's textbook was amazing, well, that's bread-n-butter analysis. I always thought of Spivak as "one-dimensional analysis" rather than calculus.
Spivak also introduces a bit of algebra, BTW. The first few chapters are really about abstract algebra and you might notice they feel very different from the latter chapters, especially after he introduces the least-upper-bound property. Spivak's "properties of numbers" (P1-P9) are actually the 9 axioms which define an algebraic object called a field. So if you thought those first few chapters were a lot of fun, well, that's algebra!
There isn't that much topology in Spivak, although I'm sure he hides some topology exercises throughout the book. Topology is sometimes called the study of "shape" and is where our most general notions of "continuous function" and "open set" live.
Here are my recommendations.
Analysis If you want to keep learning analysis, check out Introductory Real Analysis by Kolmogorov & Fomin, Principles of Mathematical Analysis by Rudin, and/or Advanced Calculus of Several Variables by Edwards.
Algebra If you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.
Topology There's really only one thing to recommend here and that's Topology by Munkres.
If you're a high-school student who has read through Spivak in your own, you should be fine with any of these books. These are exactly the books you'd get in a more advanced undergraduate mathematics degree.
I might also check out the Chicago undergraduate mathematics bibliography, which contains all my recommendations above and more. I disagree with their elementary/intermediate/advanced categorization in many cases, e.g., Rudin's Principles of Mathematical Analysis is categorized as "elementary" but it's only "elementary" if your idea of doing math is pursuing a PhD. Baby Rudin (as it's called) is to first-year graduate analysis as Spivak is to first-year undergraduate calculus — Rudin says as much right in the introduction.
Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
I'm 2 years into a part time physics degree, I'm in my 40s, dropped out of schooling earlier in life.
As I'm doing this for fun whilst I also have a full time job, I thought I would list what I'm did to supplement my study preparation.
I started working through these videos - Essence of Calculus as a start over the summer study whilst I had some down time. https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr
Ive bought the following books in preparation for my journey and to start working through some of these during the summer prior to start
Elements of Style - A nice small cheap reference to improve my writing skills
https://www.amazon.co.uk/gp/product/020530902X/ref=oh_aui_detailpage_o02_s00?ie=UTF8&psc=1
The Humongous Book of Trigonometry Problems https://www.amazon.co.uk/gp/product/1615641823/ref=oh_aui_detailpage_o08_s00?ie=UTF8&psc=1
Calculus: An Intuitive and Physical Approach
https://www.amazon.co.uk/gp/product/0486404536/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1
Trigonometry Essentials Practice Workbook
https://www.amazon.co.uk/gp/product/1477497781/ref=oh_aui_detailpage_o05_s00?ie=UTF8&psc=1
Systems of Equations: Substitution, Simultaneous, Cramer's Rule
https://www.amazon.co.uk/gp/product/1941691048/ref=oh_aui_detailpage_o05_s00?ie=UTF8&psc=1
Feynman's Tips on Physics
https://www.amazon.co.uk/gp/product/0465027970/ref=oh_aui_detailpage_o07_s00?ie=UTF8&psc=1
Exercises for the Feynman Lectures on Physics
https://www.amazon.co.uk/gp/product/0465060714/ref=oh_aui_detailpage_o08_s00?ie=UTF8&psc=1
Calculus for the Practical Man
https://www.amazon.co.uk/gp/product/1406756725/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1
The Feynman Lectures on Physics (all volumes)
https://www.amazon.co.uk/gp/product/0465024939/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1
I found PatrickJMT helpful, more so than Khan academy, not saying is better, just that you have to find the person and resource that best suits the way your brain works.
Now I'm deep in calculus and quantum mechanics, I would say the important things are:
Algebra - practice practice practice, get good, make it smooth.
Trig - again, practice practice practice.
Try not to learn by rote, try understand the why, play with things, draw triangles and get to know the unit circle well.
Good luck, it's going to cause frustrating moments, times of doubt, long nights and early mornings, confusion, sweat and tears, but power through, keep on trucking, and you will start to see that calculus and trig are some of the most beautiful things in the world.
It just comes from the way we define sums of infinite sums, aka series. .999... is just shorthand for (.9+.09+.09+.009...), which is an infinite sum. We define the sum of a series to be equal to the limit of the partial sums. The limit is rigorously defined, and you can read the definition on wikipedia if you google "epsilon delta". The limit of an infinite sum, if it exists, is unique. For this infinite sum, that limit is exactly 1. By the way we define infinite sums, .999... is therefore exactly equal to 1.
It's not so bad when you remember that all real numbers have a representation as a non-terminating decimal. 0.5 can be written as 0.4999... and 1/3 can be written as 0.333... and pi can be written as 3.14159... for example.
And lastly, if .999... and 1 are different real numbers, then there must exist a number between them. This is because of an axiom we have called trichotomy: for any two real numbers a and b, exactly one of the following is true: a<b, a=b, a>b. If a=/=b, then there exists a real number between them, because the real numbers have a property called "dense". It is easy to prove that here is no such number between .999... and 1, real or otherwise. Therefore .999... is exactly equal to 1.
e: The sum (.9+.09+.009...) is bigger than every real number less than 1. You can check if you want. The smallest number that is greater than every real number less than 1 is 1 itself. We get this from an axiom called the "least upper bound property". Therefore .999... is at least 1. Using our rigorous definition of a limit, we find that it is exactly 1.
e2: Apostol's Calculus vol 1 is a fantastic place to start learning about rigorous math shit. Chapter one starts you out with axioms for real numbers, and about half way through chapter 1 you prove the whole thing about repeating decimals corresponding to rational numbers. It is slow and easy to follow. Other people recommend Spivak but I haven't seen it so idk.
The rate of your learning is defined by your determination. If you don't give up then you will learn the material.
Look at the book that is required and only learn what you need in the class. Don't learn everything in the book either. Just learn what you need to do well and refer to the books when you get confused.
Note don't try to learn everything that's below. Only use it to learn what you actually need. This can be overwhelming at first but just set aside a set time to study this.
EDIT I added more books and courses.
OCW
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/
http://ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/index.htm
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/
http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
Helpful books
http://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321390539/ref=sr_1_3?s=books&amp;ie=UTF8&amp;qid=1312542911&amp;sr=1-3
http://www.amazon.com/Understanding-Probability-Chance-Rules-Everyday/dp/0521540364
http://www.amazon.com/gp/product/048663518X/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=486539851&amp;pf_rd_s=lpo-top-stripe-1&amp;pf_rd_t=201&amp;pf_rd_i=0155510053&amp;pf_rd_m=ATVPDKIKX0DER&amp;pf_rd_r=0YXJR9EVHCH9PCBDN372
Khan Academy
http://khan-academy.appspot.com/#calculus
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/probability--part-1?playlist=Old%20Algebra
http://www.youtube.com/user/keithpeterb#p/u/19/dS2p_APpcnI
http://khan-academy.appspot.com/video/linear-algebra--introduction-to-vectors?playlist=Linear%20Algebra
EDIT: I knew nothing about topological quantum computation about 1.5 years ago but then I took a independent study in college and I was assigned 1-3 papers a week to read. Eventually I got it a few months ago. What got me through it was not giving up...
OCW
Single Variable Calculus
Multivariable Calculus
Differential Equasions
Linear Algebra
Helpful books
Mathematical Proofs: A Transition to Advanced Mathematics
Understanding Probability: Chance Rules in Everyday Life
Linear Algebra
Other Resources
Kahn Academy
Probability (keithpeterb on youtube)
Probability (Kahn)
Linear Algebra (Kahn)
There, Fixed up your links.
This is the work of Abraham Wald. If you're interested in survivorship bias, and thinking mathematically in general, please consider reading this book, which discusses this exact story, among others. I just read it last week, and I recommend it.
I was in the same position as you in high school (and am finishing my math major this semester). Calculus is not "math" in the sense you're referring to it, which is pure mathematics, without application, just theory and logic. Calculus, as it is taught in high school, is taught as a tool, not as a theory. It is boring, tedious, and has no aesthetic appeal because it is largely taught as rote memorization.
Don't let this bad experience kill your enthusiasm. I'm not sure what specifically to recommend to you to perk your enthusiasm, but what I did in high school was just click around Wikipedia entries. A lot of them are written in layman enough terms to give you a glimpse and you inspire your interest. For example, I remember being intrigued by the Fibonacci series and how, regardless of the starting terms, the ratio between the (n-1)th and nth terms approaches the golden ratio; maybe look at the proof of that to get an idea of what math is beyond high school calculus. I remember the Riemann hypothesis was something that intrigued me, as well as Fermat's Last Theorem, which was finally proved in the 90s by Andrew Wiles (~350 years after Fermat suggested the theorem). (Note: you won't be able to understand the math behind either, but, again, you can get a glimpse of what math is and find a direction you'd like to work in).
Another thing that I wish someone had told me when I was in your position is that there is a lot of legwork to do before you start reaching the level of mathematics that is truly aesthetically appealing. Mathematics, being purely based on logic, requires very stringent fundamental definitions and techniques to be developed first, and early. Take a look at axiomatic set theory as an example of this. Axiomatic set theory may bore you, or it may become one of your interests. The concept and definition of a set is the foundation for mathematics, but even something that seems as simple as this (at first glance) is difficult to do. Take a look at Russell's paradox. Incidentally, that is another subject that captured my interest before college. (Another is Godel's incompleteness theorem, again, beyond your or my understanding at the moment, but so interesting!)
In brief, accept that math is taught terribly in high school, grunt through the semester, and try to read farther ahead, on your own time, to kindle further interest.
As an undergrad, I don't believe I yet have the hindsight to recommend good books for an aspiring math major (there are plenty of more knowledgeable and experienced Redditors who could do that for you), but here is a list of topics that are required for my undergrad math degree, with links to the books that my school uses:
And a couple electives:
And a couple books I invested in that are more advanced than the undergrad level, which I am working through and enjoy:
Lastly, if you don't want to spend hundreds of dollars on books that you might not end up using in college, take a look at Dover publications (just search "Dover" on Amazon). They tend to publish good books in paperback for very cheap ($5-$20, sometimes up to $40 but not often) that I read on my own time while trying to bear high school calculus. They are still on my shelf and still get use.
My main hobby is reading textbooks, so I decided to go beyond the scope of the question posed. I took a look at what I have on my shelves in order to recommend particularly good or standard books that I think could characterize large portions of an undergraduate degree and perhaps the beginnings of a graduate degree in the main fields that interest me, plus some personal favorites.
Neuroscience: Theoretical Neuroscience is a good book for the field of that name, though it does require background knowledge in neuroscience (for which, as others mentioned, Kandel's text is excellent, not to mention that it alone can cover the majority of an undergraduate degree in neuroscience if corequisite classes such as biology and chemistry are momentarily ignored) and in differential equations. Neurobiology of Learning and Memory and Cognitive Neuroscience and Neuropsychology were used in my classes on cognition and learning/memory and I enjoyed both; though they tend to choose breadth over depth, all references are research papers and thus one can easily choose to go more in depth in any relevant topics by consulting these books' bibliographies.
General chemistry, organic chemistry/synthesis: I liked Linus Pauling's General Chemistry more than whatever my school gave us for general chemistry. I liked this undergraduate organic chemistry book, though I should say that I have little exposure to other organic chemistry books, and I found Protective Groups in Organic Synthesis to be very informative and useful. Unfortunately, I didn't have time to take instrumental/analytical/inorganic/physical chemistry and so have no idea what to recommend there.
Biochemistry: Lehninger is the standard text, though it's rather expensive. I have limited exposure here.
Mathematics: When I was younger (i.e. before having learned calculus), I found the four-volume The World of Mathematics great for introducing me to a lot of new concepts and branches of mathematics and for inspiring interest; I would strongly recommend this collection to anyone interested in mathematics and especially to people considering choosing to major in math as an undergrad. I found the trio of Spivak's Calculus (which Amazon says is now unfortunately out of print), Stewart's Calculus (standard text), and Kline's Calculus: An Intuitive and Physical Approach to be a good combination of rigor, practical application, and physical intuition, respectively, for calculus. My school used Marsden and Hoffman's Elementary Classical Analysis for introductory analysis (which is the field that develops and proves the calculus taught in high school), but I liked Rudin's Principles of Mathematical Analysis (nicknamed "Baby Rudin") better. I haven't worked my way though Munkres' Topology yet, but it's great so far and is often recommended as a standard beginning toplogy text. I haven't found books on differential equations or on linear algebra that I've really liked. I randomly came across Quine's Set Theory and its Logic, which I thought was an excellent introduction to set theory. Russell and Whitehead's Principia Mathematica is a very famous text, but I haven't gotten hold of a copy yet. Lang's Algebra is an excellent abstract algebra textbook, though it's rather sophisticated and I've gotten through only a small portion of it as I don't plan on getting a PhD in that subject.
Computer Science: For artificial intelligence and related areas, Russell and Norvig's Artificial Intelligence: A Modern Approach's text is a standard and good text, and I also liked Introduction to Information Retrieval (which is available online by chapter and entirely). For processor design, I found Computer Organization and Design to be a good introduction. I don't have any recommendations for specific programming languages as I find self-teaching to be most important there, nor do I know of any data structures books that I found to be memorable (not that I've really looked, given the wealth of information online). Knuth's The Art of Computer Programming is considered to be a gold standard text for algorithms, but I haven't secured a copy yet.
Physics: For basic undergraduate physics (mechanics, e&m, and a smattering of other subjects), I liked Fundamentals of Physics. I liked Rindler's Essential Relativity and Messiah's Quantum Mechanics much better than whatever books my school used. I appreciated the exposition and style of Rindler's text. I understand that some of the later chapters of Messiah's text are now obsolete, but the rest of the book is good enough for you to not need to reference many other books. I have little exposure to books on other areas of physics and am sure that there are many others in this subreddit that can give excellent recommendations.
Other: I liked Early Theories of the Universe to be good light historical reading. I also think that everyone should read Kuhn's The Structure of Scientific Revolutions.
Think about basic high school math. You might have forgotten a few very specific ideas to solve a few very specific problems, but it's likely you remember almost all of it.
Why? Because you used it exhaustively in your basic undergrad math courses. Setting a derivative = 0 often demanded you factor, even if your directions never specifically said "find all solutions to this equation."
So, maybe you forgot a specific application of something like finding the principal value given blah blah compounded continuously, but you certainly know how to rearrange equations to solve for a variable.
Using something was practice, and so it was ingrained into your head (plus, after years of doing it, it's simple and downright monotonous).
But what about now? Were you extensively using Weierstrass's M-test on series in later classes? If you say yes, I won't believe you. Can you still find the integral of an obnoxious complex-valued function using residue theorems? Did you use these extensively in other classes? Doubtful, but possible.
This is the problem you are facing. I STRONGLY DOUBT you've been underexposed, but I HIGHLY AGREE with the possibility that you've forgotten.
So here's the important question: CAN you go back and relearn things? You say "progress is slow," but this is not a real answer to my question. Given one hour each day, can you, in 3 days, Mon/Wed/Fri, reteach yourself to determine if a metric space is compact? If you say Yes, then you are in a great position! There are many who sit through the class in one week and still have no clue! If you say No, then you're not necessarily in a BAD position (though you might be), you're just possibly in NO position.
So, here's the idea: you can't get good at upper level math (which will be considered lower level MATH math when you're going through grad school) by simply figuring it out. You got good at lower level math through practice; this is how you will get good at upper level math.
So what if progress is "slow"? Speed is subjective, but it's far more important that you CAN solve abstract problems rather than being able to blast through them--speed will develop later, and I know many PhD students at great schools who don't always remember what the subgroups of some strange group are or even how to find them.
So, let's answer, now, your REAL question: are you in for a rude awakening?
Yes, you are. But not for the reason you suspect. When you are in grad school, your faculty will (or it BETTER) have higher expectations of what you know vs. what you can do, and they're more concerned with what you can do than they are with what you know (forget something? Look it up. Forget how to do something? Looking it up may not help you...).
The fact that you are making ANY progress at ALL is enough to show that you are capable of doing things, even if you don't know things.
But are you in for a rude awakening because things are going to be hard because you've forgotten so much knowledge and thus you might have made a mistake because you'll never get up to speed? No. Most of my graduate level courses redefined things defined for me back as an undergrad, since at that level it gets difficult to figure out what students know and what they don't know based on where they came from.
But let's not build false hope and try and stay grounded in reality by this--
Check out this book: http://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071
Tinker through it and, when you're done, retake the MGRE. If all goes well, you're fine. If not, then you may very well not be. Don't rely entirely on that book to fill in gaps: use it for the TOPICS it presents, read through it, and when you're confused go find ANOTHER source relevant to the current chapter to fill in the gap.
But don't be crazy: I specifically never went through chapters 5,6,7,8,12,13,15,16 until I was in grad school. So, rather, figure out what you did as an undergrad, and go through THOSE relevant chapters in this book to get you up to speed with the ideas, and maybe dabble in some other chapters as time allows.
You should read this. I found it here a month ago on this subreddit, and it really stuck with me. I love those stories that help round out abstract concepts I've been thinking of.
More generally though... simple algebra used to be for the greatest thinkers alive. 'Ars Magna'. "The Great Art" written by Geromalo Cardano in the 1600s or whatever was the first European mathematical work that advanced beyond what was known by the Greeks... it gave a partial solution to how to find solutions to homogenous cubic polynomials (ax^3 + bx^2 + cx + d = 0)
he solved it with hilarious methods. Galileo used some incredibly painful notation where you're juggling ratios instead of... you know. Doing algebra the way we think of it. Fibonacci tried to encourage a switch to our standard number system, because arithmatic is RADICALLY easier when dealing with a simple base ten system instead of whatever crappy roman numeral type language they were using before. Took them 400 years to adopt our modern number system from the time the 'better' alternative was introduced.
All this is to say... you're absolutely right. The crystal core of the ideas we use can radically change our reach. What was impossible with one way of working becomes elementary when you can look at it right. But you've got a few layers of problems here. First... what's the right way of looking at it? I just read Judea Pearl's "Causality", and it's fascinating seeing a branch of math that's still so young, that there are arguments about what the definitions and axioms should even be. It's still a bubbling cauldron of ideas more so than an established branch. But even once you've gotten the 'right' way of looking at things (often there are many possible ways, you need to pick the right one for the job) now you're left with the arguably harder task of communication. How do you build a bridge to efficiently transmit a new way of thinking? I love 3blue1brown just because his whole shtick is finding new ways to graphically describe concepts that most people only vaguely understanding. The article I linked above (Ars Longa, Vita Brevis: 'long art, short life') breaks down the emergence of an art as being in 3 tiers... the inventors, the teachers, and the teacher teachers. The 'best' teachers I think are what you're asking about partly, but the right 'inventors' (what is the perfect framing that should be taught?) is part of the problem too.
Anyway, a related article you might also enjoy... thought as technology. A cool little exploration by Michael Nielson about the fact that 'how to think about things' is itself a technology, just one that's a pain in the ass to pass on compared to physical goods. He had some cool things to say on the topic you might also enjoy.
Also also... from a math perspective, I highly recommend you check out Alcock's how to think about analysis if you're looking for something fun to read. It's a very, very light introduction to real analysis, looking at the foundations of calculus, limits, series and convergence and so on. If you're interested in the 'heart' of what it means to learn math, I think you'll find that to be a pretty fun, approachable little book. You'll be able to blow through it in a couple weeks, but it'll give you some good framing for continuing the journey, if you're interested in doing so.
Munkres' book is the standard intro to topology. If you have no experience in it at all, it has a good intro to most everything you'll need to know in Point-Set Topology and the second part is a fairly intuitive intro to Algebraic Topology. Once you are familiar with Point-Set Topology, you can also learn from Hatcher.
The most important thing is to do the problems, you'll just be another buzzword-filled physics student if you don't prove anything.
A lot of it is probably confirmation bias, but yes, it does happen.
HP used to have expiry dates on their cartridges claiming they degraded printing after a certain time: http://www.hp.com/pageyield/articles/uk/en/InkExpiration.html
Another example from the software development world: Red Gate announced that one of their products (Reflector) would no longer be free starting from the next version and disabled all existing free copies, a move that upset many developers: http://www.infoq.com/news/2011/02/NET-Reflector-Not-Free
College textbooks are the most literal example of planned obsolescence; the new editions often contain very few new material and cost a lot while all older versions can be bought for almost nothing... and of course most classes require the new version.
For instance, Kenneth Rosen's "Discrete Mathematics and its Applications" currently sells for $125 if you want the [latest edition] (http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073383090/), $100 for the one before that and $16 for an older one even though the number of pages only increased by 100 each time. Thankfully, my teacher gave us the page numbers for both the latest and the second-latest editions...
And here's the free version:
https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_apa_i_V1UDDb4JBGWFX
> Mathematical Logic
It's not exactly Math Logic, just a bunch of techniques mathematicians use. Math Logic is an actual area of study. Similarly, actual Set Theory and Proof Theory are different from the small set of techniques that most mathematicians use.
Also, looks like you have chosen mostly old, but very popular books. While studying out of these books, keep looking for other books. Just because the book was once popular at a school, doesn't mean it is appropriate for your situation. Every year there are new (and quite frankly) pedagogically better books published. Look through them.
Here's how you find newer books. Go to Amazon. In the search field, choose "Books" and enter whatever term that interests you. Say, "mathematical proofs". Amazon will come up with a bunch of books. First, sort by relevance. That will give you an idea of what's currently popular. Check every single one of them. You'll find hidden jewels no one talks about. Then sort by publication date. That way you'll find newer books - some that haven't even been published yet. If you change the search term even slightly Amazon will come up with completely different batch of books. Also, search for books on Springer, Cambridge Press, MIT Press, MAA and the like. They usually house really cool new titles. Here are a couple of upcoming titles that might be of interest to you: An Illustrative Introduction to Modern Analysis by Katzourakis/Varvarouka, Understanding Topology by Shaun Ault. I bet these books will be far more pedagogically sound as compared to the dry-ass, boring compendium of facts like the books by Rudin.
If you want to learn how to do routine proofs, there are about one million titles out there. Also, note books titled Discrete Math are the best for learning how to do proofs. You get to learn techniques that are not covered in, say, How to Prove It by Velleman. My favorites are the books by Susanna Epp, Edward Scheinerman and Ralph Grimaldi. Also, note a lot of intro to proofs books cover much more than the bare minimum of How to Prove It by Velleman. For example, Math Proofs by Chartrand et al has sections about doing Analysis, Group Theory, Topology, Number Theory proofs. A lot of proof books do not cover proofs from Analysis, so lately a glut of new books that cover that area hit the market. For example, Intro to Proof Through Real Analysis by Madden/Aubrey, Analysis Lifesaver by Grinberg(Some of the reviewers are complaining that this book doesn't have enough material which is ridiculous because this book tackles some ugly topological stuff like compactness in the most general way head-on as opposed to most into Real Analysis books that simply shy away from it), Writing Proofs in Analysis by Kane, How to Think About Analysis by Alcock etc.
Here is a list of extremely gentle titles: Discovering Group Theory by Barnard/Neil, A Friendly Introduction to Group Theory by Nash, Abstract Algebra: A Student-Friendly Approach by the Dos Reis, Elementary Number Theory by Koshy, Undergraduate Topology: A Working Textbook by McClusckey/McMaster, Linear Algebra: Step by Step by Singh (This one is every bit as good as Axler, just a bit less pretentious, contains more examples and much more accessible), Analysis: With an Introduction to Proof by Lay, Vector Calculus, Linear Algebra, and Differential Forms by Hubbard & Hubbard, etc
This only scratches the surface of what's out there. For example, there are books dedicated to doing proofs in Computer Science(for example, Fundamental Proof Methods in Computer Science by Arkoudas/Musser, Practical Analysis of Algorithms by Vrajitorou/Knight, Probability and Computing by Mizenmacher/Upfal), Category Theory etc. The point is to keep looking. There's always something better just around the corner. You don't have to confine yourself to books someone(some people) declared the "it" book at some point in time.
Last, but not least, if you are poor, peruse Libgen.
I highly recommend reading "Mathematical Proofs: A Transition to Advanced Mathematics" by Gary Chartrand et. al. It helped me get a better understanding of how to write a proof as well as organize my own thoughts.
Here's the Amazon link: Mathematical Proofs: https://www.amazon.com/dp/0321797094/ref=cm_sw_r_cp_apa_i_V1UDDb4JBGWFX
Without knowing much about you, I can't tell how much you know about actual math, so apologies if it sounds like I'm talking down to you:
When you get further into mathematics, you'll find it's less and less about doing calculations and more about proving things, and you'll find that the two are actually quite different. One may enjoy both, neither, or one, but not the other. I'd say if you want to find out what higher level math is like, try finding a very basic book that involves a lot of writing proofs.
This one is aimed at high schoolers and I've heard good things about it, but never used it myself.
This one I have read (well, an earlier edition anyway) and think is a phenomenal way to get acquainted with higher math. You may protest that this is a computer science book, but I assure you, it has much more to do with higher math than any calculus text. Pure computer science essentially is mathematics.
Of course, you are free to dive into whatever subject interests you most. I picked these two because they're intended as introductions to higher math. Keep in mind though, most of us struggle at first with proofwriting, even with so-called "gentle" introductions.
One last thing: Don't think of your ability in terms of your age, it's great to learn young, but there's nothing wrong with people learning later on. Thinking of it as a race could lead to arrogance or, on the other side of the spectrum, unwarranted disappointment in yourself when life gets in the way. We want to enjoy the journey, not worry about if we're going fast enough.
Best of luck!
Here's a great reference that I didn't know about before grad school: "All the Mathematics You Missed: But Need to Know for Graduate School" https://www.amazon.com/dp/0521797071/ref=cm_sw_r_cp_api_-Zqryb5W5PQEA
It's not for learning new subjects, rather it's useful for seeing context and figuring out where your weak points are so that you can brush up using more thorough references.
if you want determinants, Shilov's is supposed to be "Determinants done right" I wouldn't recommend the other Dover LA book by Stoll
http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/product-reviews/048663518X/
-----------
Anyway: Free!
http://www.math.ucdavis.edu/~anne/linear_algebra/
http://www.math.ucdavis.edu/~linear/linear.pdf
http://www.cs.cornell.edu/courses/cs485/2006sp/LinAlg_Complete.pdf (Dawkins notes that were recently pulled off lamar.edu site, gentle intro like Anton's)
http://joshua.smcvt.edu/linearalgebra/
http://www.ee.ucla.edu/~vandenbe/103/reader.pdf
http://www.math.brown.edu/%7Etreil/papers/LADW/LADW.pdf
https://math.byu.edu/~klkuttle/Linearalgebra.pdf
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Or, google "positive definite matrix" or "hermitian" or "hessian" or some term like that and it will show you lecture notes from dozens of universities after the inevitable wikipedia and Wolfram hits
This book has a nice ELI5-style chapter on these voting systems. And it's just generally a really good book.
https://smile.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535/ref=sr_1_1?ie=UTF8&amp;qid=1496252174&amp;sr=8-1&amp;keywords=how+not+to+be+wrong
I don't think that is a very compelling argument, unless we believe mathematicians can do no notational wrong :-) The imprecise, ambiguous, sometimes obfuscatory notation that arises in multivariable calculus and the calculus of variations is a well known and frequently discussed issue. I think we underestimate the difficulty it causes to students, especially to students coming from other disciplines who aren't steeped in the mathematical vernacular.
It's been problematic enough that there are some high profile and semi-accepted attempts to refine the notation, such as the functional notation used in Spivak's Calculus on Manifolds, which is based in an earlier attempt from the 50s if I remember correctly. Another presentation of physics motivated in large part by fixing the notation is Sussman & Wisdom's Structure and Interpretation of Classical Mechanics which adopts Spivak's notation, and also uses computer programs to describe algorithms more precisely.
There's a nice little book, All the Mathematics You Missed: But Need to Know for Graduate School, that serves well as an answer to your question. It's pretty well-written, and lives up to the title.
In my opinion, the ideal undergraduate has had introductory courses in real analysis/advanced calculus, algebra, general topology, differential geometry of curves and surfaces, complex analysis, and combinatorics. Furthermore, more than one semester of linear algebra would be preferred.
I don't say this to be discouraging: Most people don't really have any idea what doing Physics at a high level looks like. I decided in High School that I wanted to be a physicist, and as luck would have it I'm a graduate student and I still enjoy it, but truth be told, the exposure you have in High School doesn't really prepare you for the reality. All that to say: There's no reason to decide at thirteen years old that you need a PhD in Physics! Maybe once you learn math beyond trig you'll decide it isn't for you, or maybe you'll love math and want to switch to a math degree.
All right, now that that's out of the way... You said you're learning trig, that's good, you need it. You also need some basic algebra skills. Then try to teach yourself basic calculus (limits, derivatives, integrals). Then you want to learn Linear Algebra and at least Ordinary Differential Equations.
You can also do some basic physics reading before you've learned the essentials. I really like George Gamow's books for this - he was a very well know and important physicist who also happened to write very accessible books that are very much for lay people but that also don't shy away completely from the math. I really enjoyed this one in particular.
For mathematics, I love Dover books - they're cheap AND good. Shilov, I've found, is clear and readable. This might not be introductory level, but it's inexpensive and let's you see what you're getting yourself into.
Last bit of advice for Physics is what one of my old high school teachers used to say - draw, label, and you can't go wrong. It's still mostly true.
I'm not so sure this is a fundamental difference, so much as a distinction in who is looking at each field. For the most part, category theory is studied by those who are looking to make advances in knowledge. Sure, the things researchers are looking at can be complex. But if you look at current research in abstract algebra, it's equally difficult to get up to speed and comprehend. The reason abstract algebra can be seen as simpler is that there is also introductory material, aimed at undergraduates, and even the general population.
Is it fundamentally impossible to produce such introductory material in category theory? Of course not! Several people have made serious and credible attempts. For example, here and here
Proceeding with knowledge gaps is something everyone has to do. In your case, you're going to have to improvise a lot. What I tend to do is put a black box on any confusing detail and write it off as
"this blackbox let's me do X." If a definition is confusing for me, I replace the definition with an example that I understand and leave it be. (Some definitions have this sort of infinite regression to it; to understand this definition, you need to understand these other 3 definitions, which requires you to understand these 9 definitions and so on.)
Normally people have to take classes and pass an exam so you have at least 1 year to build that knowledge.
I don't recommend trying to learn what took others years to learn in 1 month, that's just unrealistic. Talk to your advisor; a lot of times you don't need to know the subject 100%, just some parts of it.
For analysis, you might not need to know everything about it, just maybe what a Hilbert space is and some standard results. For complex, honestly I think that class was more to teach people how to do analysis (the proofs are very elegant and it really give you experience on how one ought to go about proving something in classical analysis), as far as results goes, I only know the residue theorem and Riemann mapping theorem. For algebra, I guess I know what all the structures are... but don't remember much else.
Oh, there's this book that supposedly give a good outline on math you need to know.
Anyone have any book recommendations for a 26 year old? No topic in particular, not necessarily financial/business or otherwise, just any suggestions?
I'm currently reading:
https://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535
I'm not far into it, but it's basically on how to properly apply mathematics and logic to problem-solving. It's not exactly a new strategy for life or anything, but it's probably a good idea to read if you're analytical. I got it off Bill Gates reading list.
https://www.amazon.com/How-Lie-Statistics-Darrell-Huff/dp/0393310728
Found through the reading list- This one I've finished and can't recommend enough. It's from the 50's and it's intended reader were investment bankers. The main suggestion is hide yourself from bad information because you can't eliminate the impact it'll have on your decision making, and we aren't exactly equipped to know what's good or bad if we don't have experience in that realm already. It's a lot of common stuff people use stats for to push a product service policy etc.
https://www.amazon.com/Starship-Troopers-Robert-Heinlein/dp/0441783589/
I'm really into it. I love sci-fi. I don't necessarily love philosophy, but I'm really enjoying this book. It's hard for me to read a lot of at once but I don't ever want to put it down. The mindset of the character and narration really gets me. Since reading this, I've heard or noticed many many recommendations for Heinlein, though I'm unsure. He seems to be a proponent of fascism, but I guess he could just be writing down the fantasy of the particular fascist society he created and not necessarily saying "ya know this is how we should be" I don't know. I see conflicting things.
The textbook is usually Discrete Mathematics and It's Applications. Don't let the bookstore fool you, the sell the "UCF Edition" for an inflated price. The only difference is an additional introduction. Here is an Amazon link where you should be able to find a reasonably priced used copy. Alternatively, here is a link to a PDF copy that you can have for free! Enjoy =)
Pick up mathematics. Now if you have never done math past the high school and are an "average person" you probably cringed.
Math (an "actual kind") is nothing like the kind of shit you've seen back in grade school. To break into this incredible world all you need is to know math at the level of, say, 6th grade.
Intro to Math:
These books only serve as samplers because they don't even begin to scratch the surface of math. After you familiarized yourself with the basics of writing proofs you can get started with intro to the largest subsets of math like:
Intro to Abstract Algebra:
There are tons more books on abstract/modern algebra. Just search them on Amazon. Some of the famous, but less accessible ones are
Intro to Real Analysis:
Again, there are tons of more famous and less accessible books on this subject. There are books by Rudin, Royden, Kolmogorov etc.
Ideally, after this you would follow it up with a nice course on rigorous multivariable calculus. Easiest and most approachable and totally doable one at this point is
At this point it's clear there are tons of more famous and less accessible books on this subject :) I won't list them because if you are at this point of math development you can definitely find them yourself :)
From here you can graduate to studying category theory, differential geometry, algebraic geometry, more advanced texts on combinatorics, graph theory, number theory, complex analysis, probability, topology, algorithms, functional analysis etc
Most listed books and more can be found on libgen if you can't afford to buy them. If you are stuck on homework, you'll find help on [MathStackexchange] (https://math.stackexchange.com/questions).
Good luck.
Conceptual Mathematics by Lawvere and Schanuel is a good low level introduction to category theory (and a bit of set theory) if you are feeling shaky on those grounds. From there lots of books open up to you.
The best books I know on how to "think" like a functional programmer are all written by Richard Bird. http://www.amazon.com/gp/product/1107452643/ref=pd_lpo_sbs_dp_ss_1?pf_rd_p=1944579842&amp;pf_rd_s=lpo-top-stripe-1&amp;pf_rd_t=201&amp;pf_rd_i=0134843460&amp;pf_rd_m=ATVPDKIKX0DER&amp;pf_rd_r=090NKMWKY6078Z0WPCTW http://www.amazon.com/Pearls-Functional-Algorithm-Design-Richard/dp/0521513383
Not much is available in book form, especially that I can recommend on the FRP front.
Dependent types is a broad area, you're going to find yourself reading a lot of research papers. You might be able to get by with something more practical like Chlipala's Certified Programming with Dependent Types, but if you want a more theoretical treatment then perhaps Zhaohui Luo's Computation and Reasoning might be a better starting point.
/u/another_user_name posted this list a while back. Actual aerospace textbooks are towards the bottom but you'll need a working knowledge of the prereqs first.
Non-core/Pre-reqs:
Mathematics:
Calculus.
1-4) Calculus, Stewart -- This is a very common book and I felt it was ok, but there's mixed opinions about it. Try to get a cheap, used copy.
1-4) Calculus, A New Horizon, Anton -- This is highly valued by many people, but I haven't read it.
1-4) Essential Calculus With Applications, Silverman -- Dover book.
More discussion in this reddit thread.
Linear Algebra
3) Linear Algebra and Its Applications,Lay -- I had this one in school. I think it was decent.
3) Linear Algebra, Shilov -- Dover book.
Differential Equations
4) An Introduction to Ordinary Differential Equations, Coddington -- Dover book, highly reviewed on Amazon.
G) Partial Differential Equations, Evans
G) Partial Differential Equations For Scientists and Engineers, Farlow
More discussion here.
Numerical Analysis
5) Numerical Analysis, Burden and Faires
Chemistry:
Physics:
2-4) Physics, Cutnel -- This was highly recommended, but I've not read it.
Programming:
Introductory Programming
Programming is becoming unavoidable as an engineering skill. I think Python is a strong introductory language that's got a lot of uses in industry.
Core Curriculum:
Introduction:
Aerodynamics:
Thermodynamics, Heat transfer and Propulsion:
Flight Mechanics, Stability and Control
5+) Flight Stability and Automatic Control, Nelson
5+)[Performance, Stability, Dynamics, and Control of Airplanes, Second Edition](http://www.amazon.com/Performance-Stability-Dynamics-Airplanes-Education/dp/1563475839/ref=sr_1_1?ie=UTF8&amp;qid=1315534435&amp;sr=8-1, Pamadi) -- I gather this is better than Nelson
Engineering Mechanics and Structures:
3-4) Engineering Mechanics: Statics and Dynamics, Hibbeler
6-8) Analysis and Design of Flight Vehicle Structures, Bruhn -- A good reference, never really used it as a text.
G) Introduction to the Mechanics of a Continuous Medium, Malvern
G) Fracture Mechanics, Anderson
G) Mechanics of Composite Materials, Jones
Electrical Engineering
Design and Optimization
Space Systems
There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.
To illustrate my point:
Linear Algebra:
Linear Algebra Through Geometry by Banchoff and Wermer
3. Here's more rigorous/abstract Linear Algebra for undergrads:
Linear Algebra Done Right by Axler
4. Here's more advanced grad level Linear Algebra:
Advanced Linear Algebra by Steven Roman
-----------------------------------------------------------
Calculus:
Calulus by Spivak
3. Full-blown undergrad level Analysis(proof-based):
Analysis by Rudin
4. More advanced Calculus for advance undergrads and grad students:
Advanced Calculus by Sternberg and Loomis
The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
Here's how you start studying real math NOW:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
Discrete Math by Susanna Epp
How To prove It by Velleman
Intro To Category Theory by Lawvere and Schnauel
There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
Good Luck, buddyroo.
I didn't read your other post. But one suggestion is that if you're having trouble in a grad level class, figure out what undergrad prereq you're missing. For example I was weak in multivariable calculus so I had a heck of a time in differential geometry. The problem wasn't with the grad level material, it was my lack of mastery of the corresponding undergrad material.
So figure out what you missed in your undergrad years and work on that.
There's a book I wish had been available when I was in grad school.
All the Mathematics You Missed: But Need to Know for Graduate School
Also do you join study groups with other students? That's a great way to learn.
It really depends which direction in mathematics you want to go. Even as a math major, I didn't really understand how vast it was until I got into abstract math.
My favorite way to learn is browse Amazon for "Dover Books on Mathematics." They are generally had for a penny + shipping if you don't mind buying used.
A good intro into modern mathematics: https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247
The standard textbook, which doesn't require much background (just calculus and a bit of set theory) is Topology by James R. Munkres.
Topology stands at the base of many mathematical subjects, but I don't know of many real world applications of general topology per se. Algebraic topology and knot theory have applications in biology, astronomy and I'm sure plenty else.
If you are getting your degree in math or computer science, you will probably have to take a course on "Discrete math" (or maybe an "introduction to proofs") in your first year or two (it should be by your 3rd semester). Unfortunately, this will probably be the first time you will take a course that is more about the why than the how. (On the bright side, almost everything after this will focus on why instead of how.) Depending on how linear algebra is taught at your university, and the order you take classes in, linear algebra may be also be such a class.
If your degree is anything else, you may have no formal requirement to learn the why.
For the math you are learning right now, analysis is the "why". I'm not sure of a good analysis book, but there are two calculus books which treat the subject more like a gentle introduction to analysis-- Apostol's and Spivak's. Your library might have a copy you can check out. If not, you can probably find pdfs (which are probably[?] legal) online.
I found Axler's Linear Algebra Done Right to be a very easy to digest introduction to abstract linear algebra.
None of these I've finished, but they're on the backburner whenever I have free time.
A Singular Mathematical Promenade (Etienne Ghys)
Music: A Mathematical Offering (Dave Benson)
Nonlinear Dynamics and Chaos (Strogatz)
well my favorite subjects in graduate school were differential geometry, particularly the theory of smooth manifolds, and functional analysis, in particular distributions. once i got a job writing software and dealing with hardware systems, i tried to keep up with my math (a losing battle somewhat) to see what connections i could make, and i eventually found the book conceptual mathematics: a first introduction to categories. i was also at the same time trying to pick up haskell, so between haskell and the category theory book and my job and my mathematics background, i started to realize that there are some connections between what people do in software and systems and the math. then i came across the book/paper category theory for scientists.
i'm now convinced that category can serve as a fantastic foundation for applied mathematics. when people think of applied mathematics, they immediately think discrete, combinatorial mathematics or throw differential equations at whatever problem is at hand. but i think there's a lot of the more abstract mathematics that can be applied, and i think (or at least agree with the authors of the materials i linked to) that category theory can help with this. you should also take a look at the work of robert ghrist as well, who applies algebraic topology to many engineering problems.
This:
http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X
is a good book as an introduction for a math student!
The classic textbook for a first course in topology is Topology by Munkres. It's a very good book.
Michael Starbird offers his topology "book" free of charge on his website. Here's the link. It's really closer to lecture notes for the course, and it's intended for an inquiry-based learning (IBL) course. What this means is that all of the proofs are omitted. The reader is expected to prove each result themselves. This obviously works much better in a group setting.
If you see any book titled "algebraic topology," I would recommend you ignore it for now. Algebraic topology courses assume you've at least had the one semester course in point-set topology (i.e. the books I linked) and one or two semesters in abstract algebra.
The standard/classic intro undergrad textbook is Munkres.
I actually never took a proper Topology course, I've just been forced to pick up a lot of it along the way. This book has been helpful for that. It's very friendly for reading/self-study.
If you don't want to buy a $60 book, I'm sure you can find it online somewhere, though I learn a lot better when trying to teach myself from a book I can easily flip through rather than a pdf in any form.
We need to make a few definitions.
A group is a set G together with a pair of functions: composition GxG -> G and inverse G -> G, satisfying certain properties, as I'm sure you know.
A topological group is a group G which is also a topological space and such that the composition and inverse functions are continuous. It makes sense to ask if a topological group for example is connected. Every group is a topological group with the discrete topology, but in general there is no way to assign an interesting (whatever that means) topology to a group. The topology is extra information that comes with a topological group.
A Lie group is more than a topological group. A Lie group is a group G that is also a smooth manifold and such that the composition and inverse are smooth functions (between manifolds).
In the same way that
O(n)is the set of matrices which fix the standard Euclidean metric onR^n, the Lorentz groupO(3,1)is the set of invertible 4x4 matrices which fix the Minkowski metric onR^4. The Lorentz group inherits a natural topology from the set of all 4x4 matrices which is homeomorphic toR^16. It is some more work to show that the Lorentz group in fact smooth, that is, a Lie group.It is easy to see the Lorentz group is not connected: it contains orientation preserving (det 1) matrices and orientation reversing (det -1) matrices. All elements are invertible (det nonzero), so the preimage of
R+andR-under the determinant are disjoint connected components of the Lorentz group.There are lots of references. Munkres Topology has a section on topological groups. Stillwell's Naive Lie Theory seems like a great undergraduate introduction to basic Lie groups, although he restricts to matrix Lie groups and does not discuss manifolds. To really make sense of Lie theory, you also need to understand smooth manifolds. Lee's excellent Introduction to Smooth Manifolds is an outstanding introduction to both. There are lots of other good books out there, but this should be enough to get you started.
I speak of this famous calculus book: https://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Which is a "theoretical" approach to Calculus rather than a mechanical approach.
You won't get a hang of anything until and unless you practice. Since you are having Object Oriented Programming, go on and make a project. This will give you a sense of accomplishment and on the way you will learn a lot of things.
Talking about data structures, you will need the concepts of this course everywhere. I would suggest you to strengthen your basics by refering to CLRS or some other resource, that is totally your choice. But, implement the data structure you have learned. There are a lot of resources out there, I am listing some of my favorites:
>https://www.youtube.com/playlist?list=PL2_aWCzGMAwI3W_JlcBbtYTwiQSsOTa6P
>https://www.coursera.org/specializations/algorithms
I would also suggest you to read discrete mathematics. The book that I use is
>https://www.amazon.com/Discrete-Mathematics-Applications-Seventh-Higher/dp/0073383090/ref=sr_1_1?ie=UTF8&amp;qid=1492831532&amp;sr=8-1&amp;keywords=discrete+mathematics+kenneth+rosen
You can also go through the discrete mathematics course from MIT OCW.
In case you need some help, PM me. I'll be more than happy to help :)
I don't see why you couldn't start with the standard graduate math text on topology, Munkres. If you have the formal maturity to do proofs, you can just start here. Analysis and abstract algebra not necessary.
There's also Zomorodian, which I wouldn't consider a complete introduction to topology in a mathematical sense, but the intended audience here is exactly you. YMMV.
Anti-disclaimer: I do have personal experience with all the below books.
I really enjoyed Lee for Riemannian geometry, which is highly related to the Lorentzian geometry of GR. I've also heard good things about Do Carmo.
It might be advantageous to look at differential topology before differential geometry (though for your goal, it is probably not necessary). I really really liked Guillemin and Pollack. Another book by Lee is also very good.
If you really want to dig into the fundamentals, it might be worthwhile to look at a topology textbook too. Munkres is the standard. I also enjoyed Gamelin and Greene, a Dover book (cheap!). I though that the introduction to the topology of R^n in the beginning of Bartle was good to have gone through first.
I'm concerned that I don't see linear algebra in your course list. There's a saying "Linear algebra is what separates Mathematicians from everyone else" or something like that. Differential geometry is, in large part, about tensor fields on manifolds, and these are studied by looking at them as elements of a vector space, so I'd say that linear algebra is something you should get comfortable with before proceeding. (It's also great to study it before taking quantum.) I can't really recommend a great book from personal experience here; I learned from poor ones :( .
Also, there are physics GR books that contain semi-rigorous introductions to differential geometry, even if these sections are skipped over in the actual class. Carroll is such a book. If you read the introductory chapter and appendices, you'll know a lot. On the differential topology side of things, there's Schutz, which is a great book for breadth but is pretty material dense. Schwarz and Schwarz is a really good higher level intro to special relativity that introduces the mathematical machinery of GR, but sticks to flat spaces.
Finally, once you have reached the mountain top, there's Hawking and Ellis, the ultimate pinnacle of gravity textbooks. This one doesn't really fall under the anti-disclaimer from above; it sits on my shelf to impress people.
Also about the relationship between statistical analyses and building narratives.
There's a super good book that breaks a lot of this down in detail: https://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535
Your professors really aren't expecting you to reinvent groundbreaking proofs from scratch, given some basic axioms. It's much more likely that you're missing "hints" - exercises often build off previous proofs done in class, for example.
I appreciated Laura Alcock's writings on this, in helping me overcome my fear of studying math in general:
https://www.amazon.com/How-Study-as-Mathematics-Major/dp/0199661316/
https://www.amazon.com/dp/0198723539/ <-- even though you aren't in analysis, the way she writes about approaching math classes in general is helpful
If you really do struggle with the mechanics of proof, you should take some time to harden that skill on its own. I found this to be filled with helpful and gentle exercises, with answers: https://www.amazon.com/dp/0989472108/ref=rdr_ext_sb_ti_sims_2
And one more idea is that it can't hurt for you to supplement what you're learning in class with a more intuitive, chatty text. This book is filled with colorful examples that may help your leap into more abstract territory: https://www.amazon.com/Visual-Group-Theory-Problem-Book/dp/088385757X
You could try Abbott's Understanding Analysis. Quite a few students seem to like this book.
One concrete suggestion I can give you is when faced with a theorem or definition, try first to understand what it means in 'words' and then try to reason why it may be true, again in 'words'. I've noticed that often what trips students up is the symbolism -- often when I see incorrect answers from bright students, 10 to 1, its because they've got caught up in symbols and are now mentally running around in circles. This, I feel, is the unfortunate transition-pangs from school math to real math.
Remember math is not about symbols, formulas or equations, its about the concepts and ideas that hide behind those things.
If you're looking for other texts, I would suggest Spivak's Calculus and Calculus on Manifolds. At first the text may seem terse, and the exercises difficult, but it will give you a huge advantage for later (intermediate-advanced) undergraduate college math.
It may be a bit obtuse to recommend you start with these texts, so maybe your regular calculus texts, supplemented with linear algebra and differential equations, should be approached first. When you start taking analysis and beyond, though, these books are probably the best way to return to basics.
Not sure what level you're approaching it from, but Steve Strogatz's Nonlinear Dynamics and Chaos is a pretty good upper-level undergraduate introduction to the topic.
I think the most important part of being able to see beauty in mathematics is transitioning to texts which are based on proofs rather than application. A side effect of gaining the ability to read and write proofs is that you're forced to deeply understand the theory of the math you're learning, as well as actively using your intuition to solve problems, rather than dry route calculations found in most application based textbooks. Based on what you've written, I feel you may enjoy taking this path.
Along these lines, you could start of with Book of Proof (free) or How to Prove It. From there, I would recommend starting off with a lighter proof based text, like Calculus by Spivak, Linear Algebra Done Right by Axler, or Pinter's book as you mentioned. Doing any intro proofs book plus another book at the level I mentioned here would have you well prepared to read any standard book at the undergraduate level (Analysis, Algebra, Topology, etc).
I'd like to give you my two cents as well on how to proceed here. If nothing else, this will be a second opinion. If I could redo my physics education, this is how I'd want it done.
If you are truly wanting to learn these fields in depth I cannot stress how important it is to actually work problems out of these books, not just read them. There is a certain understanding that comes from struggling with problems that you just can't get by reading the material. On that note, I would recommend getting the Schaum's outline to whatever subject you are studying if you can find one. They are great books with hundreds of solved problems and sample problems for you to try with the answers in the back. When you get to the point you can't find Schaums anymore, I would recommend getting as many solutions manuals as possible. The problems will get very tough, and it's nice to verify that you did the problem correctly or are on the right track, or even just look over solutions to problems you decide not to try.
Basics
I second Stewart's Calculus cover to cover (except the final chapter on differential equations) and Halliday, Resnick and Walker's Fundamentals of Physics. Not all sections from HRW are necessary, but be sure you have the fundamentals of mechanics, electromagnetism, optics, and thermal physics down at the level of HRW.
Once you're done with this move on to studying differential equations. Many physics theorems are stated in terms of differential equations so really getting the hang of these is key to moving on. Differential equations are often taught as two separate classes, one covering ordinary differential equations and one covering partial differential equations. In my opinion, a good introductory textbook to ODEs is one by Morris Tenenbaum and Harry Pollard. That said, there is another book by V. I. Arnold that I would recommend you get as well. The Arnold book may be a bit more mathematical than you are looking for, but it was written as an introductory text to ODEs and you will have a deeper understanding of ODEs after reading it than your typical introductory textbook. This deeper understanding will be useful if you delve into the nitty-gritty parts of classical mechanics. For partial differential equations I recommend the book by Haberman. It will give you a good understanding of different methods you can use to solve PDEs, and is very much geared towards problem-solving.
From there, I would get a decent book on Linear Algebra. I used the one by Leon. I can't guarantee that it's the best book out there, but I think it will get the job done.
This should cover most of the mathematical training you need to move onto the intermediate level physics textbooks. There will be some things that are missing, but those are usually covered explicitly in the intermediate texts that use them (i.e. the Delta function). Still, if you're looking for a good mathematical reference, my recommendation is Lua. It may be a good idea to go over some basic complex analysis from this book, though it is not necessary to move on.
Intermediate
At this stage you need to do intermediate level classical mechanics, electromagnetism, quantum mechanics, and thermal physics at the very least. For electromagnetism, Griffiths hands down. In my opinion, the best pedagogical book for intermediate classical mechanics is Fowles and Cassidy. Once you've read these two books you will have a much deeper understanding of the stuff you learned in HRW. When you're going through the mechanics book pay particular attention to generalized coordinates and Lagrangians. Those become pretty central later on. There is also a very old book by Robert Becker that I think is great. It's problems are tough, and it goes into concepts that aren't typically covered much in depth in other intermediate mechanics books such as statics. I don't think you'll find a torrent for this, but it is 5 bucks on Amazon. That said, I don't think Becker is necessary. For quantum, I cannot recommend Zettili highly enough. Get this book. Tons of worked out examples. In my opinion, Zettili is the best quantum book out there at this level. Finally for thermal physics I would use Mandl. This book is merely sufficient, but I don't know of a book that I liked better.
This is the bare minimum. However, if you find a particular subject interesting, delve into it at this point. If you want to learn Solid State physics there's Kittel. Want to do more Optics? How about Hecht. General relativity? Even that should be accessible with Schutz. Play around here before moving on. A lot of very fascinating things should be accessible to you, at least to a degree, at this point.
Advanced
Before moving on to physics, it is once again time to take up the mathematics. Pick up Arfken and Weber. It covers a great many topics. However, at times it is not the best pedagogical book so you may need some supplemental material on whatever it is you are studying. I would at least read the sections on coordinate transformations, vector analysis, tensors, complex analysis, Green's functions, and the various special functions. Some of this may be a bit of a review, but there are some things Arfken and Weber go into that I didn't see during my undergraduate education even with the topics that I was reviewing. Hell, it may be a good idea to go through the differential equations material in there as well. Again, you may need some supplemental material while doing this. For special functions, a great little book to go along with this is Lebedev.
Beyond this, I think every physicist at the bare minimum needs to take graduate level quantum mechanics, classical mechanics, electromagnetism, and statistical mechanics. For quantum, I recommend Cohen-Tannoudji. This is a great book. It's easy to understand, has many supplemental sections to help further your understanding, is pretty comprehensive, and has more worked examples than a vast majority of graduate text-books. That said, the problems in this book are LONG. Not horrendously hard, mind you, but they do take a long time.
Unfortunately, Cohen-Tannoudji is the only great graduate-level text I can think of. The textbooks in other subjects just don't measure up in my opinion. When you take Classical mechanics I would get Goldstein as a reference but a better book in my opinion is Jose/Saletan as it takes a geometrical approach to the subject from the very beginning. At some point I also think it's worth going through Arnold's treatise on Classical. It's very mathematical and very difficult, but I think once you make it through you will have as deep an understanding as you could hope for in the subject.
Attention Publishers
This is why readers hate you. Note the version number. Seventh Edition? Really, how much has calculus changed in the past 20 years? The past 50? Or 100? I only graduated 4 years ago, and this is the second time they've cranked out a new version of the book since my freshman year.
Of course they quit printing the older editions, because they can cripple the market for used textbooks and force everyone to buy new versions. So they go and re-hash and reword a chapter here and there and pretend it's a "new" book somehow.
I seriously doubt it takes until the 4th, 5th, or 6th printing of a book for the publisher to recoup their investment; if it does, I think the only reason is that they're writing themselves such large checks.
How to Ace Calculus: The Streetwise Guide is charming. It does an excellent job scaffolding intuitive understanding without unnecessarily sacrificing rigor. It took me at least three attempts to properly spell the word "unnecessarily" in the previous sentence.
Extremely delayed edit: It also has the marked advantage of being quite cheap.
If you're currently at the pre-calc level, you could probably get away with learning from khan academy for a little while. After that (and building some familiarity with proof writing), you'd be ready for some of the pure math classes like abstract algebra and real analysis. For those courses, you'll probably want to check out some Open Courseware. You'd want to treat it like a real class; watch the lectures online and read from the textbooks, while working on problem sets.
While you're working your way through the khan academy stuff, you might want to check out Stewart's calculus book; it's pretty solid for making your way through the calculus sequence.
I'd ask around for a good linear algebra book, since I haven't encountered a decent one that's at that level.
Everyone has to take MATH 150--MATH 152's prerequisite isn't Calculus 12. So after 150, you're at the same level as everyone else.
A tip: make sure when studying, you understand every part of what's being taught. You won't be able to just memorize this stuff. If you don't get something, spend a bit of time trying to figure it out, move forward if the following information doesn't rely on what you're passing, but come back to it later and try again and again till you understand what that thing is, how it works, and why. YouTube the name of what you're having trouble with, cause there are going to be several tutorials from people on there per topic.
You'll have to put in the hours, though, and study smart. Remember: being a student is your job, and 3 courses is full time (equivalent to 9-5 Mon-Fri). SFU uses the "flipped classroom" where you're supposed to read the sections of the textbook before class, the lecture reinforces and clarifies the most important stuff, then you self-study till you understand it 100%.
The rule of thumb for all classes is 2-3 hours of study for every hour in lecture. That means for MATH 150 you should expect to spend 8-12 hours studying on your own outside of class.
Engineering requires 12 credits/semester, so you'd have at least 13 in the semester you take 150--That means 26-39 hours of studying on your own outside class i.e. 6 hours a day 7 days a week, 6.5 hours every day but Sat/Sun, or 8 hours a day Mon-Fri.
Here are a couple useful resources:
Also, FYI for YouTube, if you hit shift+> on YouTube, it speeds up the video 1.25x 1.5x 2.0x -- which has really saved my ass, cause I don't have patience for slow talkers. (Conversely, shift+< slows the video back down)
CS182 is a discrete mathematics course. It has a lot to do with logic and proofs, and less to do with algebra and calculus. Most have never really seen what you will be covering. If you can, I would get the book and work through some of the problems before the start of the semester.
CS240 is similar to CS180, but it is taught in C — a much lower-level language. Once again, I recommend getting the book (I assume it will be The C Programming Language) and doing some of the exercises. Java syntax comes from C/C++, so that part will be somewhat familiar. C is pretty barebones, though. There are no classes, only functions. There is no
ArrayList,LinkedList, etc. You have to build it all yourself. And when you allocate memory usingmalloc()(similar to callingnew), you have to remember to free it when you’re done usingfree(). There is no garage collection.Good luck!
I don't think you'll "spoil" what you'll learn later. If anything, seeing the material before will help you understand cooler stuff during the class next year. There's a lot of remarks and subtle examples I missed the first time I went through the standard undergrad math topics, that I only learned later.
But if you still want to avoid the topics you'll see in class, you could try some point-set topology (e.g. Munkres Topology). It would be beneficial for the real analysis class too. For differential geometry, I'd recommend Jänich Vector Analysis, which says it only needs calculus and linear algebra as prereqs.
I would say book of proof is the easiest to get you started and its free online with solutions!
I would start with the above its the easiest to read for an introduction to proofs books I have come across yet it still presents everything you need.
If you want some more challenging problems I would recommend
A transition to advanced mathematics
Understanding Analysis by Abbott is a book that is more gentle than most.
Understanding Analysis is a very nice book I used to get a good grasp on the concepts behind real analysis. It goes at a very nice pace, perfect for the analysis novice.
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
Yeah, definitely the best book I've read on differential forms was Spivaks Calculus on Manifolds. Its very readable once you have a solid foundational calculus background and is pretty small given what it covers (160pp). If you need to know this stuff then this is definitely the right place to learn it.
once you get into partial differential equations, you'll be able to understand them. the basic ideas are pretty simple. there's just a bunch of computational overhead
this is a great book: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0813349109/ref=dp_ob_title_bk
it's informal and pretty easy to read. I don't remember it being so expensive though. i could've sworn i paid $20 for it
My suggestion: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0813349109/ref=sr_1_1?ie=UTF8&amp;qid=1494109595&amp;sr=8-1&amp;keywords=strogatz+nonlinear+dynamics+and+chaos
Should be approachable for someone who has only completed Calc I.
I'm not sure about PDE's, but ODE's are more than just existence and uniqueness theorems. You could argue that the modern study of ODE's is now dynamical systems.
Strogatz's Nonlinear Dynamics and Chaos is a classic if you want to know what applied dynamical systems is like. A more formal text that still captures some interesting ideas is Hale and Kocak's Dynamics and Bifurcations.
Reading textbooks is, of course, a huge time commitment. So perhaps go talk to the dynamical systems people in your department and ask them what is interesting about ODE's. Hell, even go talk to the numerical analysis and do the same for PDE's. Assuming you haven't taken a numerical analysis class, you might be surprised how "pure" numerical analysis feels.
As everyone else is saying, Gilbert Strang's book. He also has a [great course][1] on MIT's OCW.
[1]:http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/index.htm
I used Joe Friel's Going Long: Training for Triathlon's Ultimate Challenge and Joe Friel's The Triathlete's training Bible Very in-depth books on how to set up a training plan and schedule your time.
Congrats! And sorry about the DNF.
My opinion (for whatever its worth i guess), if your right on the edge of cut off times then you have to look at 3 things: age, weight, time spent training.
Unfortunately not much we can do about age, at a certain point no one is finishing a half ironman. I assume that you are not at that age yet.
Weight is probably the hardest thing to adjust. You can't out run a bad diet. So knowing nothing about your weight, are you satisfied with your weight or do you think that there is room for improvement?
Time spent training is the easy stuff! Woooo! More specifically, effective training and an effective training plan is probably your biggest gap. I (and others) suggest a book called The Triathlete's Training Bible by Joel Friel. This gets into how to spend your time to be more effectively training with self guided training plans etc etc. If you give more information about what you did to train for this specific event then maybe we could have more in-depth conversation about what you should be doing.
https://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198/ref=sr_1_2?ie=UTF8&amp;qid=1491248736&amp;sr=8-2&amp;keywords=triathletes+training+bible
Here's an easy read that I liked: Concepts of Modern Mathematics by Ian Stewart. It gives a pretty broad overview. And you can't beat the price of those Dover paperbacks.
You may also be interested in a more thorough exploration of the history of the subject. Try History of Mathematics by Carl Boyer.
Textbooks (calculus): Fundamentals of Physics: http://www.amazon.com/Fundamentals-Physics-Extended-David-Halliday/dp/0470469080/ref=sr_1_4?ie=UTF8&amp;qid=1398087387&amp;sr=8-4&amp;keywords=fundamentals+of+physics ,
Textbooks (calculus): University Physics with Modern Physics; http://www.amazon.com/University-Physics-Modern-12th-Edition/dp/0321501217/ref=sr_1_2?ie=UTF8&amp;qid=1398087411&amp;sr=8-2&amp;keywords=university+physics+with+modern+physics
Textbook (algebra): [This is a great one if you don't know anything and want a book to self study from, after you finish this you can begin a calculus physics book like those listed above]: http://www.amazon.com/Physics-Principles-Applications-7th-Edition/dp/0321625927/ref=sr_1_1?ie=UTF8&amp;qid=1398087498&amp;sr=8-1&amp;keywords=physics+giancoli
If you want to be competitive at the international level, you definitely need calculus, at least the basics of it.
Here is a good book: http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?ie=UTF8&amp;qid=1398087834&amp;sr=8-1&amp;keywords=calculus+kline
It is quite cheap and easy to understand if you want to self teach yourself calculus.
Another option would be this book:http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&amp;qid=1398087878&amp;sr=8-1&amp;keywords=spivak
If you can finish self teaching that to yourself, you will be ready for anything that could face you in mathematics in university or the IPhO. (However it is a difficult book)
Problem books: Irodov; http://www.amazon.com/Problems-General-Physics-I-E-Irodov/dp/8183552153/ref=sr_1_1?ie=UTF8&amp;qid=1398087565&amp;sr=8-1&amp;keywords=irodov ,
Problem Books: Krotov; http://www.amazon.com/Science-Everyone-Aptitude-Problems-Physics/dp/8123904886/ref=sr_1_1?ie=UTF8&amp;qid=1398087579&amp;sr=8-1&amp;keywords=krotov
You should look for problem sets online after you have finished your textbook, those are the best recourses. You can get past contests from the physics olympiad websites.
There are a lot of good classics on /u/thebenson's list. I want to highlight the books that are good for what you'll be learning, and give you a sense of how the sequence works. And I'll add a few.
Calculus books: Thomas' Calculus, Calculus by James Stewart (not multivariable), and this cheap easy read by Morris Kline.
Have you learned calculus in the past? It sounds like you'll need it for at least one of those courses, but either way, it will definitely help you conceptually for the others. You should really try to get solid on this before you need to use it.
Intro physics books: Fundamentals of Physics (Halliday & Resnick), Physics for Scientists and Engineers (Serway & Jewett), Physics for Scientists and Engineers (Tipler & Mosca), University Physics (Young), and Physics for Scientists and Engineers (Knight) are all good. Gee, they get really unoriginal with the names, huh?
Each of these books assumes no background in physics, but you do need to use calculus. If you're going to take a class in basic mechanics that doesn't involve any calculus, you may find it more useful to get a book at that level. The only such book that I'm familiar with is Physics: Principles with Applications by Giancoli. I know there are many others, but I can't speak for them.
Mathematical methods: Greenberg is way more than you need here. I think you would find Engineering Mathematics by Stroud & Booth more useful as a reference, since it covers a lot of the less advanced stuff that you may need a refresher on.
Sequence: it's typical to start learning physics by learning about Newtonian mechanics, with or without calculus. After that, one often goes on to thermodynamics or to electricity and magnetism. It sounds like this is roughly how your program is going to work.
If you are learning mechanics with calculus, you can expect E&M to be even heavier on the calculus and thermodynamics to be less so. More calculus is not a bad thing. People often get scared of it, but it actually makes things easier to understand.
It is very typical that you will use only one book (from the intro books above) for all of these topics. You shouldn't need to get any books on specific topics.
**
The other books on /u/thebenson's list are all great textbooks, but I think you should avoid them for now. They generally assume a healthy background in basic physics, and they may not be very relevant to the physics you'll be studying.
But I do want to give some mention to Spacetime Physics* by Taylor and Wheeler, since I don't want to imply that this is a background-heavy book. On the contrary, this is one of the most beginner-friendly physics books ever written, and it is my favorite introduction to special relativity. Special relativity is probably not something you need to learn about right now, but if you have any interest, I seriously recommend finding an old used copy of this book—it's a fun read aside from any other uses!
These were the most enlightening for me on their subjects:
This should keep you busy, but I can suggest books in other areas if you want.
Math books:
Algebra: http://www.amazon.com/Algebra-I-M-Gelfand/dp/0817636773/ref=sr_1_1?ie=UTF8&amp;s=books&amp;qid=1251516690&amp;sr=8
Calc: http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1356152827&amp;sr=1-1&amp;keywords=spivak+calculus
Calc: http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X
Linear algebra: http://www.amazon.com/Linear-Algebra-Modern-Introduction-CD-ROM/dp/0534998453/ref=sr_1_4?ie=UTF8&amp;s=books&amp;qid=1255703167&amp;sr=8-4
Linear algebra: http://www.amazon.com/Linear-Algebra-Dover-Mathematics-ebook/dp/B00A73IXRC/ref=zg_bs_158739011_2
Beginning physics:
http://www.amazon.com/Feynman-Lectures-Physics-boxed-set/dp/0465023827
Advanced stuff, if you make it through the beginning books:
E&M: http://www.amazon.com/Introduction-Electrodynamics-Edition-David-Griffiths/dp/0321856562/ref=sr_1_1?ie=UTF8&amp;qid=1375653392&amp;sr=8-1&amp;keywords=griffiths+electrodynamics
Mechanics: http://www.amazon.com/Classical-Dynamics-Particles-Systems-Thornton/dp/0534408966/ref=sr_1_1?ie=UTF8&amp;qid=1375653415&amp;sr=8-1&amp;keywords=marion+thornton
Quantum: http://www.amazon.com/Principles-Quantum-Mechanics-2nd-Edition/dp/0306447908/ref=sr_1_1?ie=UTF8&amp;qid=1375653438&amp;sr=8-1&amp;keywords=shankar
Cosmology -- these are both low level and low math, and you can probably handle them now:
http://www.amazon.com/Spacetime-Physics-Edwin-F-Taylor/dp/0716723271
http://www.amazon.com/The-First-Three-Minutes-Universe/dp/0465024378/ref=sr_1_1?ie=UTF8&amp;qid=1356155850&amp;sr=8-1&amp;keywords=the+first+three+minutes
Learn math first. Physics is essentially applied math with experiments. Start with Calculus then Linear Algebra then Real Analysis then Complex Analysis then Ordinary Differential Equations then Partial Differential Equations then Functional Analysis. Also, if you want to pursue high energy physics and/or cosmology, Differential Geometry is also essential. Make sure you do (almost) all the exercises in every chapter. Don't just skim and memorize.
This is a lot of math to learn, but if you are determined enough you can probably master Calculus to Real Analysis, and that will give you a big head start and a deeper understanding of university-level physics.
I've already made a comment but I just remembered that this book exists:
https://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071
You might find it helpful
Link to Dave Ramsey on credit cards
I am not a fan of Dave Ramsey in many specific cases and this is one of them.
First, having access to a ready line of credit is important to financial security if you do not have access to a similar amount of immediate cash. Even forms of liquid capital may require to much time for conversion in an emergency. This can be overcome, obviously, with a large emergency savings pool but then this savings isn't working for you in an index fun, etc. As such, having access to an emergency line of credit is important even if you never plan on using a credit card day to day.
Second, building credit is necessary for long term savings on loans and mortgages. While it is possible to build credit without a credit card it is more difficult.
Third, avoiding rewards is leaving money on the table similar to not contributing to a 401k when match is available.
Ramsey's advice is often about eliminating options for risky behavior which is one way to reduce your possible debt burden, but it is not the only way. The more obvious way, which requires personal self discipline.
Dave Ramsey quotes:
"Even by paying the bills on time, you are not beating the system!". It isn't about beating the system, it is about using the system as intended and getting the rewards the system put in place to encourage your use of their credit card over others. Credit card companies make their profit off vendors and consumers. Credit card companies bank on a pool of consumers having some who do not pay their bills on time and some who do, similar to insurance, and offset their risk with rewards with one group over another. The problem with Ramsey's statement is that we are making individual decisions as individual actors within the context of a "system" built around a large pool of participants. The two are disjointed ideas and make no sense in the context of each other.
"A study of credit card use at McDonald’s found that people spent 47% more when using credit instead of cash." This is one of those statements I would refer people to How Not to Be Wrong: The Power of Mathematical Thinking where a statistic has been taking out of context to support a point but is, likely, unrelated. We live in a relatively cashless society and people are more likely to make larger purchases on a card rather than with cash so relative size of purchases will always favor a credit card.
"Personal finance is 80% behavior. You need to cut out habits that make you spend more. You do not build wealth with credit cards. Use common sense." And this is completely true. Personal finance is about personal behavior and creating good habits. If you habitually pay off your credit card month over month, never spending more credit than you have cash reserves, then you are at no greater risk than if you used cash for those same purchases.
How to Prove It: A Structured Approach by Velleman is good for developing general proof writing skills.
How to Think About Analysis by Lara Alcock beautifully deconstructs all the major points of Analysis(proofs included).
It's common to have some difficulty adjusting from lower-level courses with a computational emphasis to upper-level courses with an emphasis on proof. Fortunately, this phenomenon is well known, and there are a number of books aimed at bridging the gap between the two types of courses. A few such books are listed below.
I hope that helps!
The 1st Course In Math Analysis by Brannan
Analysis I by Terrence Tao
Yet Another Intro To Real Analysis by Bryant
Understanding Analysis Stephen Abbott
Elementary Analysis: The Theory of Calculus Kenneth A. Ross
Metric Spaces by Robert Reisel
A Problem Text in Advanced Calculus by John Erdman. PDF
Advanced Calculus by Shlomo Sternberg and Lynn Loomis.PDF
You are missing Abstract Algebra that usually comes before or after Real Analysis. As for that 4chan post, Rudin's book will hand anyone their ass if they havent seen proofs and dont have a proper foundation (Logic/Proofs/Sets/Functions). Transition to Higher Math courses usually cover such matters. Covering Rudin in 4 months is a stretch. It has to be the toughest intro to Real Analysis. There are tons of easier going alternatives:
Real Mathematical Analysis by Charles Pugh
Understanding Analysis by Stephen Abbot
A Primer of Real Functions by Ralph Boas
Yet Another Introduction to Analysis
Elementary Analysis: The Theory of Calculus
Real Analysis: A Constructive Approach
Introduction to Topology and Modern Analysis by George F. Simmons
...and tons more.
It's probably not possible to review everything you need, but getting more experience with proofs is a good start. This course might be helpful:
https://www.coursera.org/course/matrix
and these texts are great examples of mathematical thinking in prose:
Grinstead and Snell's Introduction to Probability:
https://math.dartmouth.edu/~prob/prob/prob.pdf
Apostol's Calculus I and II:
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Apostol and Spivak are the best calculus texts I know; paperback versions of each exist.
Topology by James Munkres
There's really no easy way to do it without getting yourself "in the shit", in my opinion. Take a course on multivariate calculus/analysis, or else teach yourself. Work through the proofs in the exercises.
For a somewhat grounded and practical introduction I recommend Multivariable Mathematics: Linear Algebra, Calculus and Manifolds by Theo Shifrin. It's a great reference as well. If you want to dig in to the theoretical beauty, James Munkres' Analysis on Manifolds is a bit of an easier read than the classic Spivak text. Munkres also wrote a book on topology which is full of elegant stuff; topology is one of my favourite subjects in mathematics,
By the way, I also came to mathematics through the study of things like neural networks and probabilistic models. I finally took an advanced calculus course in my last two semesters of undergrad and realized what I'd been missing; I doubt I'd have been intellectually mature enough to tackle it much earlier, though.
How Not To Be Wrong explores this vignette in some detail - highly recommend the book!
I'm doing that, I guess, if you call 'advanced maths' anything proof-based (which is, generally, what people mean). I use the internet, my brain, and a lot of books. It was hard for sure. Only way to do it is to enjoy it and not burn yourself out working too hard.
This book is how I got started and probably the easiest way into anything proof based: http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605.
Ofcourse you might not want to do analysis especially if you have't done any calc yet. At that level people (I think) do stuff like http://www.artofproblemsolving.com/. Also khan academy, MiT OCW, and competition-oriented books like https://www.google.com/webhp?sourceid=chrome-instant&amp;ion=1&amp;espv=2&amp;ie=UTF-8#q=complex%20numbers%20from%20a%20to%20z.
That said if you can work through that analysis book it'll open the doors to tons of undergrad level math like Abstract Algebra, for example.
Just keep at it?
Apostol's classic calculus textbook, used at Caltech and MIT. The Art of Problem Solving textbook for calculus. The Stanford and Harvard-MIT Math Tournaments have calculus subject tests. The college-level Putnam competition has calculus problems, in addition to linear algebra, abstract algebra, etc.
This is a compilation of what I gathered from reading on the internet about self-learning higher maths, I haven't come close to reading all this books or watching all this lectures, still I hope it helps you.
General Stuff:
The books here deal with large parts of mathematics and are good to guide you through it all, but I recommend supplementing them with other books.
Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
Abstract Algebra: One of the most important, and in my opinion fun, subjects in mathematics. Deals with algebraic structures, which are roughly sets with operations and properties of this operations.
There are many other beautiful fields in math full of online resources, like Number Theory and Combinatorics, that I would like to put recommendations here, but it is quite late where I live and I learned those in weirder ways (through olympiad classes and problems), so I don't think I can help you with them, still you should do some research on this sub to get good recommendations on this topics and use the General books as guides.
Start With "Foundations Of Analysis" By Edmund Landau
http://www.amazon.com/Foundations-Analysis-AMS-Chelsea-Publishing/dp/082182693X
It's a tiny book, but is very good at explaining basic abstract algebra.
Here is the description from Amazon:
"Why does $2 \times 2 = 4$? What are fractions? Imaginary numbers? Why do the laws of algebra hold? And how do we prove these laws? What are the properties of the numbers on which the Differential and Integral Calculus is based? In other words, What are numbers? And why do they have the properties we attribute to them? Thanks to the genius of Dedekind, Cantor, Peano, Frege and Russell, such questions can now be given a satisfactory answer. This English translation of Landau's famous Grundlagen der Analysis-also available from the AMS-answers these important questions."
With the above book you should then have enough knowledge to move on to calculus.
I recommend the two volume series called "Calculus" by Tom M. Apostol.
The first volume is single variable calculus and the second is multivariate calculus
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051/ref=sr_1_4?ie=UTF8&amp;s=books&amp;qid=1239384587&amp;sr=1-4
http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Applications/dp/0471000078/ref=sr_1_3?ie=UTF8&amp;s=books&amp;qid=1239384587&amp;sr=1-3
This might be of interest, Spivak's Calculus on Manifolds.
Strogatz Nonlinear Dynamics and Chaos covers phase space, phase portraits, and linear stability analysis in great detail with examples from many disciplines including physics. It's probably a good place to start, but I don't think it has very much that's specifically on turbulent fluids. For that, you'll probably want a more focused textbook. Hopefully, someone more knowledgeable can recommend one.
You're English is great.
I'd like to reemphasize /u/Plaetean's great suggestion of learning the math. That's so important and will make your later career much easier. Khan Academy seems to go all through differential equations. All of the more advanced topics they have differential and integral calculus of the single variable, multivariable calculus, ordinary differential equations, and linear algebra are very useful in physics.
As to textbooks that cover that material I've heard Div, Grad, Curl for multivariable/vector calculus is good, as is Strang for linear algebra. Purcell an introductory E&M text also has an excellent discussion of the curl.
As for introductory physics I love Purcell's E&M. I'd recommend the third edition to you as although it uses SI units, which personally I dislike, it has far more problems than the second, and crucially has many solutions to them included, which makes it much better for self study. As for Mechanics there are a million possible textbooks, and online sources. I'll let someone else recommend that.
Here's a couple of books I'd recommend.
amazon
amazon
You might check out the Minneapolis area for a tri club. I'm certain there is a good one up there. Some clubs have New Triathlete programs that can be really good.
I recommend reading the Triathlete's Training Bible (http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198) which quite extensively covers the base training period.
If I recall correctly, he speaks about doing lots of leg and core strength training, swimming drills concentrating heavily on technique, hill repeats on the treadmill, etc... Things that would serve as a good base for other training later on.
Yes!
If you don't know any calculus Stewart Calculus is the typical primer in colleges. Combine this with Khan Academy for easy mode cruise control.
After that, you want to look at The Big Orange Book, which is essentially the bible for undergrad astrophysics and 100% useful beyond that. This book could, alone, tell you everything you need to know.
As for other topics like differential equations and linear algebra you can shop around. I liked Linear Algebra Done Right for linear personally. No recommendations from me on differential equations though, never found a book that I loved.
Calculus: An Intuitive and Physical Approach (Second Edition) (Dover Books on Mathematics) https://www.amazon.com/dp/0486404536/ref=cm_sw_r_cp_apip_qmMduBiBKxeqD
This book currently. I learned precalculus using Kahn academy over the year along with trig.
Definitely agree with the people recommending Calculus Made Easy by Silvanus P. Thompson. Often imitated, never equalled.
Another book similar to that is The Calculus for the Practical Man by J.E. Thompson. Besides its fame for being the book that Richard Feynman used to teach himself calculus, it has a completely nonstandard proof that the derivative of sin(x) is cos(x), using an argument based on arc length, which I haven't seen in any other book.
For more modern books I'd recommend Kline's book, which is underrated in my opinion. I'd avoid Spivak's book, which I feel is vastly overrated; it makes calculus even drier than the standard books do.
The course I took as an undergraduate used Friedberg, Insel and Spence. I remember liking it fine, but it's insultingly expensive. Find it in a library or get a used copy if you can. If you're looking for a bargain, it can't hurt to try Shilov. He's Russian, so the book is very terse, but covers a lot of ground.
http://www.amazon.com/Book-Abstract-Algebra-Edition-Mathematics/dp/0486474178/ref=sr_1_1?ie=UTF8&amp;qid=1394386195&amp;sr=8-1&amp;keywords=pinter+abstract+algebra)
This book in conjunction with this book should keep you busy.
Consider using Anki for stuff you want to review periodically and already understand.
Interviews with mathematicians from MIT (haven't read it, but it is leisurely):
http://www.amazon.com/Recountings-Conversations-Mathematicians-Joel-Segel/dp/1568817134
Some good magazines from AMS:
http://www.amazon.com/Whats-Happening-Mathematical-Sciences-Mathermatical/dp/0821849999
If you want to learn some math in a leisurely way (although it does get pretty deep at times):
http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247
A good book on the history of mathematics:
http://www.amazon.com/Mathematics-Nonmathematician-Dover-explaining-science/dp/0486248232
I'll definitely check out that Poincare book, it looks good!
Many thanks for the suggestions!
For the interested, I bought this book for GT:
http://www.amazon.com/Introductory-Graph-Theory-Gary-Chartrand/dp/0486247759
I also was tempted by the following book:
http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247
I think buying a book feels better than sex. (I can compare.)
If you're on a budget, check out Calculus: An Intuitive and Physical Approach by Morris Kline.
:D
http://betterexplained.com/
http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?ie=UTF8&amp;qid=1422649729&amp;sr=8-1&amp;keywords=calculus+an+intuitive&amp;pebp=1422649747330&amp;peasin=486404536
The first site is fun, because it teaches you how to intuitively understand math. I love it. Second is a book that makes you think. Read the reviews for it. I really hope it helps because it's helped me, and I didn't even like math that much in the beginning, now I'm all excited for it :D
Perhaps you might find Shilov's Linear Algebra or Roman's Advanced Linear Algebra to be useful. Both of them treat bilinear and quadratic forms.
I think Shilov does actually discuss Gram-Schmidt orthonormalization, but he doesn't call it that, and it seems to be spread over several sections in chapters 7 and 8. Roman might be better for that. Anyway, you can peruse both of these at libgen.
Shilov gives a rigorous, determinant-heavy treatment of LA in his $10 book. All the nice properties of determinants are verified in the first chapter
There are a couple of easy-ish sources on category theory that are good to have under your belt.
Category Theory for Programmers is available for free: https://github.com/hmemcpy/milewski-ctfp-pdf
It's not amazing, but it's good for programmers who want to start having basic intuitions about category theory.
Lawvere's Conceptual Mathematics is enjoyable and accessible
https://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=mp_s_a_1_1?keywords=conceptual+mathematics&amp;qid=1568389352&amp;s=gateway&amp;sr=8-1
To answer your general question: in my experience, your question is less about math and maybe more about chasing something you think has the answers. You'll meander as long as you feel like something is lacking.
I've seen this a lot with people who have massive textbook collections. A massive collection of textbooks is debt, and it provokes anxiety. You may have to figure out some squishy human stuff in addition to the technical math stuff.
http://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071
Munkres is a great resource to learn topology if you want to actually learn the material and as for complex I don't have a good suggestion for it, but since you're trying to study for the GRE I would suggest checking out All the Mathematics You Missed but Need to Know For Graduate School by Thomas Garrity. The link I added leads to the amazon page where you can buy it for pretty cheap. It's a great book that contains the two subjects that you want to study and many more topics. I myself am using it to study for the GRE and am finding it very helpful in learning the subjects I haven't touched.
http://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071
People will give me flack for this but I think Stewart is a great text for an intro to calc, and moreover, one that a person with little math experience can feasibly use for self study. Obviously buying it new is expensive but I've heard rumors of PDF's flying around on torrent sites and stuff, and there's always a few used copies of it in like a 1 mile radius of wherever you are. Working through the first 8 chapters and maybe chapter 11 (infinite sequences and series) will give you a pretty thorough understanding of all of a first year calculus course, and the sections on multivariable calculus aren't bad either. Once you actually know some basics you'll want to find a more advanced text, but I find myself turning back to this text constantly when I need to remember how to do something basic that I've forgotten from first year.
Do the problems. You'll get stuck on lots of them. /r/learnmath is great for that—if you post a problem from this book up there you'll have a detailed answer in about 45 seconds. http://math.stackexchange.com is also great for that.
As for statistics, there's only so far you can go in traditional statistics without knowing any calculus. You can learn the extreme basics like descriptive statistics and basic probability, but at some point, probability theory requires that you know how to take a derivative or an integral, so you'll need to have those skills under your belt. So I'd start on Stewart's book and just try to work through it.
You should spend some lovely evenings with my friend, Stewart. If you find my friend Stewart too hard on you, take some exercises from my little friend Thomas! And if you want even more fun, my friend Piskunov has some lovely exercises for you!
And ask your questions here :-)
I have not taken the class yet (I'm taking 161 and 225 in January), but I looked at the syllabi already and here's the textbook for the class:
http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073383090/ref=sr_1_1?ie=UTF8&amp;qid=1417826968&amp;sr=8-1&amp;keywords=Rosen+Discrete+Math
You may want to go ahead and pick this up and start looking through it prior to January. I already grabbed a copy; I finish Calculus II tomorrow at my community college and I am going to be starting Rosen very soon.
This book is also commonly recommended:
http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328/ref=sr_1_1?ie=UTF8&amp;qid=1417827137&amp;sr=8-1&amp;keywords=Epp
I'm not sure what your math background is, but one of the most important success factors (in my experience) in math classes is a lot of practice. If you start working through either of those books now, you'll probably be in a good place once class starts in January.
We could also probably get a study group going on in here; I'm pretty comfortable with math, so I am happy to help out anyone else who needs help.
I took a Discrete Mathematics class in College, and this was the textbook the professor recommended:
http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073383090/ref=la_B001IGOE0C_1_1/189-4818032-4394023?s=books&amp;ie=UTF8&amp;qid=1382239523&amp;sr=1-1
I can't honestly say I ever touched it, because the class was actually very easy, and you could study for it using only the professor's lecture notes.
MIT's OCW also has some material available (it includes video lectures, assignments and a textbook):
http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010
I love Jack Lee's series on manifolds:
Introduction to Topological Manifolds
Introduction to Smooth Manifolds
I've heard Munkres' Topology is fantastic as an introduction to general topology, but never read it myself.
Read How Not to be Wrong a bit ago and am currently reading Thinking Fast and Slow. Both lighter reads, Thinking Fast and Slow is a bit thicker, but both cover ways of using basic logic, quantitative reasoning, and probability.
Thinking Fast and Slow does an incredible job of explaining how the mind can work both for and against you without getting too technical, definitely recommend that. How Not to be Wrong is a bit lighter.
edit: lol both of the recommendations have already showed up in the thread
I recently started reading How Not To Be Wrong (The Power of Mathematical Thinking), by Jordan Ellenberg, and while the material is probably way too simple for most on this thread, it's very engaging and informative, relating real world examples to simple math concepts. It's especially good at pointing out how math is used and abused by people to come to inaccurate or sometimes completely false conclusions.
But I think math geniuses aside, everyone can get something out of this book. It's good.
That's certainly one of the issues. There is a great book that covers this and other topics. How Not To Be Wrong by Jordan Ellenberg.
It's possible without college, but it's not possible without education (leaving aside the incredibly rare exceptions like being a professional athlete.) That education can be apprenticeships; it can be on the job training (which is very hard to get in the US); it can be self taught; it can be college. Usually college is easiest.
Mathematics actually has very wide applicability although I'll grant you that many or most courses don't go out of their way to make that clear.
However, I'm not suggesting you should follow a math program. But you will need some form of education that's in demand to not live paycheck to paycheck. (This was much less true 40 years ago but it's true today, and getting more true with each passing year.)
I didn't struggle with Real Analysis mostly because it addresses all your "why?" questions from the get-go.
How to Think About Analysis by Lara Alcock is a nice book that walks you through the Analysis skeleton in a very short time especially if you have no problem with quantifiers.
I am struggling with Linear Algebra right now because of high school style "shut up and do these useless exercises" attitude of most LA books.
I found a book on LA:Real Linear Algebra by Fekete that seems to kick ass(deals with most of your "why" questions) if you are struggling with that as well. I found a free copy online, but it's shit quality, so I had to buy it for that exorbitant price. I think it's worth it since I am tired of crap Linear Algebra books.
I never understand the voting on this sub. Some unrelated posts are upvoted, but tangentially related posts downvoted. Hell, even two similar topics on the main page of this sub get different votes.
@OP, How To Think About Analysis by Lara Alcock.
What's up, man. I failed geometry twice (sophomore and junior year) in high school. I barely graduated high school with a 2.0 gpa. I am now a senior studying math and computer science (going to be getting masters in math). I am at the top of my class, and I will be graduating with a ~3.91 GPA.
Math, just like anything else, is about practice and perseverance. I thought I sucked at math (and basically everyone told me I was more of an "english" kind of guy). But when I got to college, I found that I really enjoyed the challenge, and I found the material interesting as hell. So I worked my ass off at it.
If you work hard (some may need to work harder than others!) and persevere, then you will be fine. There will definitely be challenges, but that's what makes math so fun.
edit: Also, unless you are a math major, I can't imagine you will be getting into too much rigorous theory. You will likely continue mostly just be doing calculations (Calc 1, Calc 2 and Calc 3). That is how it is at my university, at least. However, if you are a math major, it can't hurt to get a head start on writing simple proofs. For that, I recommend the following book: https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094/ref=sr_1_4?s=books&amp;ie=UTF8&amp;qid=1474297774&amp;sr=1-4&amp;keywords=a+transition+to+advanced+mathematics
Seriously, that book is such a fucking good introductory text. It helped me so much.
How do you learn proofs? Do you just memorize them straight up? Can you prove simple things in Set Theory and Point Set Topology on your own? There are only so many techniques for proving things. You absolutely need to master them. After that all you have to remember is a few definitions/theorems/lemmas and an odd trick here and there.
It could also be notation/symbols constantly throwing you off.
Anyway, I like these 2 books below:
Mathematical Proofs by Gary Chartrand et al.
Mathematical Writing by Franco Vivaldi.
I agree with all the suggestions to start with How to Prove It by Velleman. It's a great start for going deeper into mathematics, for which rigor is a sine qua non.
As you seem to enjoy calculus, might I also suggest doing some introductory real analysis? For the level you seem to be at, I recommend Understanding Analysis by Abbott. It helped me bridge the gap between my calculus courses and my first analysis course, together with Velleman. (Abbott here has the advantage of being more advanced and concise than Spivak, but more gentle and detailed than baby Rudin -- two eminent texts.)
Alternatively, you can start exploring some other fascinating areas of mathematics. The suggestion to study Topology by Munkres is sound. You can also get a friendly introduction to abstract algebra by way of A Book of Abstract Algebra by Pinter.
If you're more interested in going into a field of science or engineering than math, another popular approach for advanced high schoolers to start multivariable calculus (as you are), linear algebra, and ordinary differential equations.
Apostol's Calculus
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Calculus Made Easy -- Can't get much better as far as bang for the buck. Follow it up something more rigorous. Maybe, Calculus, Vol 1 by Apostol. The problem with Apostol, as most calculus texts, is price.
For the price and material, you really can't go wrong with Mathematics for the Non-mathematician
Preview here
Strogatz talks about the mathematical details of simpler models of synchronization in his book Nonlinear Dynamics and Chaos. I highly recommend this book: it teaches a wonderful, qualitative way to look at ODEs. The approach is really intuitive, and I wish that I saw it in undergrad. This is also somewhat unrelated, but I know someone who met him, and Strogatz is a super nice guy.
People like “linearizing” problems because it’s a simplification that makes them much easier to solve. It’s like when your Southern African colleague with an unpronounceable name full of clicks gets called “Steve” around the office. That’s not really his name, but eh... if you squint it seems close enough.
If you want to learn about the gnarliness of non-linear problems, Steven Strogatz’s book Nonlinear Dynamics and Chaos is fun and pretty accessible.
This Book Is a great read. Explains every part of training and competing at your best.
You should also pick up a copy of The Triathlete's Training Bible. It's a great read with lots of good training & nutrition advice.
You'll usually find the following recommended:
I've personally used Friedberg's text, and I found it to be pretty well written.
If you are still an undergrad and your school offers a "how to prove stuff and how to think about abstract maths" course take it anyway. No matter how far along you have come.
An example text for such a course is this one:
https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188
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As for Linear Algebra (the most useful part of all higher mathematics for sure (R/math: if you disagree, fight me on this one...i'll win) ) I will tell you i learned a LOT from these two texts:
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https://www.amazon.com/Linear-Algebra-Introduction-Mathematics-Undergraduate/dp/0387940995
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https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/3319110799/ref=pd_lpo_sbs_14_img_0?_encoding=UTF8&psc=1&refRID=APH3PQE76V9YXKWWGCR9
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If it's too simple, stop wasting your time and start reading something more your speed. Say, Linear Algebra Done Right by Axler. If you're still unimpressed, try Advanced Linear Algebra by Roman. If you can solve most problems in this book cold, just drop everything you're doing now and walk straight into the nearest best grad school.
Strictly speaking it's "Analysis in Several Variables" and it uses the Spivak "Calculus on Manifolds" book.
http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219/ref=sr_1_1?ie=UTF8&amp;qid=1314643509&amp;sr=8-1
OK, then let's try this again, this time using more calculus and less topology-specific results. I'm going to be using LaTeX markup here; see the sidebar to /r/math for a free browser plugin that'll translate my code into readable mathematics.
The following is from Michael Spivak's Calculus on Manifolds, and it's pretty close to the result you want, but with more restrictions in terms of differentiability and such:
(a) Let
[; f \colon \mathbf{R}^2} \to \mathbf{R} ;]be a continuously differentiable function. Show that[; f ;]is not 1-1. Hint: If, for example,[; D_1 f(x,y) \neq 0 ;]for all[; (x,y) ;]in some open set[; A, ;]consider[; g \colon A \to \mathbf{R}^2 ;]defined by[; g(x,y) = \left( f(x,y), y \right). ;](b) Generalize this result to the case of a continuously differentiable function
[; f \colon \mathbf{R}^n \to \mathbf{R}^m ;]with[; m&lt;n. ;]The basic idea for (a) is that if there were such an continuously differentiable injection
[; f \colon \mathbf{R}^2 \to \mathbf{R}, ;]then (1) we can find some subset[; A \subseteq \mathbf{R}^2 ;]such that (depending on your convention for notation)[; D_1 f(x,y) = \partial_1 f(x,y) = \partial_x f(x,y) = \frac{\partial f}{\partial x} (x,y) \neq 0 ;]for all
[; (x,y) \in A, ;]and (2) the function[; g \colon A \to \mathbf{R}^2 ;]must have a local continuously differentiable inverse. (This is by the Inverse Function Theorem.)The problem, however, arises when you consider the actual form of a local inverse for
[; g, ;]since[; g^{-1} ;]will be independent of the second coordinate. Accordingly,[; g ;]cannot be injective, whence[; f ;]cannot be injective.I imagine the generalization to part (b) is similar. The important thing here is that given a function
[; f \colon \mathbf{R}^m \times \mathbf{R}^n \to \mathbf{R}^m, \text{ where } m&lt;n, ;]one can construct the associated function
[; \begin{align*}<br /> g \colon \mathbf{R}^m \times \mathbf{R}^n &amp;\to \mathbf{R}^m \times \mathbf{R}^n\\<br /> (\mathbf{x}, \mathbf{y}) &amp;\mapsto \left( f(\mathbf{x},\mathbf{y}), \mathbf{y} \right).<br /> \end{align*} ;]In the above example, we're considering the case
[; m=n=1, ;]and we're considering the equivalence[; \mathbf{R}^1 \times \mathbf{R}^1 \simeq \mathbf{R}^2. ;]The advantage is that
[; g ;]now maps between two spaces of the same dimension, so one can often apply the Inverse Function Theorem. (In fact, this is a common way to deduce the Implicit Function Theorem from the Inverse Function Theorem, so you see this technique often enough that it's worth your time to remember it.)These exercises require stronger assumptions—i.e., continuous differentiability rather than mere continuity—but perhaps this'll at least be a bit more accessible because it doesn't invoke quite so much topology. Hope this helps, and good luck!
For vector calculus, you might enjoy the less formal British text Div, Grad, Curl, and All That by H. M. Schey; for group theory in brief, consider the free textbook Elements of Abstract and Linear Algebra by Edwin H. Connell.
Alternatives to Schey's book include the much more formal Calculus on Manifolds by Michael Spivak, which does have more exercises than Schey but uses most of them to develop the theory, rather than as the mindless drills that fill an ordinary textbook; Michael E. Corral's free textbook Vector Calculus isn't huge but is written closer to an ordinary textbook.
Start here: https://www.complexityexplorer.org/
then here: https://www.amazon.com/Nonlinear-Dynamics-Student-Solutions-Manual/dp/0813349109/ref=asc_df_0813349109/?tag=hyprod-20&amp;linkCode=df0&amp;hvadid=312168166316&amp;hvpos=1o1&amp;hvnetw=g&amp;hvrand=11660544257293770322&amp;hvpone=&amp;hvptwo=&amp;hvqmt=&amp;hvdev=c&amp;hvdvcmdl=&amp;hvlocint=&amp;hvlocphy=9006604&amp;hvtargid=pla-455692727025&amp;psc=1
In essence what you are interested in is "attractor reconstruction (Takens Theorem)", "measuring the lypaunov exponents", or "finding the correlation dimension". Search around for these things or look them up in a nonlinear dynamics textbook and it should get you on your way.
Check out this paper for a good overview of each of these terms, what they mean, and what they can tell you about your timeseries.
It gives a nice runthrough of the things that you can do with a simple time series to detect any chaos in the signal. They also provide some software which can run their analysis on your own time series.
I also would recommend the book: Nonlinear dynamics and Chaos by Steven Strogatz. Its a fantastic book that lays out a primer for chaotic systems, and its relatively short and not too maths heavy for a textbook.
Finally, this website has some nice pictures of analysis of a number of different chaotic systems that might give a better idea of where you can get started in this area.
For anyone interested in this topic, I can recommend two sources for newcomers.
Conversational, largely non-technical: Chaos: Making a New Science by James Gleick
Technical (requires knowledge of ordinary differential equations, but highly readable): Nonlinear Dynamics and Chaos by Steven H. Strogatz
You need calculus, linear algebra and some differential equations. Real analysis is extremely helpful but not completely necessary. Here is a good book on an introduction to the subject:
http://www.amazon.com/gp/aw/d/0813349109/ref=dp_ob_neva_mobile
You might also want to search for your question on MSE. One advice that I recall is to consider Gilbert Strang's book Introduction to Linear Algebra along with his videos on OCW. He has an offbeat style both in his book and in his videos that might be unappealing to some people but the reason is that he really tries to make his students understand rather than remember. Also note that his target audience is typically engineers so proofs are present in his book but not the emphasis of his course.
I think before you start you should ask yourself what you want to learn. If you're into programming or want to become a sysadmin you can learn everything you need without taking classes.
If you're interested in the theory of cs, here are a few starting points:
Introduction to Automata Theory, Languages, and Computation
The book you should buy
MIT: Introduction to Algorithms
The book you should buy
Computer Architecture<- The intro alone makes it worth watching!
The book you should buy
Linear Algebra
The book you should buy <-Only scratches on the surface but is a good starting point. Also it's extremely informal for a math book. The MIT-channel offers many more courses and are a great for autodidactic studying.
Everything I've posted requires no or only minimal previous education.
You should think of this as a starting point. Maybe you'll find lessons or books you'll prefer. That's fine! Make your own choices. If you've understood everything in these lessons, you just need to take a programming class (or just learn it by doing), a class on formal logic and some more advanced math classes and you will have developed a good understanding of the basics of cs. The materials I've posted roughly cover the first year of studying cs. I wish I could tell you were you can find some more math/logic books but I'm german and always used german books for math because they usually follow a more formal approach (which isn't necessarily a good thing).
I really recommend learning these thing BEFORE starting to learn the 'useful' parts of CS like sql,xml, design pattern etc.
Another great book that will broaden your understanding is this Bertrand Russell: Introduction to mathematical philosophy
If you've understood the theory, the rest will seam 'logical' and you'll know why some things are the way they are. Your working environment will keep changing and 20 years from now, we will be using different tools and different languages, but the theory won't change. If you've once made the effort to understand the basics, it will be a lot easier for you to switch to the next 'big thing' once you're required to do so.
One more thing: PLEASE, don't become one of those people who need to tell everyone how useless a university is and that they know everything they need just because they've been working with python for a year or two. Of course you won't need 95% of the basics unless you're planning on staying in academia and if you've worked instead of studying, you will have a head start, but if someone is proud of NOT having learned something, that always makes me want to leave this planet, you know...
EDIT: almost forgot about this: use Unix, use Unix, and I can't emphasize this enough: USE UNIX! Building your own linux from scratch is something every computerscientist should have done at least once in his life. It's the only way to really learn how a modern operating system works. Also try to avoid apple/microsoft products, since they're usually closed source and don't give you the chance to learn how they work.
| $350 on amazon
wat? the newest edition (2009) of his undergrad book is available for about 70 dollars, which is probably a steal.
http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Gilbert/dp/0980232716/ref=sr_1_1?ie=UTF8&amp;qid=1459450518&amp;sr=8-1&amp;keywords=gilbert+strang
I realized as I was writing this reply, I'm not sure if you're interested in a general linear algebra reference material recommendation, or more of a computer graphics math recommendation. My reply is all about general linear algebra, but I don't think matrix decompositions or eigensolvers are used in real-time computer graphics (but what do I know lol), so probably just focusing on the transformations chapter in Mathematics for 3D Game Programming and Computer Graphics would be good. If it feels like you're just memorizing stuff, I think that's normal, but keep rereading the material and do examples by hand! If you really understand how projection matrices work, then the transformations should make more sense and seem less like magic.
I took Linear Algebra last semester and we used http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Edition/dp/0980232716, I would highly recommend it. Along with that book, I would recommend watching these video lectures, http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/, given by the author of the book. I've never watched MIT's video lectures until I watched these in preparation for an interview, because I always thought they would be dumb, but they're actually really great! I will say that I used the pause button furiously because the lectures are very dense and I had to think about what he was saying!
In my opinion, the most important topics to focus on would be the definition of a vector space, the four fundamental subspaces, how the four fundamental subspaces relate to the fundamental theorem of linear algebra, all the matrix decompositions in that book, pivot variables and special solutions...I just realized I'm basically listing all of the chapters in the book, but I really do think they are all very important! The one thing you might not want to focus on is the chapter on incidence matrices. However, in my class, we went over PageRank in detail and I think it was very interesting!
I think the book I used was by Gilbert Strang. He also has some video lectures, apparently. However, I think most of my real understanding of linear algebra (after being introduced to the formalism) came from some combination of upper division classes (classical mechanics, mathematical methods, linear algebra in the math dept). Maybe quantum mechanics was when I just got used to it..
I think I'd suggest complex analysis if you've already been introduced to the basic formalism of linear algebra because you have to use linear algebra a shit ton in quantum mechanics so you'll get good at it just from sheer exposure, imo.
Thanks! Link for anyone interested
If your priority is training for the Tri, a muscle building program like SL will not be very helpful.
You would be much better off following an endurance program that peaks on your event date. You still have a couple months to establish base and then another couple months added anaerobic and intervals.
Read this entire book- it will help you plan a good peak - http://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198/ref=dp_ob_title_bk
I'm not sure what kind of shape you're in, but I'm guessing that the ironman requires a lot more planning just to finish it. I'd suggest getting a copy of this book which will help you plan out and train for all three sports.
Depending on the area you're in, I'd suggest joining a club that does group worksouts (runs, rides, swims, etc). Very useful for all sorts of things, but especially for organized pool workouts. If you're in the DC area, I'll suggest (Team Z)[http://www.triteamz.com/], but I'm sure there are other teams out there.
Can you run on the deck of the ship?
If you are already pretty fit (which I assume you are since you are in the Navy), you shouldn't have too much of an issue finishing an Oly. If you are shooting for a specific time goal you will be a bit more constrained however.
You have quite a bit of time until early summer so I would build up a strong aerobic base and maybe incorporate a bit of weights in for lower body and upper body. I would be careful with maximal weights at this point. Try to go for low weight and a lot of reps. Try to avoid putting on a ton of mass -- keep it lean.
Joe Friel writes some amazing books that you would find very interesting and helpful in structuring your plan. See the Triathlete's Training Bible.
I would say it is absolutely doable. Joe Friel says tris are a swim warm up, a bike race, and a jog to the finish. So you being a cyclist, yes. Yes you can do it.
I keep referencing this site and keep referring back to it. I'm making my own plan, but I started with this as my template: http://www.beginnertriathlete.com/Scott%20Herrick/halfim/preparing_for_your_first_half_ir.htm
I bought these books this past weekend and I'm learning a lot from them:
http://www.amazon.com/Triathlete-Magazines-Complete-Triathlon-Book/dp/0446679283/ref=cm_lmf_img_7
http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198/ref=cm_lmf_img_2
The Joe Friel Books are great. The Triathletes Training Bible by Joe Friel is fantastic (https://www.amazon.com/Triathletes-Training-Bible-Joe-Friel/dp/1934030198) in addition I found a subscription to training peaks with a training plan to be great for accountability.
Not much, the nice thing for upper math courses is they do a good job of building up from bare bones. If you have some linear algebra and a multivariable calc course you should be good. The big requirement is however mathematical maturity. You should be able to read, understand, and write proof.
A very basic intro to proofs course is usually taught to first year math students, this covers set notations, logic, and some basic proof techniques. A common reference is "How to prove it: a structured approach", I learned from Intro to mathematical thinking. The latter isn't as liked, it does seem to cover some material that I think should be taught early. A lot of classical number theory and algebra, for example fundamental theorem of arithmetic, and Fermat's little (not last) theorem are proven. Try to find a reference for that stuff if you can.
It's really important to do a proof based linear algebra class. It helps build the maturity I mentioned and will make life easier with topology. But even more importantly teaching linear algebra in a more abstract way is important for a physics undergrad as it can serve as a foundation for functional analysis, the theory upon which quantum mechanics is built. And in general it is good to stop thinking of vectors as arrows in R^n as soon as possible. A great reference is Axler's LADR.
Again not strictly required, but it helps build maturity and it serves as a good motivation for many of the concepts introduced in a topology class. You will see the practical side of compact sets (namely they are closed and bounded sets in R^(n)), and prove that using the abstract definition (which is the preferred one in topology). You will also prove some facts about continuous functions which will motivate the definition of continuity used in topology, and generally seeing proofs about open sets will show you why open sets are important and why you may wish to look at spaces described only by their open sets (as you will in topology). The reference for real analysis is typically Rudin, but that can be a little tough (I'm sorry, I can't remember the easier book at the moment)
Edit: I will remove this as it doesn't meet the requirements for an /r/askscience question, we usually answer questions about the science rather than learning references. If you feel my answer wasn't comprehensive enough feel free to ask on /r/math or /r/learnmath
Usual hierarchy of what comes after what is simply artificial. They like to teach Linear Algebra before Abstract Algebra, but it doesn't mean that it is all there's to Linear Algebra especially because Linear Algebra is a part of Abstract Algebra.
Example,
Linear Algebra for freshmen: some books that talk about manipulating matrices at length.
Linear Algebra for 2nd/3rd year undergrads: Linear Algebra Done Right by Axler
Linear Algebra for grad students(aka overkill): Advanced Linear Algebra by Roman
Basically, math is all interconnected and it doesn't matter where exactly you enter it.
Coming in cold might be a bit of a shocker, so studying up on foundational stuff before plunging into modern math is probably great.
Books you might like:
Discrete Mathematics with Applications by Susanna Epp
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Building Proofs: A Practical Guide by Oliveira/Stewart
Book Of Proof by Hammack
Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al
How to Prove It: A Structured Approach by Velleman
The Nuts and Bolts of Proofs by Antonella Cupillary
How To Think About Analysis by Alcock
Principles and Techniques in Combinatorics by Khee-Meng Koh , Chuan Chong Chen
The Probability Tutoring Book: An Intuitive Course for Engineers and Scientists (and Everyone Else!) by Carol Ash
Problems and Proofs in Numbers and Algebra by Millman et al
Theorems, Corollaries, Lemmas, and Methods of Proof by Rossi
Mathematical Concepts by Jost - can't wait to start reading this
Proof Patterns by Joshi
...and about a billion other books like that I can't remember right now.
Good Luck.
This webpage has a solid list of recommended textbooks: https://mathblog.com/mathematics-books/
For Linear Algebra, Linear Algebra Done Right (3rd Ed.).
Where I teach they use Linear Algebra by Lay for the introductory class. I'm not sure what level you need but Linear Algebra Done Right is also commonly recommended; could be more abstract than what you need?
Check out:
This one is pretty good
Mathematics for the Nonmathematician, disgustingly eurocentric but still good, Concepts of Modern Mathematics gives an overview of some higher maths, and I have the set The World of Mathematics which I occasionally read a random chapter, it covers lots of ground.
Hi,
I'm a TA in my school's CS theory course (a mixture of discrete math, and the automata, languages and complexity topics most CS theory courses cover).
As others have said, "theory" is pretty broad, so there are an awful lot of resources you could look at. As far as textbooks go, we use two - Sipser's Introduction to the Theory of Computation (which others have recommended), and the freely available textbook Mathematics for Computer Science, by Lehman, Leighton and Meyer - which concentrates more on the "discrete math" side of things. Both seem fine to me. Another discrete-math–focused set of notes is by James Aspnes (PDF here) and seems to have some good introductions to these topics.
If you feel that you're "terrible at studying for these types of courses", it might be worth stepping back a bit and trying to find some sort of an intro to university-level math that resonates for you. A few books I've recommended to people who said they were "terrible at uni-level math", but now find it quite interesting, are:
These aren't specific to computer science, but might be of help. Good luck!
A good intro book on calculus I found helpful was Calculus: A Physical and Intuitive Approach by Morris Kline. Jumping right into Spivak, while doable, is not for the faint of heart. (But one should definitely approach it eventually!)
Edit: spelling
https://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536
An invaluable book when I took calculus the second time: Precalculus Mathematics in a Nutshell
I took calc a second time, because I had taken it previously over ten years before. My instructor at the time was quite the hardass and didn't allow calculators on his tests or homework. I remember doing integration by parts where problems would take two whole sheets of handwritten work.
Consequently, I have a bit of a "been there, done that" attitude towards calculus...
EDIT - My instructor was a big fan of Kline
What text are you using?
Edit: Most calc II or multivariable textbooks that I've encountered (e.g.: this one, this one, this one, or this one) are full of examples, problems, and sections dealing with physical applications, if that's what you mean by outside the classroom.
From what I recollect, Calc II was mostly about developing facility with integration techniques, with some extensions of the concept of integration to boot. Although some of the material may seem to be of little relevance, think of it as an important stepping stone. It is preparing you for some super interesting subjects (like line integrals on vector fields!) that are used to model physical systems.
I dont know much about boot camp, but it sounds like having a physical book will be your best bet.
Personally, my favorite text book to use is Calculus: an Intutitive Approach by Morris Kline, but you might want something more advanced than that.
Also, this book is a tough piece of work, for sure, but it's very helpful. It probably goes deeper than your class will, and may present ideas/methods in a different way, but if you grapple w/ this one, it'll really help you figure out L.A.
There are some recommendations on Amazon :
>I find it ironic that my two favourite Linear Algebra texts are this book and the Axler, for they are exact opposites: Axler shuns determinants, and Shilov starts with them and builds much of his theory off them. However, there is no book I have found that has such a deep and clear exposition of determinants. The first chapter alone makes this book worth buying.
http://www.amazon.com/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1346872221&amp;sr=1-1&amp;keywords=linear+algebra
I would suggest this book for more advanced reading : http://www.amazon.com/gp/product/0415267994/ref=cm_cr_mts_prod_img
^ That book is really good. It starts with linear algebra topics and moves into functional analysis.
Here are some great books that I believe you may find helpful :)
and last but definitely not least:
Later on:
Yeah, I've just never been shown a problem where this stuff gives deep insight, and until I see one and understand it these are just gonna be arbitrary definitions that slide right out of my brain when I'm done reading them. I'll definitely give the book a look - is it motivated with examples?
The only book I have on category theory is Conceptual Mathematics: A First Introduction to Categories, and I must say, I'm not a fan of it - too intuitive, not detailed enough, not well organized, not formal enough - should have gone for MacLane instead.
Category theory is an overkill. If you think you're gonna have an easier time with it, you're mistaken. Category Theory is an extreme generalization of abstract math. Although, there's a very nice intro that you can get started with: Conceptual Mathematics: A First Introduction to Categories by Schanuel and Lawvere. It's accessible to most high school students.
What you are trying to understand is trivial. Most any intro to proofs/higher math book has an explanation of the subject.
In general, you need to learn how to think logically because the way you're going right now won't get you anywhere.
Again, read a book on the very basics of logic and sets. It would contain everything you need to know. For example,
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
> ... relation between finite and infinite.
...relation between finite sets and infinite sets. Just about everything in math is a set. There are many different types of relations. Some are functions, some are equivalence relations, some are isomorphisms.
> Just because something is an adjective or property does not mean it can't be negated.
Ok. Opposite of infinite is finite. In fact, we can say that a set is finite if it is not infinite. But limit is a number and infinity is not. You can't compare apples to oranges.
> In fact almost everything has an inverse.
Relations and special kind of relations called functions have an inverse. Also, operations can be inverse.
For me, it was Conceptual Mathematics: A First Introduction to Categories. Today, I might suggest How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics or Category Theory in Context to start.
There is this online Category Theory book (PDF). Also, the book Conceptual Mathematics has been well recommended as an introduction to CT starting from the basics.
All The Mathematics You Missed is a good overview of topics that are good to know for graduate school. Not all of them are on the GRE, but the summaries of the GRE topics hit most of the key points.
This was an excellent primer I used the summer before grad school, it's all undergraduate level math. There's an analysis section and algebra section. https://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp/0521797071
Optics takes a fair amount of math. If you want to read something useful, I recommend:
1.) Something that is grey: Sculpy! From my cosplay wishlist! :D
2.) Something reminiscent of rain: This hair accessory from my Silly Fun list! I don't know if they're meant to, but the blue bits remind me of raindrops. &lt;3
3.) Something food related that is unusual: Food picks from my Silly Fun list! Maybe not super unusual in Japan, but here in America I doubt you'd see them often.
4.) Something on your list that is for someone other than yourself: This book off my Books wishlist of course! It's for my husband, who's a huge fan of the Elder Scrolls games. I like them, too, but I doubt I'd ever read this.
5.) A book I should read: The Invisible Gorilla, again, off my Books list. I read almost a third of this book while hidden in a book store one day. It's an absolutely fascinating study (or rather, collection of studies) about how much trust we place in our own faulty intuitions.
6.) An item that is less than a dollar, including shipping... that is not jewelry, nail polish, and or hair related: Barely, but this nautical star decal! Unfortunately, it's not on any of my lists.
7.) Something related to cats: Another from my Books wishlist! I'm pretty sure I already know my cat wants to kill me, but this book looks funny anyway.
8.) Something that is not useful, but so beautiful you must have it: Stationary, from my Silly Fun list. I have no one to write to, but I have an obsession with pretty stationary and cards and things. I'm usually too afraid to write on it, even, because nothing ever seems worthy of the pretty paper...
9.) A movie everyone should watch at least once in their life: From my Movies/TV list: Braveheart! Because FREEEDOOOOOOM!!!!!
10.) Something that would be useful when the zombies attack. Explain: Survival knife from my Adventure wishlist! Secluded, unpopulated areas are best for hiding from zombies, and this thing even comes with a firestarter! HOW CAN YOU SAY NO?
11.) Something that would have a profound impact on your life and help you to achieve your current goals: This book which is, strangely, on my Semi-Practical list. I'm a Math/Physics major, but I haven't been in school in quite a while. I'm about to go back very soon, and I'm a little petrified of failing out.
12.) One of those pesky Add-On items: Red Heart yarn from my Crochet wishlist!
13.) The most expensive thing on your list. Your dream item: The PS4 from my Video Games list. I'm an avid gamer. Video games are how I relax. It's one of the few things that, no matter how crappy my day was, always manages to raise my spirits and help me forget about it all.
14.) Something bigger than a bread box: Apparently bread boxes are way bigger than I thought, so I'll go with this desk off my Semi-Practical wishlist. Surely that's big enough! XD
15.) Something smaller than a golf ball: Turtle earrings off my Silly Fun list! THEY'RE SO CUTE!
16.) Something that smells wonderful: Teavana's Blueberry Bliss tea off my Silly Fun list (yet again). If you've never been in a Teavana store, go this second and just...inhale. &lt;3
17.) A (SFW) toy: Frog mitt from my Practical list. I'm fairly certain this isn't supposed to be a toy, but I get the feeling I'm going to spend more time using it as a puppet than as an oven mitt.
18.) Something that would be helpful for going back to school: This backpack from my Semi-Practical list! I want it so badly!! IT'S STUDIO GHIBLI HOW AWESOME IS THAT?
19.) Something related to your current obsession, whatever that may be: 12 Hole Ocarina from my Ocarina wishlist. It's so beautiful and it comes with a Lord of the Rings songbook and I just LOVE IT SO MUCH.
20.) Something that is just so amazing and awe-inspiring that I simply must see it. Explain why it is so grand: Shark sleeping bag from my Silly Fun wishlist! You need me to explain it's awesome?? REALLY? IT'S A SHARK SLEEPING BAG. It looks like the shark is eating you!! Plus it's called the "Chumbuddy" and that just makes me laugh way harder than it should.
Fear cuts deeper than swords!
Consider getting and working through Thomas Garrity's wonderful All the Mathematics You Missed But Need to Know for Graduate School. It's quite dense, but the goal is to help you develop intuition for all of the fields you listed and more. You won't really be able to learn a semester's worth of knowledge over the summer, but if you come into your coursework in mathematics with some intuition for what you're learning, you will have a huge leg up.
Agreed with these recommendations. I'd also suggest All the Mathematics You missed But Need for Graduate School as a useful supplement.
It is an integral of the variable x, as you point out. You can refer to, this book
> I can solve though but the thought why i am doing this is always alarming inside, go and ask any teacher or students as why they do these maths? They will say it's for Grades!
Eek, you have a terrible history of teachers.
>Don't know how many students give up maths just because of wrong Teacher.
For sure.
Starting with calculus/analysis, the book most undergraduate students in America start with is this one. Not every concept starts with real-life examples, but every chapter and section includes actual real-life examples.
For math you're going to need to know calculus, differential equations (partial and ordinary), and linear algebra.
For calculus, you're going to start with learning about differentiating and limits and whatnot. Then you're going to learn about integrating and series. Series is going to seem a little useless at first, but make sure you don't just skim it, because it becomes very important for physics. Once you learn integration, and integration techniques, you're going to want to go learn multi-variable calculus and vector calculus. Personally, this was the hardest thing for me to learn and I still have problems with it.
While you're learning calculus you can do some lower level physics. I personally liked Halliday, Resnik, and Walker, but I've also heard Giancoli is good. These will give you the basic, idealized world physics understandings, and not too much calculus is involved. You will go through mechanics, electromagnetism, thermodynamics, and "modern physics". You're going to go through these subjects again, but don't skip this part of the process, as you will need the grounding for later.
So, now you have the first two years of a physics degree done, it's time for the big boy stuff (that is the thing that separates the physicists from the engineers). You could get a differential equations and linear algebra books, and I highly suggest you do, but you could skip that and learn it from a physics reference book. Boaz will teach you the linear and the diffe q's you will need to know, along with almost every other post-calculus class math concept you will need for physics. I've also heard that Arfken, Weber, and Harris is a good reference book, but I have personally never used it, and I dont' know if it teaches linear and diffe q's. These are pretty much must-haves though, as they go through things like fourier series and calculus of variations (and a lot of other techniques), which are extremely important to know for what is about to come to you in the next paragraph.
Now that you have a solid mathematical basis, you can get deeper into what you learned in Halliday, Resnik, and Walker, or Giancoli, or whatever you used to get you basis down. You're going to do mechanics, E&M, Thermodynamis/Statistical Analysis, and quantum mechanics again! (yippee). These books will go way deeper into theses subjects, and need a lot more rigorous math. They take that you already know the lower-division stuff for granted, so they don't really teach those all that much. They're tough, very tough. Obvioulsy there are other texts you can go to, but these are the one I am most familiar with.
A few notes. These are just the core classes, anybody going through a physics program will also do labs, research, programming, astro, chemistry, biology, engineering, advanced math, and/or a variety of different things to supplement their degree. There a very few physicists that I know who took the exact same route/class.
These books all have practice problems. Do them. You don't learn physics by reading, you learn by doing. You don't have to do every problem, but you should do a fair amount. This means the theory questions and the math heavy questions. Your theory means nothing without the math to back it up.
Lastly, physics is very demanding. In my experience, most physics students have to pretty much dedicate almost all their time to the craft. This is with instructors, ta's, and tutors helping us along the way. When I say all their time, I mean up until at least midnight (often later) studying/doing work. I commend you on wanting to self-teach yourself, but if you want to learn physics, get into a classroom at your local junior college and start there (I think you'll need a half year of calculus though before you can start doing physics). Some of the concepts are hard (very hard) to understand properly, and the internet stops being very useful very quickly. Having an expert to guide you helps a lot.
Good luck on your journey!
Calculus by James Stewart.
http://www.amazon.com/Calculus-James-Stewart/dp/0538497815
Sucks man. I don't know what level of Calculus you're doing, my GF had a really rough time passing Calc 2 which was the last class she needed to finish her degree (took it 3 times).
The last time she ended up getting a pair of books and those more than anything seemed to get her over the hump of failing with 50% and into the "C" range.
https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606
https://www.amazon.com/How-Ace-Rest-Calculus-MultiVariable/dp/0716741741
Also just as a general rule, studying all night so that you're sleep deprived for a test is usually counterproductive. Doesn't matter how much you cram if your brain is fried and not working on all cylinders when it's test time.
Hey man, Calculus is a tough class. Depending on what your algebra background is, Calc 1 can be an especially challenging course. It doesn't say anything about how you'll do in your CS courses. That aside, if you're struggling w/ calc check out this book. It takes the mystery out of the major concepts of Calculus and I attribute a large part of my success in Calc 1 to this book. It doesn't read like a textbook, and I guarantee you won't regret dropping $17 on this. That aside, sorry about the shittiness.
I never said it made it a "bad" book in a deep sense. But it can quite easily explain why someone who isn't in the very narrow set of potential beneficiaries of Spivak's style might feel like the book is opaque, frustrating or unclear--adjectives we commonly associate with "bad" math books. And I also want to double down on the narrowness of Spivak's approach. The people coming away frustrated from Spivak were not looking for How To Ace Calculus, they were looking for a relatively rigorous treatment of the subject matter. What they got was the real meaning of the word rigor--that unexpected revelation is enough to cause some frustration. Frustration that I am willing to partially grant people without castigating them for not matching their expectations properly.
If you feel that ZyBooks does not do a good job in explaining the topics, then you should find other sources to help you understand the material. As you have suggested, take note of the topic and exercises and look for other sources to explain them.
Sources I used when I took Calc:
Here's my dropbox link of the last book for a preview: https://www.dropbox.com/sh/spnay16f7cybdzg/AAAOWMFGgjNqEVdG6o2lXSmba?dl=0
I've only browsed Concrete Mathematics, but others have said it might not be sufficient. My uni used Rosen's Discrete Mathematics and its Applications. I think it's a fairly standard text. Pricey, but older editions might be just as useful.
Sipser's Introduction to the Theory of Computation is the standard textbook. The book is fairly small and quite well written, though it can be pretty dense at times. (Sipser is Dean of Science at MIT.)
You may need an introduction to discrete math before you get started. In my udergrad, I used Rosen's Discrete Mathematics and Its Applications. That book is very comprehensive, but that also means it's quite big.
Rosen is a great reference, while Sipser is more focused.
I should note that topics like graph theory, combinatorics, areas otherwise under the "discrete math" category, don't really require calculus, analysis, and other "continuous math" subjects to learn them. Instead, you can get up to college level algebra, then get a book like
Discrete Mathematics and Its Applications Seventh Edition (Higher Math) https://www.amazon.com/dp/0073383090/ref=cm_sw_r_cp_api_U6Zdzb793HMA7
Or the more highly regarded but less problem set answers,
Discrete Mathematics with Applications https://www.amazon.com/dp/0495391328/ref=cm_sw_r_cp_api_d7ZdzbQ77B65P
This will be enough to tackle ideas from discrete math. I'd recommend reading a book on logic to help with proof techniques and the general idea for rigorously proving statements.
Gensler is a great one but can require a computer if you want more extensive feedback and problem sets.
since you are a computer science student, you can start with proofs in Discrete Mathematics fo this you can look at Kenneth Rosen's book, it can help you with a lot of basic concepts, constructing proofs. Its a good book for those who want to go in algorithms or theoretical cs or a even want to work on pure maths problems. I had this same confusion I wanted to do maths but also cs with it. After this you can try "The art of computer programming"(this has 4 volumes) by Donald Knuth but CLRS is a must along with Rosen's if you want to take cs and maths side by side. If you want to explore further you can look at Design of Approximation Algorithms and Randomised Algorithms. These book can help you with concepts of probability, number theory, geometry, linear algebra etc. But then if you want pure math problems then search for them, go though different journals, SIAM and Combinatorica are really good ones, search them pick a problem you like, then find text relevant to problem and try to give better solutions.
Mulțumesc mult pentru recomandări și pentru răspunsul elaborat!
Împreună cu cartea linkuită de tine am mai luat și Discrete Mathematics and Its Applications, Kenneth H Rosen.
Could be. I took discrete math last semester, and we spent a few weeks on cryptography. We used this book.
That book still got nothing on this one
My topology textbooks were Munkres, Hatcher, and Bredon.
If you want to earn credits towards an engineering degree, not that there's anything wrong with that: probability, statistics, multivariable calculus, differential equations, linear algebra
If you want to have fun and broaden your horizons: point-set topology (Munkres!), abstract algebra.
Find out which teacher(s) at your high school have mathematics degrees, and ask them for advice. Even if you want to study by yourself, see if you can work out an arrangement where they check your problem sets and give regular feedback. They may also be able to set up a seminar with like-minded students. And they will know what the local community colleges have to offer.
Topology by Munkres
If you are enjoying your Calc 3 book, I highly recommend reading Topology, which provides the foundations of analysis and calculus. Two other books I would highly recommend to you would be Abstract Algebra and Introduction to Algorithms, though I suspect you're well aware of the latter.
I liked this one by Munkres
http://www.amazon.co.uk/Topology-Featured-Titles-James-Munkres/dp/0131816292
Hey I'm very very new to machine learning.
BUT I am very familiar with your situation. School didn't teach me anything and I don't think I can take the topics I should know into the workforce.
I've been reading this book
https://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535
And it has put a lot into perspective.
A lot of my education (this is at least for me going to school in the US) has been more about rote memorization and just glossing over concepts. Not really about the logic behind it, I doubt my grade school teachers even understood the concepts better than I did. But now I'm older I'm sucking it up and actually teaching myself the basics all the way up. Going to extremes as learning the Common Core math basics (and I mean the basics!) even though I have no kids.
While it seems like a lot to relearn, your actually going to be working on understanding the concept more and less about solving the problems and getting the right answer, so it's quicker than you can believe.
I say get some books that put stats into perspective, even in a fun way like the book I'm reading. Anything putting you to sleep is cause you are forcing yourself, so read something interesting in the field even if it's for people without any stats knowledge.
Go back and see your old coursework from new eyes. Do side projects and analyze things on your own and ask for help in forums.
Well, that's what I'm doing at least with all math and CS topics.
Yeah, school sucks. I think I understand why (I think) Mark Twain said "I don't let schooling get in the way of my education"
In the grand scheme of math: jack shit. But who's to stop you after 2 months of studying?
What do you know so far? Are you comfortable with inequalities and math induction?
Check out the books below for a nice intro to Real Analysis:
How to Think About Analysis by Lara Alcock.
A First Course in Mathematical Analysis by D. A. Brannan.
Numbers and Functions: Steps to Analysis by R. P. Burn.
Inside Calculus by George R. Exner .
Discrete And Continuous Calculus: The Essentials by R. Scott McIntire.
Good Look.
...here's a book I recommend
https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
I know someone else on /r/math has met the author
The only previous knowledge I really used when I took intro to proofs were some factoring methods that were helpful with proofs by induction, although they weren't necessary. That said, reviewing exponent/log laws, and certain methods of factoring couldn't hurt.
An intro to proofs course should be fairly self contained, meaning any necessary axioms and definitions should be covered in the course. Those examples that you gave are exactly the type of things that should be proven and not knowing them beforehand should be fine. The important thing is being able to understand and reproduce the proofs on your own, and with a bit of experience you will be able to intuitively reason whether a statement is true or false. This intuitive reasoning will also become much more important than memorizing later in the course when you come across statements you've never seen before that aren't immediately obvious.
I would recommend getting very comfortable with logic and basic set theory. I also highly recommend this book if you want some extra reading material (pdf). It's still one of my favorite math books. Hope that helps.
Both 215 and 220 need plenty of practice. So as long as you set time aside for that you should be well on your way.
For 220 I would review some basic proof techniques (contradiction, contra-positive, induction) but not worry too much about knowing the details. In general we were never ask to prove anything where we couldn't apply the basics from a proof we had already learned.
We used Mathematical Proofs: A Transition to Advanced Mathematics (https://www.amazon.com/dp/0321797094), which was a very clear text with plenty of practice problems. If you have time I would recommend reading chapter 2 and 3.
Hello,
You can take a step back and learn about the philosophy and history of math. Once you learn some famous mathematicians and what their discoveries mean you will have more interest. One book is "What is Mathematics"
Then you can pickup some difficult texts. Doing a million problems mechanically is useless, except perhaps to pass tests. Get some idea of how to construct and read and appreciate a mathematical proof. Learn how to write proofs and prove common theorems and what those theorems mean. I recommend this https://www.amazon.ca/gp/aw/d/0321797094 to give you an overview.
Finally nothing beats taking advanced classes in university.
If this all seems a bit too much then maybe you can pick up something specifically for your purpose like 3D Math Primer for Graphics and Game Development, 2nd Edition 3D Math Primer for Graphics and Game Development, 2nd Edition https://www.amazon.ca/dp/1568817231/
Get yourself to high school math level first (understand the unit circle, exponents, algebra, trigonometry) and you can move up from there.
No problem. FWIW my intro to proofs class used this book and I thought it served its purpose well
https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094
>My first goal is to understand the beauty that is calculus.
There are two "types" of Calculus. The one for engineers - the plug-and-chug type and the theory of Calculus called Real Analysis. If you want to see the actual beauty of the subject you might want to settle for the latter. It's rigorous and proof-based.
There are some great intros for RA:
Numbers and Functions: Steps to Analysis by Burn
A First Course in Mathematical Analysis by Brannan
Inside Calculus by Exner
Mathematical Analysis and Proof by Stirling
Yet Another Introduction to Analysis by Bryant
Mathematical Analysis: A Straightforward Approach by Binmore
Introduction to Calculus and Classical Analysis by Hijab
Analysis I by Tao
Real Analysis: A Constructive Approach by Bridger
Understanding Analysis by Abbot.
Seriously, there are just too many more of these great intros
But you need a good foundation. You need to learn the basics of math like logic, sets, relations, proofs etc.:
Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers
Discrete Mathematics with Applications by Epp
Mathematics: A Discrete Introduction by Scheinerman
http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605
You can thank me later- it's really good. Also, a full solutions manual can be found with some googlefu.
I haven't heard of some of the lesser known books, but I just wanted to point out that Algebra Chapter 0 by Aluffi is a very advanced book (in comparison to other books on the list), and that you may want a more gentle introduction to Abstract Algebra before attempting that book. (Dummit and Foote is very standard, and there's plenty other good ones as well that are better motivated). Baby Rudin is also gonna be a tough one if you have no background in Analysis, even though it is concise and elegant I think it's best appreciated after knowing some analysis (something at the level of maybe Understanding Analysis by Abbott).
I know the symbols are scary! But you will be introduced to them gradually. Right now, everything probably looks like a different language to you.
Your university will either have an entire "Methods of Proof" course that proves basic results in number theory or some course (like real analysis) in which you learn methods of proof whilst immersed in a given course. In a course like this, you will learn what all those symbols you have been seeing mean, as well as some of the terminology.
Try reading an introductory analysis book (this one is a very easy read, as analysis books go). Or something like this. Or this
Anyways, don't be afraid! Everything looks scary right now but you really do get eased into it. Just enjoy the ride! Or you can always change your major to statistics! (I'm a double math/stat major, and I know tons of math majors who found the upper division stuff just wasn't for them and were very happy with stats).
I also really struggled with real analysis in the beginning. Stephen Abbot's Understanding Analysis saved my ass, I went from "reconsidering my career choice" to passing the course with a pretty good grade thanks to that book.
http://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/0387950605/ref=sr_1_1?ie=UTF8&amp;qid=1426932693&amp;sr=8-1&amp;keywords=understanding+analysis
Here's my radical idea that might feel over-the-top and some here might disagree but I feel strongly about it:
In order to be a grad student in any 'mathematical science', it's highly recommended (by me) that you have the mathematical maturity of a graduated math major. That also means you have to think of yourself as two people, a mathematician, and a mathematical-scientist (machine-learner in your case).
AFAICT, your weekends, winter break and next summer are jam-packed if you prefer self-study. Or if you prefer classes then you get things done in fall, and spring.
Step 0 (prereqs): You should be comfortable with high-school math, plus calculus. Keep a calculus text handy (Stewart, old edition okay, or Thomas-Finney 9th edition) and read it, and solve some problem sets, if you need to review.
Step 0b: when you're doing this, forget about machine learning, and don't rush through this stuff. If you get stuck, seek help/discussion instead of moving on (I mean move on, attempt other problems, but don't forget to get unstuck). As a reminder, math is learnt by doing, not just reading. Resources:
math on irc.freenode.net
Here are two possible routes, one minimal, one less-minimal:
Minimal
Less-minimal:
NOTE: this is pure math. I'm not aware of what additional material you'd need for machine-learning/statistical math. Therefore I'd suggest to skip the less-minimal route.
Proofs: Hammack's Book of Proof. Free and contains solutions to odd-numbered problems. Covers basic logic, set theory, combinatorics, and proof techniques. I think the third edition is perfect for someone who is familiar with calculus because it covers proofs in calculus (and analysis).
Calculus: Spivak's Calculus. A difficult but rewarding book on calculus that also introduces analysis. Good problems, and a solution manual is available. Another option is Apostol's Calculus which also covers linear algebra. Knowledge of proofs is recommended.
Number Theory: Hardy and Wright's An Introduction to the Theory of Numbers. As he explains in a foreword to the sixth edition, Andrew Wiles received this book from his teacher in high school and was a starting point for him. It also covers the zeta function. However, it may be too difficult for absolute beginners as it doesn't contain any problems. Another book is Stark's An Introduction to Number Theory which has a great section on continued fractions. You should have familiarity with proof before learning number theory.
I'm currently on this journey as well! I'm a programmer teaching my self rigorous maths, so I can definitely help you out.
I find it's best to simultaneously look at several resources on topics such as proofs, so you can get a few perspectives on the same essential topics and have an easier time of finding something.
As a preliminary to proofing, I would suggest a survey of basic logic and Set Theory. I picked up my Set Theory from google searches and the introduction in Apostol's Calculus, and wiki articles on logic and set operations.. It's really easy to learn enough set theory and logic to begin understanding rigorous proofs.
To learn the proofing skills needing for Real Analysis I recommend
a) "Foundations of Analysis" by Edmund Landau.
b) Math 378: Number Systems: An Axiomatic Approach
For an actual book on real analysis, there can be no greater book than Apostol's Calculus.
The popular opinion by some mathematical elite is that Stewart dumbs down calculus, focuses too much on applications, and not enough on theory, which is important for those moving beyond to real analysis and other upper division courses. You should read the reviews of Spivak's or Apostol's calculus text books to see what I mean.
If you're looking at it from a mathematical "I want to prove things" standpoint, I'd recommend Apostol. I've also heard good things about Spivak, although I've never read that book.
If you're looking at it from an engineering "Just tell me how to do the damn problem" perspective, I'm no help to you.
http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-Algebra/dp/0471000051
Here you go. Apostol wrote this classic a while back, and it's currently used at MIT. It treats integration before differentiation. It is mathematically more mature than anything most engineers will ever encounter.
This book is decent: https://www.amazon.com/Mathematics-Nonmathematician-Dover-Books/dp/0486248232
Mathematics for the Nonmathematician by Morris Kline
I swear by this. As someone who has always been a reading person, math textbooks drive me crazy. Stupid bold text, boxed problems, and cluttered graphs distracted me from the poorly written explanatory paragraphs. If I was lucky, my math teacher would be good at explaining a concept verbally as well as visually. Many have already recommended Khan, who is much better with the visual than the verbal.
Morris Kline was a professor at NYU in the 50s, 60s and 70s, a time when textbooks were still more like books than illustrated guides. His writing is clear and concise, which is a must for math, but it is also filled with examples from the real world (including history, art, and engineering).
This book was specifically written for Liberal Arts majors at NYU, not math majors. I'm a Biology major, so somewhere in between as far as technical math goes. I bought this before taking Pre-Calculus in the summer from 8am to 11am. I had never been great at math, just as good as the time I put in doing problems (which was not much). After reading through it, I was excited about trigonometric functions, imaginary numbers, exponentials, and the like. He puts things into a conceptual framework that is very attractive to a "big picture" person like myself.
Buy this book. Buy it now. 28 reviews on amazon and 4.5 stars. 10 bucks. Do it.
Completely backwards; I learned earlier disc math in bits and pieces from various college algebra books (e.g., this, there were others but I can't recall their titles) and the bulk of my disc math/comp sci theory/graph theory/big-O from this one which I have no reason whatsoever to believe is the best way to do it and which dramatically over-emphasizes grammars and Turing machines compared to WGU curriculum and, I think, college curricula in general. It was also kind of a heavy lift since my disc math was weak coming into it, I think I spent more time on the one-chapter 'review' of discrete math than I did on any other three chapters in the book.
that is the gambler's ruin problem
keep doubling your bet until you win
it is why casino's put a upper limit on bet size.
morris kline's Mathematics for the Nonmathematician
https://www.amazon.com/Mathematics-Nonmathematician-Morris-Kline/dp/0486248232
is an oldie but a goodie. he was a famous critic of how schools teach math. he thought schools focused too much on theory without connecting math to the real world.
but to answer your question, if you are studying a particular issue, wikipedia often will give you the historical context. then you can look up the problem that was originally being solved, e.g. the student-t distribution and guinness beer.
Mathematics for the Nonmathematician (very cheap atm - $3.99 )
https://www.amazon.com/Mathematics-Nonmathematician-Morris-Kline/dp/0486248232/ref=sr_1_1?ie=UTF8&amp;qid=1522215994&amp;sr=8-1&amp;keywords=Mathematics+for+the+Nonmathematician
If you get hooked on math later, consider "Mathematics: Its Content, Methods and Meaning (3 Volumes in One)".
https://www.amazon.com/Mathematics-Content-Methods-Meaning-Volumes/dp/0486409163/ref=sr_1_1?ie=UTF8&amp;qid=1522216524&amp;sr=8-1&amp;keywords=kolmogorov+mathematics
or the Princeton companion to mathematics - https://press.princeton.edu/titles/8350.html
Cool youtube channels:
3blue1brown
PBS Infinite Series
patrickJMT
Welch Labs
http://www.amazon.com/Mathematics-Nonmathematician-Dover-Books/dp/0486248232
https://www.khanacademy.org/
Before the Princeton Companion to Mathematics, there were:
What Is Mathematics? by Courant and Robbins
Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrent'ev
Concepts of Modern Mathematics by Ian Stewart
We got told to buy this one, but after looking at the reviews and the price - I think i'll give it a miss...
>54 of 61 people found the following review helpful
>1.0 out of 5 stars Just awful. November 16, 2011
>53 of 61 people found the following review helpful
>1.0 out of 5 stars Irresponsible Publishing! September 7, 2011
>17 of 19 people found the following review helpful
>1.0 out of 5 stars Worst math textbook. Ever. February 11, 2012
I've found Discrete Mathematics and Its Applications to be easily the most useful textbook I've owned throughout my CS degree. I highly recommend it.
This was recommended by my very adept discrete math prof (don't worry, he's not the author). The excerpts I've used are good. The reviews make it seem pretty hit or miss, but textbooks tend to be that way on amazon.
I once took a first-year course in logic, starting with the propositional calculus. All these years later, I still regard it as the most important thing I ever did. Proof-writing became [almost] easy after that. It wasn't always easy to put the pieces together, but at least I had a blueprint. I knew that if I could clearly define a contrapositive, or understand how set identities like DeMorgan's Laws were constructed, I was on much firmer ground. I highly recommend Discrete Mathematics and Its Applications. It's such an enormous and comprehensive text, in so many subjects, that I found myself referring back to it, for something, in almost every class of my undergraduate.
I took a long, long break between undergrad and grad school (think decades). I found this GRE math prep book very helpful. (The GRE math section tests high school math knowledge), I'd take the sample tests, see where I fell short, and focus on understanding why. I also found Practical Algebra to be a good review-and-practice guide, for the fundamentals. I boned up on discrete math by buying an old copy of Rosen and the matching solutions guide. And, I watched a bunch of videos of this guy explaining various facets of the math you need for computer science.
Looks like a good workbook, but fails as an instructional book according to the reviews. Still a good share!
Logic, Number theory, Graph Theory and Algebra are all too much for you to handle on your own without first learning the basics. In fact, most of those books will probably expect you to have some mathematical maturity (that is, reading and writing proofs).
I don't know how theoretical your CS program is going to be, but I would recommend working on your discrete math, basic set theory and logic.
This book will teach you how to write proofs, basic logic and set theory that you will need: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995
I can't really recommend a good Discrete Math textbook as most of them are "meh", and "How to Prove It" does contain a lot of the material usually taught in a Discrete Math course. The extra topics you will find in discrete maths books is: basic probability, some graph theory, some number theory and combinatorics, and in some books even some basic algebra and algorithm analysis. If I were you I would focus mostly on the combinatorics and probability.
Anyway, here's a list of discrete math books. Pick the one you like the most judging from the reviews:
Don't bother trying to learn too much too soon, as you really do need to let time for the math to sink in.
I found this book quite understandable
I'm currently working through Munkres' book on Topology, and I am using the video lectures found here. I know these are in an annoying form factor, but, trust me, these are the only videos that go into any depth you will find on the internet. They use Munkres, too, which is a plus.
On the same site are video lectures for Algebraic Topology. For this, I definitely recommend buying Artin's "Algebra" (1st edition can be found cheaply, and I don't think there's really any significant difference from 2nd), and watch these video lectures by Harvard. Then, you can finally move on to the Algebraic Topology video lectures which uses the free textbook "Algebraic Topology" by Allen Hatcher.
Hope this helps.
If you want to do more math in the same flavor as Apostol, you could move up to analysis with Tao's book or Rudin. Topology's slightly similar and you could use Munkres, the classic book for the subject. There's also abstract algebra, which is not at all like analysis. For that, Dummit and Foote is the standard. Pinter's book is a more gentle alternative. I can't really recommend more books since I'm not that far into math myself, but the Chicago math bibliography is a good resource for finding math books.
Edit: I should also mention Evan Chen's Infinite Napkin. It's a very condensed, free book that includes a lot of the topics I've mentioned above.
I wish I was only taking those two. I've also got Abstract Algebra II (Ring Theory), and teaching the one class on top of that. This is my "tough" semester. The next two I'll probably only be taking 2 classes each semester, plus teaching.
What book are you using for Topo? We're using Munkres.
And what are you using for Real Analysis? I know Baby Rudin is sort of the standard, but we're using Ross.
It's discussed at length in https://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535
Superforecasting has been on my "get to soon" list since I got it last Christmas. It just got a nice nod in the latest CAS magazine.
Along the probability/math lines, other books I've enjoyed are:
Want to break your head? 0.999... = 1.
Want to get weirder? Try multiplying 0.999... by 10, which is just moving the decimal one spot to the right.
Lots more fun stuff in the chapter, Straight Logically Curved Globally from the book How Not to Be Wrong: The Power of Mathematical Thinking, by Jordan Ellenberg.
https://www.amazon.com/Red-Team-Succeed-Thinking-Enemy/dp/0465048943
https://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535
Definitely not more qualified than you but do enjoy tackling tough questions like you proposed and thinking through some mental framework that would make the political environment we are in a little less overwhelming.
Because the system you proposed would likely be based on (for the most part) universal values, it's probably in your best interest to do some light reading that will help you feel more grounded in your choices. If someone asked you why you believe wealth inequality was a bad thing, you might be able to form a more streamlined and coherent thought (outside of something simple like "it's just the right thing to do" or "because that's how I was raised" etc) a couple of good books I've enjoyed and don't require advanced degrees in psychology / philosophy are:
The Island of Knowledge
How not to be Wrong
While enticing to set up a simple acronym or mantra around your political decision making, I've always felt it's better to dig in a bit and then in turn use what you've learned to organize your values etc.
Thoughts?
Try out Jordan Ellenberg's How Not to Be Wrong: The Power of Mathematical Thinking.
http://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535
That doesn't sound right.
There is a simple mathematical explanation of insurance and risk in this book:https://www.amazon.com/How-Not-Be-Wrong-Mathematical/dp/0143127535
I'll try to give my two cents from memory.
The point of any insurance is to spread the risk across more than one person. I don't have the liquidity to pay the huge cost associated with an accident/serious health issue. So, in the unlikely event this happens, I'm screwed and would bankrupt. In contrast, rich people don't need insurance as they can pay up when the event does happen. They should keep on to their money as long as possible.
However, barring the margins taken by the insurer, on average the uninsured rich person and the insured poor person are expected to pay equal amounts (assuming for simplicity that both have the same probability of an event occurring).
So, I think that if you work out the math even an insurance company covering only two people would make sense as it reduces the probility of an insurmountable cost occurring (by a little bit).
So, rationally, I think the only stable point is for everybody to be insured that is either not rich enough to be able to take a hit (essentially these people can act as their own insurer) or too poor to pay the monthly fees (these are the people crossing their fingers no cataclysmic event pushes them over the edges).
If you're actually interested in why we don't have a tiered voting system, I'd encourage you to read more about the mathematics of voting; in particular, the chapter titled "There Is No Such Thing As Public Opinion" from the book How Not To Be Wrong.
Won't necessarily answer all of your questions but basically the answer is that a dictatorship is the only pure "election" system.
Read ( or listen to ) the book "How not to be wrong" by Jordan Ellenberg
It covers not just the stupid "do this" of math, but talks more high level abstract concepts, and discusses real world problems with mathematical implications, and talks about how Math is not some arcane magic, but is in fact a product of human intelligence, and that Math is mostly a formalised version of human natural understanding and rationalisation.
It also covers statistical reasoning, something INTPs typically don't do to well at, because we get distracted by focusing on the details, not realising we don't need the details to draw a good enough conclusion, failing to realise spending too much time on details may actually hinder, not help, the decision making process. ( Because ultimately, you any detail you think is sufficient can be subdivided into details that you don't understand at some level, and you can get side tracked working out how quantum particles work in the process of deciding whether or not you want chicken for dinner, so you need to stop at some depth )
IME, INTJs beat us at math because their statistical reasoning is more naturally adapted, and so they're more likely to follow a Mathematical Discipline than we are.
https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
Alcock is a Math Ed researcher with a huge focus on proofs in undergraduate mathematics.
I am sure this is the book you're referring to https://www.amazon.ca/Think-About-Analysis-Lara-Alcock/dp/0198723539
Yep, the stuff is quite hard and requires a lot of thinking about examples and counterexamples to understand what things mean. And you need time. You just can't learn this stuff in a cram session before an exam. A resource you might find helpful is
https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
I have a friend who swears by this book
https://www.amazon.com/Think-About-Analysis-Lara-Alcock/dp/0198723539
In case you also want some intro to Analysis(Calculus made a bit more rigorous), here's some:
How to Think About Analysis by Lara Alcock.
A First Course in Mathematical Analysis by Brannan.
There is actually a book called How to Think About Analysis which you might find useful. I have not read it myself, but I have read the author's other book and highly recommend her as an author.
Lara Alcock How to think about Analysis, How to study as a Mathematics Major
You can also check Chartrand's et al book:
https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094
Mathematical Proofs: A Transition to Advanced Mathematics was my undergrad discrete text and I've gone back to it for review over the years, I think it's a very fine text especially for an introduction to proofs.
What do you want to do, though? Is your goal to read math textbooks and later, maybe, math papers or is it for science/engineering? If it's the former, I'd simply ditch all that calc business and get started with "actual" math. There are about a million books designed to get you in the game. For one, try Book of Proof by Richard Hammack. It's free and designed to get your feet wet. Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang is my favorite when it comes to books of this kind. You'll also pick up a lot of math from Discrete Math by Susanna Epp. These books assume no math background and will give you the coveted "math maturity".
There is also absolutely no shortage of subject books that will nurse you into maturity. For example, check out [The Real Analysis Lifesaver: All the Tools You Need to Understand Proofs by Grinberg](https://www.amazon.com/Real-Analysis-Lifesaver-Understand-Princeton/dp/0691172935/ref=sr_1_1?ie=UTF8&amp;qid=1486754571&amp;sr=8-1&amp;keywords=real+analysis+lifesaver() and Book of Abstract Algebra by Pinter. There's also Linear Algebra by Singh. It's roughly at the level of more famous LADR by Axler, but doesn't require you have done time with lower level LA book first. The reason I recommend this book is because every theorem/lemma/proposition is illustrated with a concrete example. Sort of uncommon in a proof based math book. Its only drawback is its solution manual. Some of its proofs are sloppy, messy. But there's mathstackexchange for that. In short, every subject of math has dozens and dozens of intro books designed to be as gentle as possible. Heck, these days even grad level subjects are ungrad-ized: The Lebesgue Integral for Undergraduates by Johnson. I am sure there are such books even on subjects like differential geometry and algebraic geometry. Basically, you have choice. Good Luck!
For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.
Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:
Discrete Mathematics with Applications by Susanna Epp
Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang
Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers
Numbers and Proofs by Allenby
Mathematics: A Discrete Introduction by Edward Scheinerman
How to Prove It: A Structured Approach by Daniel Velleman
Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi
Some special topics(elementary treatment):
Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby
A Friendly Introduction to Number Theory Joseph Silverman
Elements of Number Theory by John Stillwell
A Primer in Combinatorics by Kheyfits
Counting by Khee Meng Koh
Combinatorics: A Guided Tour by David Mazur
Just a nice bunch of related books great to have read:
generatingfunctionology by Herbert Wilf
The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule
A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger
If you wanna do graphics stuff, you wanna do some applied Linear Algebra:
Linear Algebra by Allenby
Linear Algebra Through Geometry by Thomas Banchoff, John Wermer
Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman
Best of Luck.
That's not the standard 310 textbook. This is. Also, in 310 you go over the first 3 chapters of the 410 book. I'm not disagreeing with your comment otherwise though.
A course I took previously used this book; it has a chapter on introductory real analysis, which is what you want to get at. I would not suggest going directly to a book like Rudin, as he (in my opinion) tends to amplify the "general route" problem that you mention.
I graduated w/ degree in Math n' Physics but have been doing programming for startup for last 5+ years so many of my math skills got rusty.
While trying to get back into it went through several books and have found this to be the best if you're interested in more advanced mathematics: https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094. It's not only been an excellent review but has fleshed out some areas I was weak (in higher level courses like complex analysis, topology, group theory the methodology of proofs was assumed and often not taught).
The explanations are solid, varied, and they go through each proof they present (often w/ exhaustive step-by-step details).
From there pick a domain you're interested in and pickup the relevant undergraduate (and maybe some graduate) level books/textbooks and see if you can pick it up.
This is just my perspective, but . . .
I think there are two separate concerns here: 1) the "process" of mathematics, or mathematical thinking; and 2) specific mathematical systems which are fundamental and help frame much of the world of mathematics.
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Abstract algebra is one of those specific mathematical systems, and is very important to understand in order to really understand things like analysis (e.g. the real numbers are a field), linear algebra (e.g. vector spaces), topology (e.g. the fundamental group), etc.
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I'd recommend these books, which are for the most part short and easy to read, on mathematical thinking:
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How to Solve It, Polya ( https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X ) covers basic strategies for problem solving in mathematics
Mathematics and Plausible Reasoning Vol 1 & 2, Polya ( https://www.amazon.com/Mathematics-Plausible-Reasoning-Induction-Analogy/dp/0691025096 ) does a great job of teaching you how to find/frame good mathematical conjectures that you can then attempt to prove or disprove.
Mathematical Proof, Chartrand ( https://www.amazon.com/Mathematical-Proofs-Transition-Advanced-Mathematics/dp/0321797094 ) does a good job of teaching how to prove mathematical conjectures.
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As for really understanding the foundations of modern mathematics, I would start with Concepts of Modern Mathematics by Ian Steward ( https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Books/dp/0486284247 ) . It will help conceptually relate the major branches of modern mathematics and build the motivation and intuition of the ideas behind these branches.
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Abstract algebra and analysis are very fundamental to mathematics. There are books on each that I found gave a good conceptual introduction as well as still provided rigor (sometimes at the expense of full coverage of the topics). They are:
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A Book of Abstract Algebra, Pinter ( https://www.amazon.com/Book-Abstract-Algebra-Second-Mathematics/dp/0486474178 )
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Understanding Analysis, Abbott ( https://www.amazon.com/Understanding-Analysis-Undergraduate-Texts-Mathematics/dp/1493927116 ).
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If you read through these books in the order listed here, it might provide you with that level of understanding of mathematics you talked about.
Ah yeah you're at a more advanced stage than I thought. In that case an analysis text might appeal -- I like Abbot's Understanding Analysis but, again, it's quite pricey.
I suspect you'd love Galois theory, but I can't recommend a good text for self-study offhand.
This book.
This isn't a website but I'm really enjoying Mathematics for the Non-Mathematician by Morris Kline at the moment. It goes into the history of math which gives you a much better understanding of why math is the way it is rather than just how to do it. The history of mathematics is surprisingly fascinating. I just want to go back in time and hug the Greeks now.
If anyone is interested in serious math, but in a (somewhat) light fiction form, take a look at the books:
Rozsa Peter, Playing with Infinity [Amazon] [Goodreads]
Kline Morris, Mathematics for the Nonmathematician [Amazon] [Goodreads]
http://www.amazon.com/Mathematics-Nonmathematician-Dover-Books/dp/0486248232/ref=sr_1_8?ie=UTF8&amp;qid=1419370059&amp;sr=8-8&amp;keywords=mathematics+for
Read "Mathematics for the non-mathematician", despite the name, it is rigorous and tells the story of math, introducing each math subject as it was discovered in history - teaching from Babylon/Egyptian need to measure fields, up through the need to calculate cannonball distances, and on into the space-age and statistics: http://www.amazon.com/Mathematics-Nonmathematician-Dover-Books/dp/0486248232
I only skimmed your post but I'd advise against beginning at arithmetics. It'll be boring as fuck, you'll mostly find material intended for children and you're probably gonna lose interest. Also there really isn't much to it.
One book that paints a bigger picture is this:
https://www.amazon.com/Mathematics-Nonmathematician-Morris-Kline/dp/0486248232/
Though old, it's a pretty interesting and well written book and it covers the basics of many topics. It has countless real applications of mathematics and even a lot of history. You can find it on Library Genesis but the physical copy is 8 bones right now so I'd just go for that tbh.
From there you might want to dive deeper into whatever topic interested you most, if that's Calculus you might want to get some kind of precalculus book and then "Calculus - A physical approach" which was written by Morris Kline as well. I personally really enjoy this guy's style, can recommend his stuff, but there are a lot of other good textbooks out there. Spivak and Rudin might be suitable alternatives.
The short path is through Kline's Mathematics for NonMathematicians. I think calling it a book for non-mathematically inclined readers is a stretch (even the inclined are going to have 600+ pages to wade through) but it's definitely a solid redux of history. Just detail-lite.
Awesome stuff! Let me volunteer my time; if you ever want to ask a question of someone (who you will soon outstrip in mathematical ability if not interest), PM me.
One thing that might help you is to go through a history of math. This way you're learning math the way the world has been learning math. It may give you a better understanding of why things are done the way they are. Mathematics for the Non-mathematician is something I'd like to work my way through (anyone interested in doing an online reading group this summer?) that may help you. Johnson/Mowry: Mathematics: A Practical Odyssey is a text I worked through that discussed some of the history and went in something resembling historical order (I had the third edition). As far as getting texts for self study goes, keep an eye out for older editions; you'll save a lot on a subject that doesn't change over time.
Good luck! I'm rooting for you!
I enjoyed Concepts of Modern Mathematics when I was in high school. It might be a little basic and it's a bit uneven in places. But it's a really good lay account of the basic notions in "modern" mathematics. It doesn't really mention so much the various fields. For that, surfing Wikipedia is hard to beat.
I think there could do some really cool analysis tracing lines of thought and how they developed or comparing what was in vogue in math to world developments at the time. This book might be a good overview for modern developments and this one has a overview of the development of math through history
I just checked Amazon. It
says 1965, too but it is the 27^th printing.
I just noticed there is a DjVu copy here but it comes up as PDF on my browser.
You should check out Spivaks Calculus on Manifolds.
http://www.amazon.com/Calculus-Manifolds-Approach-Classical-Theorems/dp/0805390219
Read the first chapter or 2 and see how you like it, if you feel overwhelmed check some of the other recommendations out.
It is however a good book, and you should read it sooner or later.
Rudins principles of mathematical analysis is also excellent, however it
is not strictly multi-dimensional analysis.
Read at least chapter 2 and 3, they lay a very important groundwork.
For single variable calculus, like everyone else I would recommend Calculus - Spivak. If you have already seen mechanical caluculus, mechanical meaning plug and chug type problems, this is a great book. It will teach you some analysis on the real line and get your proof writing chops up to speed.
For multivariable calculus, I have three books that I like. Despite the bad reviews on amazon, I think Vector Calculus - Marsden & Tromba is a good text. Lots of it is plug and chug, but the problems are nice.
One book which is proofed based, but still full of examples is Advanced Calculus of Several Variables - Edwards Jr.. This is a nice book and is very cheap.
Lastly, I would like to give a bump to Calculus on Manifolds - Spivak. This book is very proofed based, so if you are not comfortable with this, I would sit back and learn from of the others first.
Steven Strogatz is a great one too:
for anyone interested in chaos, Nonlinear Dynamics and Chaos by Steven Strogatz is a great introduction and among many others topics addresses chaos in chemical reactions.
Apparently there's a new edition out too.
http://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0813349109/ref=dp_ob_title_bk
> This is an amazing book, but it mostly covers ODEs sadly. Both the style and the material covered are great. It might not be exactly what you're looking for, but it's a great read nonetheless.
This book changed my life. I was all set to become an experimental condensed matter physicist. Then I took a course based on Strogatz... and now I've been a mathematical physicist for the last ten years instead.
This will give you some solid theory on ODEs (less so on PDEs), and a bunch of great methods of solving both ODEs and PDEs. I work a lot with differential equations and this is one of my principal reference books.
This is an amazing book, but it mostly covers ODEs sadly. Both the style and the material covered are great. It might not be exactly what you're looking for, but it's a great read nonetheless.
This covers PDEs from a very basic level. It assumes no previous knowledge of PDEs, explains some of the theory, and then goes into a bunch of elementary methods of solving the equations. It's a small book and a fairly easy read. It also has a lot of examples and exercises.
This is THE book on PDEs. It assumes quite a bit of knowledge about them though, so if you're not feeling too confident, I suggest you start with the previous link. It's something great to have around either way, just for reference.
Hope this helped, and good luck with your postgrad!
I don't, but I'm in the minority of the field. It definitely required a lot of catch-up in my first couple years. If you want to try and break in I can make some suggestions for self-teaching.
Linear Algebra is the backbone of all numerical modeling. I can make two suggestions to start with:
As for the actual FEA/CFD implementations:
Note that none of these will actually teach you the the software side, but most commercial packages have very good tutorials available. These all teach the math behind what the solver is doing. You don't need to be an expert in it but should have a basic idea of what is going on.
Also, OpenFoam is a surprisingly good open source CFD package with a strong community. I'd try and use it to supplement your existing work if possible, which will give you experience and make future positions easier. Play with this while you're learning the theory, don't approach it as "read books for two years, then try and run a simulation".
I find Gilbert Strang's Introduction to Linear Algebra quite accessible, and seems to be aimed towards the practical (numerical) side of things. His video lectures are also quite good, IMHO.
> To be honest, I do still think that step 2 is a bit suspect. The inverse of
[;AA;]is[;(AA)^{-1};]. Saying that it's[;A^{-1}A^{-1};]seems to be skipping over something.I realized how right you are when you say this after I reread the chapter on Inverse Matrices in my book. I am using Introduction to Linear Algebra by Gilbert Strang btw. I'm following his course on MIT OCW.
The book saids: If
[;A;]and[;B;]are invertible then so is[;AB;]. The inverse of a product[;AB;]is[;(AB)^{-1}=B^{-1}A^{-1};].So, before I went through with step two, I would have to have proved that
[;A;]is indeed invertible.>Their proof is basically complete. You could add the step from A2B to (AA)B which is equivalent to A(AB) due to the associativity from matrix multiplication and then refer to the definition of invertibility to say that A(AB) = I means that AB is the inverse of A. So you can make it a bit more wordy (and perhaps more clear), but the basic ingredients are all there.
I will write up the new proof right here, in its entirety. Please let me know what you think and what I need to fix and/or add.
Theorem: if
[;B;]is the inverse of[;A^2;], then[;AB;]is the inverse of A.Proof: Assume
[;B;]is the inverse of[;A^2;][;B;]is the inverse of[;A^2;], we can say that[;A^2B=I;][;A^2B=I;]as[;(AA)B=I;][;(AA)B=I;]as[;A(AB)=I;]because of the associative property of matrix multiplication.[;A(AB)=I;],[;AB;]is indeed the inverse of[;A;].Q.E.D.
Do I have to include anything about the proof being correct for a right-inverse and a left-inverse?
> That's a great initiative! Probably means you're already ahead of the curve. Even if you get a step (arguably) wrong, you're still practicing with writing up proofs, which is good. Your write-up looks good to me, except for the questionability of step 2. In step 3 (and possibly others) you might also want to mention what you are doing exactly. You say "therefore", but it might be slightly clearer if you explicitly mention that you're using your assumption. You can also number everything (including the assumption), and then put "combining statement 0 and 2" to the right (where you can also go into a bit more detail: e.g. "using associativity of multiplication on statement 4").
I haven't began my studies at university yet, but I sure am glad that I exposed myself to proofs before taking an actual discrete math class. I think that very few people get exposed to proof writing in the U.S. public school system. I've completed all of the Khan Academy math courses, and the MIT OCW Math for CS course is still very difficult. I basically want to develop a very strong foundation in proof writing, and all the core courses I will take as a CS major now, and then I will hopefully have an easier time with my schoolwork once I begin in the fall. Hopefully this prior knowledge will keep my GPA high too. I really appreciate all the constructive criticism about my proof. I will try to make them as detailed as possible from now on.
I recommmend the math video lectures at the MIT [1]. Single variable calculus is 39 lectures at about 50 minutes each [2]. Go through the first ones and you'll have not only a refresher but also a head start. While you are about it, don't miss out Prof. Gilbert Strang's video lectures on Linear Algebra [3], that man is phenomenal (he teaches based on one of his own books [4].)
Resources:
1,
2,
3,
4.
Differential geometry track. I'll try to link to where a preview is available. Books are listed in something like an order of perceived difficulty. Check Amazon for reviews.
Calculus
Thompson, Calculus Made Easy. Probably a good first text, well suited for self-study but doesn't cover as much as the next two and the problems are generally much simpler. Legally available for free online.
Stewart, Calculus. Really common in college courses, a great book overall. I should also note that there is a "Stewart lite" called Calculus: Early Transcendentals, but you're better off with regular Stewart. Huh, it looks like there's a new series called Calculus: Concepts and Contexts which may be a good substitute for regular Stewart. Dunno.
Spivak, Calculus. More difficult, probably better than Stewart in some sense.
Linear Algebra
Poole, Linear Algebra. I haven't read this one but it has great reviews so I might as well include it.
Strang, Introduction to Linear Algebra. I think the Amazon reviews summarize how I feel about this book. Good for self-study.
Differential Geometry
Pressley, Elementary Differential Geometry. Great text covering curves and surfaces. Used this one in my undergrad course.
Do Carmo, Differential Geometry of Curves and Surfaces. Probably better left for a second course, but this one is the standard (for good reason).
Lee, Riemannian Manifolds: An Introduction to Curvature. After you've got a grasp on two and three dimensions, take a look at this. A great text on differential geometry on manifolds of arbitrary dimension.
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Start with calculus, studying all the single-variable stuff. After that, you can either switch to linera algebra before doing multivariable calculus or do multivariable calculus before doing linear algebra. I'd probably stick with calculus. Pay attention to what you learn about vectors along the way. When you're ready, jump into differential geometry.
Hopefully someone can give you a good track for the other geometric subjects.
Strang's book looks nice, and I noticed he has accompanying lectures which is good. I found this version, which is more or less in my price range but appears a bit outdated. https://www.amazon.com/Introduction-Linear-Algebra-Fourth-Gilbert/dp/0980232716
I think for a rigorous treatment of linear algebra you'd want something like Strang's class book:
http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Edition/dp/0980232716
For me, what was great about this book was that it approached linear algebra via practical applications, and those applications were more relevant to computer science than pure mathematics, or electrical engineering like you find in older books. It's more about modern applications of LA. It's great for after you've studied the topic at a basic level. It's a great synthesis of the material.
It's a little loose, so if you have some basic chops, it's fantastic.
To piggy back off of danielsmw's answer...
> Fourier analysis is used in pretty much every single branch of physics ever, seriously.
I would phrase this as, "partial differential equations (PDE) are used in pretty much every single branch of physics," and Fourier analysis helps solve and analyze PDEs. For instance, it explains how the heat equation works by damping higher frequencies more quickly than the lower frequencies in the temperature profile. In fact Fourier invented his techniques for exactly this reason. It also explains the uncertainty principle in quantum mechanics. I would say that the subject is most developed in this area (but maybe that's because I know most about this area). Any basic PDE book will describe how to use Fourier analysis to solve linear constant coefficient problems on the real line or an interval. In fact many calculus textbooks have a chapter on this topic. Or you could Google "fourier analysis PDE". An undergraduate level PDE course may use Strauss' textbook whereas for an introductory graduate course I used Folland's book which covers Sobolev spaces.
If you wanted to study Fourier analysis without applying it to PDEs, I would suggest Stein and Shakarchi or Grafakos' two volume set. Stein's book is approachable, though you may want to read his real analysis text simultaneously. The second book is more heavy-duty. Stein shows a lot of the connections to complex analysis, i.e. the Paley-Wiener theorems.
A field not covered by danielsmw is that of electrical engineering/signal processing. Whereas in PDEs we're attempting to solve an equation using Fourier analysis, here the focus is on modifying a signal. Think about the equalizer on a stereo. How does your computer take the stream of numbers representing the sound and remove or dampen high frequencies? Digital signal processing tells us how to decompose the sound using Fourier analysis, modify the frequencies and re-synthesize the result. These techniques can be applied to images or, with a change of perspective, can be used in data analysis. We're on a computer so we want to do things quickly which leads to the Fast Fourier Transform. You can understand this topic without knowing any calculus/analysis but simply through linear algebra. You can find an approachable treatment in Strang's textbook.
If you know some abstract algebra, topology and analysis, you can study Pontryagin duality as danielsmw notes. Sometimes this field is called abstract harmonic analysis, where the word abstract means we're no longer discussing the real line or an interval but any locally compact abelian group. An introductory reference here would be Katznelson. If you drop the word abelian, this leads to representation theory. To understand this, you really need to learn your abstract/linear algebra.
Random links which may spark your interest:
> don't think that there is a logical progression to approaching mathematics
Well, this might be true of the field as a whole, but def not true when it comes to learning basic undergrad level math after calc 1, as the OP asked about. There are optimized paths to gaining mathematical maturity and sufficient background knowledge to read papers and more advanced texts.
> Go to the mathematics section of a library, yank any book off the shelf, and go to town.
I would definitely NOT do this, unless you have a lot of time to kill. I would, based on recommendations, pick good texts on linear algebra and differential equations and focus on those. I mean focus because it is easy in mathematics to gloss over difficulties.
My recommendation, since you are self-studying, is to pick up Gil Strang's linear algebra book (go for an older edition) and look up his video lectures on linear algebra. That's a solid place to start. I'd say that course could be done, with hard work, in a summer. For a differential equations book, I'm not exactly sure. I would seek out something with some solid applications in it, like maybe this: http://amzn.com/0387978941
That is more than a summer's worth of work.
Sorry, agelobear, to be such a contrarian.
http://www.amazon.com/The-Triathletes-Training-Bible-Friel/dp/1934030198/ref=sr_1_sc_1?ie=UTF8&amp;qid=1398039218&amp;sr=8-1-spell&amp;keywords=triathalets+training+bible
Totally worth it.
A cool Mark Allen article, to get you in the mood
THE ONE TRUE BOOK (Accept no imitations)
Hal Higdon's Site
Joe Friel's book about how too piss on your bike seat, I mean, complete a triathlon
There is a lot in the way of resources for new triathletes these days. For your first tri, grab a free training plan online that matches where you are now. Read Beginner Triathlete in your free time; it's a fantastic resource, and I still refer back to its articles all the time. Train your butt off. You don't need to buy a sweet road bike up front, though you sound like you're pretty sure that you want to get into this stuff.
Feel free to skimp on some of the gear for your first race. No one wants to find out that they dislike triathlon after dumping $3k on tri gear. You can race on an old bike with platform pedals. Unless it's really cold, you don't need a wetsuit. The first race is where you truly find out if this is the sport for you. EDIT: Someone mentioned a bike fit. If you're riding an old bike, Competitive Cyclist's Bike Fit Calculator will get you pretty darn close--good enough to get through your first race. Use the road calculator mode if you don't have aerobars off the bat.
After you finish your first race, sit down and think about what you liked, what you did well with, what needs improvement. Get Joe Friel's Triathlete's Training Bible, read it cover to cover. Read it again. Figure out your long-term training plan for the rest of that season. If you start your base training in the winter/early spring and pick an early first race, you can get a full season of sprints and/or Olympics in.
Look for a triathlon club in your area or find a coach or drag a friend into the insanity of triathlon; the camaraderie is priceless in keeping your spirits up during long seasons packed full of hard training and races.
As far as spending money on triathlon "stuff" goes: Remember during your first couple seasons that gadgets and gizmos and aero gear are great, but what really makes the difference is eating well and training hard.
After that, the gear that makes your races more comfortable is the best place to spend your money (tri shorts if you don't them, cycling kit and proper running shorts for training). Then, points of contact with the bike and pool "toys" will improve your efficiency and form (new bike w/ fit if req'd, clipless pedals, shoes, aerobars, pull buoy, kickboard, fins, paddles... a bike computer probably fits in here, as well). Beyond that, you're at a wetsuit and then the "extras" like aero helmet, race wheels, power meters, GPS, HRM, tri bike, speedsuits, etc., etc. That's the approximate map for spending in my book, anyhow. There's practically no limit to the amount of stuff you can buy for triathlon, and as you train more, you'll know what needs to come next.
Sorry, I went on vacation and totally blanked about posting these for you!
Anyway, here are some books
Linear Algebra Done Right (Undergraduate Texts in Mathematics) https://www.amazon.com/dp/3319110799/ref=cm_sw_r_cp_api_1L8Byb5M5W9D3
This one is actually for analysis but depending on your appetite, it might help greatly with the proof side of your class. You can buy it here: Counterexamples in Analysis (Dover Books on Mathematics) https://www.amazon.com/dp/0486428753/ref=cm_sw_r_cp_api_GS8BybQWYBFXX
But there's also a PDF hosted here: http://www.kryakin.org/am2/_Olmsted.pdf
Working through Griffiths is a good idea, but I strongly suggest working through an abstract linear algebra book before you do anything else. It will make your life much better. Doing some of Griffiths in advance might make your homework a bit easier, but you'll be repeating material when you could be learning new things. And learning real linear algebra will benefit you in pretty much every class.
I recommend this book as your primary text and this one for extra problems and and a second opinion.
I don't know much about AI, though I do know that (there's a theme, here) linear algebra gets a starring role. So, if you're currently enjoying linear algebra, continue with that. Axler is frequently recommended, if you want a textbook to go through.
After that it's really up to you what you want to go for next, since you have many paths available. Sipser is a great intro to theoretical CS, but, again, don't spend $200 on it. Try to find it in a library, or use something like this to find a much cheaper international edition.
Edit: Forgot to mention, CLRS is the standard for algorithms, but I'm not sure how useful it is as a primary source for learning. Maybe try to borrow a copy to see if you like it.
Like 50 on amazon but could also try Abebooks and see if there's a cheaper used or international copy.
How about selected chapters from Stewart's Concepts of Modern Mathematics? It has a pretty wide range of jumping off points and is a relatively affordable Dover book. You could go into more or lesser detail on these topics based on the students' backgrounds.
Another idea would be to focus on foundations like set theory, logic, construction/progression of number systems from ℕ -> ℤ -> ℚ -> ℝ -> ℂ , and then maybe move into some philosophy of math. There could be some fun and accessible class discussion, such as having them argue for or against Platonism. [Edit: You could throw in some Smullyan puzzle book stuff for the logic portion of this for further entertainment value.]
It's a pretty difficult question to answer because only you know what you want out of this (or maybe, you don't know yourself!)
"I want to see what kind of mathematics is out there"
Try The Joy Of X. This is a super fun "guided tour" of mathematics. Each chapter surveys a different mathematical topic with examples, intutions, and fun thought experiments. You won't learn to "do the math", but you should have more of an idea of the kinds of things mathematicians think about, and some of the history of mathematics. This is easy and enjoyable, even though no mathematical background.
There's a "Hard mode" version of this called Concepts Of Modern Mathematics. The language is still light and informal, but the concepts are dealt with in more depth and abstraction -- there are fewer "real life" examples, and you will have to follow some real mathematical arguments in your head or on paper. This is more difficult, but still requires no formal mathematical background.
The other place to check of course is YouTube. 3Blue1Brown, Mathologer, Numberphile and many other channels have great exposés of mathematical concepts for the general audience, with 3Blue1Brown being my favourite for his wonderful animations.
"I want to actually improve my mathematical knowledge and skill"
This is difficult, but doable. I'm a mature mathematics student, and I was only really in with a shot of university owing to the kindness of my then fiancée supporting me while I knuckled down and learned the basics. The first step will be to brush up on what you should know from school. I'm not really sure what to recommend here; most texts targeted at this level of mathematics are targeted at... well, bored teenagers who don't want to learn mathematics, rather than keen adults possessing of some degree of patience and perseverance. I suppose Serge Lang, probably the most prolific mathematics textbook author of all time, can offer "Basic Mathematics", but this means paying Springer textbook prices, unless you enjoy marauding on the high seas. Khan Academy is a website with dozens (hundreds?) of free videos, articles, and exercises on basic mathematics
After you're up to speed on your basic algebra and geometry, the two most widely applied and important topics in mathematics beyond school-level are calculus and linear algebra (other than maybe statistics and probability). Calculus is typically learned first, but actually, it doesn't really matter which order you do these in. Exactly how to learn these topics is also a pretty difficult question, and depends what you want to get out of it. I guess post back here if "step 1" (recovering all your school-level maths) goes well?
Maths is hard, but fun. You have to do exercises and practice. You have to think deeply about difficult and abstract concepts. If you do choose the "improve actual skill" route, I'd still recommend supplementing your learning with the books and YouTube videos from the first half of this reply. Being exposed to fun new ideas regularly helps motivate you to push through the technical difficulties of learning it "properly".
Concepts of Modern Mathematics by Ian Stewart is an excellent book about modern math. As is Foundations and Fundamental Concepts of Mathematics by Howard Eves I would recommend these two along with the far more expensive Naive Set Theory by Halmos
These are different fields (programming vs math etc) however I will ask you, do you like math or programming ? If not maybe you need to get to know these quite interesting fields better. For math I would recommend one of the Dover introduction books, such as Ian Stewarts' concepts of modern math.
Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:
&nbsp;
To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)
By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.
So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.
Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.
Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.
&nbsp;
Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.
So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?
Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis
http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1405668438&amp;sr=1-1&amp;keywords=calculus+an+intuitive+and+physical+approach
Starts out with a brief history of calculus in chapter 1.
Chapter 2 is derivatives.
Chapter 3 is anti-derivatives
Chapter 4 talks about the geometric importance of the derivative...etc..
Chapter 21 talks about multivariable functions and geometric representation then 22 is over partial differentiation, 23 multiple integrals then an introduction to diff eq.
I don't know if that's what you're looking for.. but its been an excellent read so far, and it tends to be written in layman's terms(great for me) rather than math speak.
Unfortunately, many books like Spivak or Thomas are going to be very expensive, although you can find scans of them online if you look hard enough.
Dover books are cheap and are often classics, for example Calculus by Kline.
Spivak would be worth it if you plan to go on to study mathematics. It's going to have the rigor (and interesting stuff from a mathematical standpoint) that are omitted or hidden in other texts.
That's great, it reminds me a lot of Calculus by Kline. He takes a similar approach and his introduction perfectly foresaw 60 years ago the problems with math education now.
https://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536
A true classic that will make you a beast at calculus:
Calculus: An Intuitive and Physical Approach by Morris Kline
It's old-school but totally awesome. Gives you great explanations for why we use what we use in the mathematical world.
Made me the man I am today.
http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536
Whew, not looking for Stewart or spivak? That's the two ends of the spectrum as far as calculus is concerned.
Maybe check out Morris Kline. Its intuitive and sounds right up your alley (I think)! For vector calc you may need to pick up something more advanced. I hope this helps :)
http://www.amazon.com/gp/aw/review/0486404536/RTE3I14V7OSHN/ref=cm_cr_dp_mb_rvw_1?ie=UTF8&amp;cursor=1
I bought a copy of Dover's Linear Algebra (Border's Blowout) which I plan to go through after I finish A Book of Abstract Algebra.
I feel like I have a long way to go to get anywhere. :S
I meant it quite literally, something along the lines Linear Algebra by Georgi E. Shilov, but less rigorous.
> Leslie Ballentine's book
these
https://www.amazon.co.uk/Quantum-Mechanics-Development-Leslie-Ballentine/dp/9810241054
https://www.amazon.co.uk/Linear-Algebra-Dover-Books-Mathematics/dp/048663518X/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1542818674&amp;sr=1-1&amp;keywords=linear+algebra+Halmos
I've always wanted to recommend this book to someone who knows no math. I find the writing infuriating. It is a dialog but this approach to dialog totally sucks. On the other hand, this is a stunning introduction to categorical logic. It will not help you solve problems etc. but I can guarantee that this book will change your entire outlook on the world.
http://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X/ref=sr_1_1?ie=UTF8&amp;qid=1320560710&amp;sr=8-1
I love this book, personally.
I don't claim to know Category Theory, but I came across it when doing exercises in the beginning part of Chapter 0 by Aluffi. It was very terse, but still understandable. The video seems to be much more relaxed in comparison. It is even more relaxed than Awodey's book which is a much better intro to CT than Aluffi's Chapter 0. In short, it reminds me of Conceptual Mathematics: A First Introduction to Categories by Lawvere/Schnauel a little.
Maybe start with a book like this and when you hit a wall, you get the relevant textbook and do exercises.
One of my undergrad profs said that "The only way to learn mathematics is to DO mathematics."
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Hard to say without knowing your exact course (is it taught or research based?). Speak to your supervisor and/or current students to get an idea of what you'll be doing. If you can, read some relevant and current academic papers to get a grasp of where you have gaps in your knowledge.
I also recommend 2 general books:
There are probably better books for you depending on what you'll be doing. For example, my particular research involves multivariate analysis, so I have a variety of dedicated statistics books, including course materials from another school that teaches relevant topics.
I would suggest you find out more about the work to come (courses and schools can vary quite a lot), get one of those books and learn the maths you need as you go along.
As a first step, you should decide what your dream career is. You're considering a Master's degree - why not a PhD? Or maybe a second Bachelor's in a related field would be more appropriate? It all depends what you want to do as a career.
You might want to see this book: https://amzn.com/0521797071 - This book won't teach you everything, but it could help you get started. Then start looking into the math GRE (the math subject test - not the math part of the general GRE). Buy some prep books for that and try taking a practice GRE. See how much of that material you know.
Once you attempt a practice GRE , it should help you figure out how prepared or underprepared you are. At this point, you will probably want to sign up for some senior-level undergraduate math classes like Calc 3, Real Analysis, and ODEs. Once you can get an acceptable score on the GRE, you should apply to graduate programs.
If you're able to, I think you should consider a PhD program with a teaching assistantship. These programs offer a tuition waiver and a small stipend as payment for you teaching. Master's students often don't get any financial support. It's possible to complete a PhD program without getting into debt, and picking up a Master's along the way is optional.
Keep in mind that a graduate program might require you to take some undergraduate courses. If they don't require it, they might suggest it. You should take their advice and sign up for these classes. I had to take undergrad Real Analysis during my first semester as a graduate student, and everything worked out fine.
Good luck!
Most schools just use 1 textbook for calc 1-3 : http://www.amazon.com/Calculus-James-Stewart/dp/0538497815
Doesn't really matter which edition you get, you're still going to suffer through it.
A popular other book recommended by math majors/professors is
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918
You can get the pdf on "certain websites."
Videos will make you lazy and you will likely lose focus and turn to reddit or games or whatever because the professors can be really boring. Just stay focused on the text.
"Just do it."
I like this calculus book: https://www.amazon.com/Calculus-7th-James-Stewart/dp/0538497815
And this is Halliday and Resnick https://www.amazon.com/Fundamentals-Physics-David-Halliday/dp/111823071X/ref=sr_1_3?crid=9SK20IQF11Z5&amp;keywords=halliday+and+resnick+fundamentals+of+physics&amp;qid=1567697861&amp;s=books&amp;sprefix=hallida%2Cstripbooks%2C124&amp;sr=1-3
If you want a gentler introduction to calculus, with many examples, lots of intuition, diagrams, and nicer explanations, take any edition of James Stewart's Calculus - Early Transcendentals.
If you feel up to a serious challenge and want to study it as a mathematician would, get Michael Spivak's Calculus.
http://www.amazon.com/Calculus-7th-Edition-James-Stewart/dp/0538497815
King of the formulaic problems.
Cool.
Linear Algebra Don't waste your time with anything other than Lay, pretty much. Sounds like you're 100% new to LinAlg (it's not about polynomial equations) so it may be a bit tough to get off the ground working by yourself, but not impossible. It'd be worth finding a MOOC on the subject, there should be plenty. Otherwise, it's a pretty standard freshman maths course and a lot of people struggle with it (not because it's hard, just because it's different to HS maths), so there's a ton of resources on the internet.
Calculus Kinda just gotta slog away with where you're at tbh. I had Stewart as a freshman, didn't think it was overly great though. Still, that's the kind of level you need, so search for "alternatives to Stewart calculus" and anything that comes up should be appropriate. I wouldn't be able to tell you which to pick though.
Stats Basically, completing both of the above is pretty much a prerequisite for being able to understand linear regression properly, so don't expect to gain much by diving straight into stats. You could probably find a "business analytics" style textbook that would let you do more stats without understanding what's really going on under the hood, but if you want to stick with it in the long term you'll benefit more from getting stuff right at the beginning.
For 241, my favorite 'guide' was this book! . It's got everything you need and is a fun/easy read. There's also one for calc 2/3 if you'll be taking those.
I might have some past exams but i'll have to check. Who's your professor?
It's really smart to be playing to your strengths: if you excel at language and writing, then read a book that talks about math in more detail. Textbooks are good for problems and for reference, but I find them very hard to read. They use equations where they should be using words.
Go to your local library, and look in the math section until you find something interesting. I found this book when I was struggling with calculus: How to Ace Calculus: The Streetwise guide. It was smart, funny, and really explained topics in ways I could relate to.
That's the kind of thing I would look for if I were you. Good luck! I hope you see post in all the ~430 comments!
Well the good news is that we have more resources available now than even 5 years ago. :) I'm in calc 1 right now, and was having trouble putting the pieces together into a whole that made sense. A few of my resources are classroom specific but many would be great for anyone not currently in a class.
Free:
www.khanacademy.org
free video lectures and practice problems on all manner of topics, starting with elementary algebra. You can start at the beginning and work your way through, or just start wherever.
http://ocw.mit.edu/index.htm
free online courses and lessons from MIT (!!) where you can watch lectures on a subject, do practice problems, etc. Use just for review or treat it like a course, it's up to you.
Cheap $$
http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606/ref=sr_1_1?ie=UTF8&amp;qid=1331675661&amp;sr=8-1
$10ish shipped for a book that translates calculus from math-professor to plain english, and is funny too.
http://www.amazon.com/Calculus-Lifesaver-Tools-Excel-Princeton/dp/0691130884/ref=pd_cp_b_1
$15 for a book that is 2-3x as thick as the previous one, a bit drier, but still very readable. And it covers Calc 1-3.
I also had about a 12 year break between HS and college, and like you got through Trig just fine and then found myself drowning in Calc 1. Here's what helped me:
-attended another section of the class with another professor
-books that translated the mathy language into intuition
(http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606 and http://www.amazon.com/Calculus-Easy-Way-Douglas-Downing/dp/0764129201/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1415864089&amp;sr=1-1&amp;keywords=calculus+the+easy+way)
-MIT OCW videos
-Khan Academy
Good luck. If you make it through this.. well, I'm not going to say it's easy going after, but you will know how to be confused and work through that confusion, and that is a priceless skill in the rest of the curriculum.
Calculus: https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606 and https://www.amazon.com/How-Ace-Rest-Calculus-MultiVariable/dp/0716741741 will get you right up to speed.
How to Ace Calculus
Got an A in Calculus (regular, not business) with this book, and I was really rusty at math.
Oh, and accounting is NOT math intensive...at all. If you can do + - * / and use a calculator, then you're fine.
While not a replacement text (you need more problems!), this is pretty swell for single variable and they even have a follow up text.
How to Ace Calculus: The Streetwise Guide
This book seems silly, but it's honestly great for learning Calculus, especially the second time: https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606
(I read it in 1999 when I went from HS -> College, and the college I went into assumed you had already passed calc, and freshmen all had to start with second year calc. The professors recommended all incoming students refresh before the start of class, and I'm glad they did, because that book retaught some things I don't think I learned correctly the first time, made a huge difference).
Several books helped me with calc II
http://www.amazon.com/Calculus-Lifesaver-Tools-Princeton-Guides/dp/0691131538
Good refresher
http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606
Itunes Video Series
http://itunes.apple.com/us/podcast/calculus-lifesaver-lecture/id266853222
When I was (approximately) in 8th grade I read https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606 and I loved it. :)
This isn't an online resource, but this book is awesome for learning Calc 1.
Hey, if you or anybody is having trouble with Calc 1, check out this book: How to Ace Calculus: The Streetwise Guide
It's a math book that is actually fun to read and will take you through the key points of Calc 1 with no bullshit. Lots of fun little jokes and illustrations. It's pretty short and cheap. It helped me a lot back when I learned that stuff.
Not a textbook, but check out these:
http://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606
http://www.amazon.com/gp/product/0716741741/ref=pd_lpo_k2_dp_sr_1?pf_rd_p=304485901&amp;pf_rd_s=lpo-top-stripe-1&amp;pf_rd_t=201&amp;pf_rd_i=0716731606&amp;pf_rd_m=ATVPDKIKX0DER&amp;pf_rd_r=0KX1ZS8ARF7D5EB6AF92
I'm planning on relearning calculus also. The books that were recommended to me were:
http://www.amazon.com/gp/aw/d/1592575129?pc_redir=1412262976&amp;robot_redir=1
http://www.amazon.com/gp/aw/d/0716731606/ref=pd_aw_sims_3?pi=SL500_SY115&amp;simLd=1
They're not exactly textbooks, but they appear to be good guides. Best of luck.
Would probably have to say Calculus on Manifolds by Spivak.
I think you were looking for things that weren't necessarily textbooks, but I think this book is still popular...amongst analysis courses.
Introduction to Linear Algebra by Serge Lang.
http://www.amazon.com/Introduction-Linear-Algebra-Serge-Lang/dp/3540780602
Or Introduction to Linear Algebra by Gilbert Strang
http://www.amazon.com/Introduction-Linear-Algebra-Fourth-Edition/dp/0980232716
I have not used the Strang book, but I here it is all right for non-mathematicians.
Buy a training plan off amazon and follow it rigorously.
Something like
this
or this
> Calculus has a huge foundation in mathematical analysis that at most universities takes roughly half a year to a year of graduate/upper-undergrad study to develop (at least this is how it is at my university).
Graduate/upper undergrad? At Copenhagen University (KU) material corresponding roughly to Abbott's Understanding Analysis is covered in the first year. Plus some linear algebra and other stuff.
KU does have the advantage that it doesn't have to teach any engineers. They are all over at DTU in Lyngby learning to use maths to compute things leaving the mathematics department at KU to focus on teaching maths students to prove things.