# Best calculus books according to redditors

We found 592 Reddit comments discussing the best calculus books. We ranked the 205 resulting products by number of redditors who mentioned them. Here are the top 20.

We found 592 Reddit comments discussing the best calculus books. We ranked the 205 resulting products by number of redditors who mentioned them. Here are the top 20.

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

Calculusis 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.

AnalysisIf 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.AlgebraIf you want to check out abstract algebra, check out Dummit & Foote's Abstract Algebra and/or Pinter's A Book of Abstract Algebra.TopologyThere'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-SolvingCalculusLinear AlgebraDifferential EquationsNumber TheoryProof-WritingAnalysisComplex AnalysisFunctional AnalysisPartial Differential EquationsHigher-dimensional Calculus and Differential GeometryAbstract AlgebraGeometryTopologySet Theory and LogicCombinatorics / Discrete MathGraph TheoryP. 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&amp;psc=1

The Humongous Book of Trigonometry Problems https://www.amazon.co.uk/gp/product/1615641823/ref=oh_aui_detailpage_o08_s00?ie=UTF8&amp;psc=1

Calculus: An Intuitive and Physical Approach

https://www.amazon.co.uk/gp/product/0486404536/ref=oh_aui_detailpage_o09_s00?ie=UTF8&amp;psc=1

Trigonometry Essentials Practice Workbook

https://www.amazon.co.uk/gp/product/1477497781/ref=oh_aui_detailpage_o05_s00?ie=UTF8&amp;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&amp;psc=1

Feynman's Tips on Physics

https://www.amazon.co.uk/gp/product/0465027970/ref=oh_aui_detailpage_o07_s00?ie=UTF8&amp;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&amp;psc=1

Calculus for the Practical Man

https://www.amazon.co.uk/gp/product/1406756725/ref=oh_aui_detailpage_o09_s00?ie=UTF8&amp;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&amp;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

allreal 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 least1. 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.

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.> 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.

If you want to learn how calculus actually works (rather than just how to do computations), I highly recommend working through Spivak's

Calculus. Spivak builds up calculus from the foundations with mathematical rigor and actual proofs, explaining (and proving) what's really going on. (That includes properly developing sequences and limits.) The exercises are also excellent; many of them require real thought and insight, instead of the usual "repeat the steps you were just told fifty times" exercises that fill up mainstream calculus textbooks.Also, from a more sophisticated perspective,

dxis a differential form.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.

My favorite book: http://www.amazon.com/Foundations-Analysis-Graduate-Studies-Mathematic/dp/082182693X

Available online at: http://www.futuretg.com/FTHumanEvolutionCourse/FTFreeLearningKits/01-MA-Mathematics,%20Economics%20and%20Preparation%20for%20University/006-MA06-UN01-05-Analysis/Additional%20Resources/Edmund%20Landau%20-%20Foundations%20of%20analysis.%20The%20arithmetic%20of%20whole,%20rational,%20irrational%20and%20complex%20numbers.pdf [PDF]

One of the most fun things I did when I was first learning about proofs was proving the basic facts about algebra from axioms. Where I first read about these ideas was the first chapter of Spivak's Calculus. This would be a very high level book for an 18 year old, but if you decide to look at it, don't be afraid to take your time a little.

Another option is just picking up an introduction to proof, like Velleman's How to Prove It. This wil lteach you the basics for proving anything, really, and is a great start if you want to do more math.

If you want a free alternative to that last one, you can look at The Book of Proof by Richard Hammack. It's well-written although I think it's shorter than How to Prove It.

There would have been a time that I would have suggested getting a curriculum

text book and going through that, but if you're doing this for independent work

I wouldn't really suggest that as the odds are you're not going to be using a

very good source.

Going on the typical

Arithmetic > Algebra > Calculus

****## Arithmetic

Arithmetic refresher. Lots of stuff in here - not easy.

I think you'd be set after this really. It's a pretty terse text in general.

*

****## Algebra

Algebra by Chrystal Part I

Algebra by Chrystal Part II

You can get both of these algebra texts online easily and freely from the search

`chrystal algebra part I filetype:pdf`

`chrystal algebra part II filetype:pdf`

I think that you could get the first (arithmetic) text as well, personally I

prefer having actual books for working. They're also valuable for future

reference. This

`filetype:pdf`

search should be remembered and used liberallyfor finding things such as worksheets etc (eg

`trigonometry worksheet<br /> filetype:pdf`

for a search...).Algebra by Gelfland

No where near as comprehensive as chrystals algebra, but interesting and well

written questions (search for 'correspondence series' by Gelfand).

## Calculus

Calculus made easy - Thompson

This text is really good imo, there's little rigor in it but for getting a

handle on things and bashing through a few practical problems it's pretty

decent. It's all single variable. If you've done the algebra and stuff before

this then this book would be easy.

Pauls Online Notes (Calculus)

These are just a solid set of Calculus notes, there're lots of examples to work

through which is good. These go through calc I, II, III... So a bit further than

you've asked (I'm not sure why you state up to calc II but ok).

Spivak - Calculus

If you've gone through Chrystals algebra then you'll be used to a formal

approach. This text is only single variable calculus (so that might be calc I

and II in most places I think, ? ) but it's extremely well written and often

touted as one of the best Calculus books written. It's very pure, where as

something like Stewart has a more applied emphasis.

**## Geometry

I've got given any geometry sources, I'm not too sure of the best source for

this or (to be honest) if you

reallyneed it for the above. If someone hasgood geometry then they're certainly better off, many proofs are given

gemetrically as well and having an intuition for these things is only going to

be good. But I think you can get through without a formal course on it.... I'm

not confident suggesting things on it though, so I'll leave it to others. Just

thought I'd mention it.

****Calculus Made Easy by Silvanus Thompson.

Its Prologue

Considering how many fools can calculate, it is surprising that it

should be thought either a diﬃcult or a tedious task for any other fool

to learn how to master the same tricks.

Some calculus-tricks are quite easy. Some are enormously diﬃcult.

The fools who write the textbooks of advanced mathematics—and they

are mostly clever fools—seldom take the trouble to show you how easy

the easy calculations are. On the contrary, they seem to desire to

impress you with their tremendous cleverness by going about it in the

most diﬃcult way.

Being myself a remarkably stupid fellow, I have had to unteach

myself the diﬃculties, and now beg to present to my fellow fools the

parts that are not hard. Master these thoroughly, and the rest will

follow. What one fool can do, another can.

And a link to a physical copy

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

mathright 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.

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

whythan thehow. (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.

Calculus on Manifolds is an elementary treatment that only assumes basic mathematical maturity along with say a year of calculus. A classic from Michael Spivak, who these days is mostly known for his rigorous calculus book.

You can construct the naturals, integers, rationals, reals, and the complex numbers all in terms of sets. The constructions for everything except the reals are elementary, and the reals aren't too hard, just more involved. There's a short book by Landau that does all of these, you should check it out.

Cardinals are defined in terms of ordinals, which are defined in terms of order types and well ordered sets.

Most things that you will deal with on a regular basis can be described in terms of sets. However, due to Russel's paradox, sometimes we want to talk about things that can't (consistently) be considered sets. These objects often show up in category theory, often as objects that are "too big" to a set (see proper class).

I'm sure someone who knows more about category theory than I do can give you lots of example of categories that aren't sets.

Okay. The book ''Calculus'' by Michael Spivak link is an introduction to Calculus, but using precise/rigorous notions that do away with the vague and imprecise infinitesimals. This should be what you want.

If you want something a bit more hardcore, but self-contained, I can recommend that you look at Rudin's The Principles of Mathematical Analysis, which should be in any library.

Calculus done rigorously usually goes under the name of Analysis/ Real Analysis, there should be tons of books in any math library on this subject if you feel uncomfortable with the two above.

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.

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)

You're not really doing higher math right now as much as you're learning tricks to solve problems. Once you start proving stuff that'll be a big jump. Usually people start doing that around Real Analysis like your father said. Higher math classes almost entirely consist of proofs. It's a lot of fun once you get the hang of it, but if you've never done it much before it can be jarring to learn how. The goal is to develop mathematical maturity.

Start learning some geometry proofs or pick up a book called "Calculus" by Spivak if you want to start proving stuff now. The Spivak book will give you a massive head start if you read it before college. Differential equations will feel like a joke after this book. It's called calculus but it's really more like real analysis for beginners with a lot of the harder stuff cut out. If you can get through the first 8 chapters or so, which are the hardest ones, you'll understand a lot of mathematics much more deeply than you do now. I'd also look into a book called Linear Algebra done right. This one might be harder to jump into at first but it's overall easier than the other book.

No, his single variable book.

I do plan on reading Calculus on Manifolds eventually, though.

Practical Algebra: A Self-Teaching Guide

really helped me a couple of years ago when I had to get up to speed on algebra quickly.

Beyond that, you can hardly do better in the best-bang-for-the-buck department than the Humongous Books series. 1000 problems in each book, annotated and explained, and he has an entertaining style.

The Humongous Book of Algebra Problems: Translated for People Who Don't Speak Math

The Humongous Book of Geometry Problems: Translated for People Who Don't Speak Math

The Humongous Book of Calculus Problems: For People Who Don't Speak Math

I hear Spivak's book is pretty challenging. I think that it is considered more challenging because it is strongly focused on the mathematical arguments rather than the mechanical computations.

For a challenge in applying your Calc II knowledge: take a look at Forman Acton's Real Computing Made Real. Chapter Zero on Tools of the Trade and Sketching Functions can be pretty challenging (lots of Taylor expansions, crazy algebra, etc.).

If she's bright and interested enough you might want to consider getting her an entry level college calculus book such as Spivak's.

It won't pose a replacement to the technical approach of high school, but it will illuminate a lot.

I think this is a better approach than trying to tie connections between calculus and other areas of math, because calculus has an inherent beauty of its own which could be very compelling when taught with the right philosophical approach.

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).

If you're looking for other texts, I would suggest Spivak's

CalculusandCalculus 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.

You should really start with a good introductory analysis text before trying to tackle topology. It's more familiar territory and less generalized. I think I know a book that will have exactly what you are looking for. I'd recommend picking up Analysis: With an Introduction to Proof by Steven Lay.

It assumes no previous experience with writing proofs, so the first few chapters introduce some basic logic and set theory. Then, you will rigorously define the notion of a function before going through a very nice topology primer. It's not the ultimate analysis reference book by any means, but it's a great starter for beginners and self-study.

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.

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.

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.

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)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.

I recommend you start studying proofs first. How to Prove It by Velleman is a great book for new math students. I went through the first three chapters myself before my first analysis course, and it made all the difference.

As you are taking a class than combines analysis and calculus, you might benefit from Spivak's book Calculus, which despite it's title, is precisely a combination of calculus and real analysis.

Intro Calculus, in American sense, could as well be renamed "Physics 101" or some such since it's not a very mathematical course. Since Intro Calculus won't teach you how to think you're gonna need a book like How to Solve Word Problems in Calculus by Eugene Don and Benay Don pretty soon.

Aside from that, try these:

Excursions In Calculus by Robert Young.

Calculus:A Liberal Art by William McGowen Priestley.

Calculus for the Ambitious by T. W. KORNER.

Calculus: Concepts and Methods by Ken Binmore and Joan Davies

You can also start with "Calculus proper" = Analysis. The Bible of not-quite-analysis is:

[Calculus by Michael Spivak] (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1413311074&amp;sr=1-1&amp;keywords=spivak+calculus).

Also, Analysis is all about inequalities as opposed to Algebra(identities), so you want to be familiar with them:

Introduction to Inequalities by Edwin F. Beckenbach, R. Bellman.

Analytic Inequalities by Nicholas D. Kazarinoff.

As for Linear Algebra, this subject is all over the place. There is about a million books of all levels written every year on this subject, many of which is trash.

My plan would go like this:

1. Learn the geometry of LA and how to prove things in LA:

Linear Algebra Through Geometry by Thomas Banchoff and John Wermer.

Linear Algebra, Third Edition: Algorithms, Applications, and Techniques

by Richard Bronson and Gabriel B. Costa.

2. Getting a bit more sophisticated:

Linear Algebra Done Right by Sheldon Axler.

Linear Algebra: An Introduction to Abstract Mathematics by Robert J. Valenza.

Linear Algebra Done Wrong by Sergei Treil.

3. Turn into the LinAl's 1% :)

Advanced Linear Algebra by Steven Roman.

Good Luck.

>When university starts, what can I do to ensure that I can compete and am just as good as the best mathematics students?

Read textbooks for mathematics students.

For example for Linear Algebra I heard that Axler's book is very good (I studied Linear Algebra in another language, so I can't really suggest anything from personal experience). For Calculus I personally suggest Spivak's book.

There are many books that I could suggest, but one of the greatest books I've ever read is The Art and Craft of Problem Solving.

Hey!

So, the topics you listed are all covered in a Calculus I class. There are some texts that are specific to calc I, but most (in my experience) have

the whole shebang,up through Calc III and maybe into some basic diff. eqns.Larson's Calculus of a Single Variable is availible for $13 as an E-book, if you're okay with that. This version only goes through Calc I, but it's a bit cheaper than the full book. I personally don't love this book, but a lot of people swear by it. It gives lots of application examples, but I don't think they do a great job showing how they work through solutions. This is best

as a supplement to a class that uses problems from that book.My personal favourite is Dover's Calculus: An Intuitive and Physical Approach. This book is much more theorem-oriented and I think it stands better alone than Larson's calculus. I taught myself from this book.

Learning proofs can mean different things in different contexts. First, a few questions:

The sort of recommendations for a pre-university student are likely to be very different from those for a university student. For example, high school students have a number of mathematics competitions that you could consider (at least in The United States; the structure of opportunities is likely different in other countries). At the university level, you might want to look for something like a weekly problem solving seminar. These often have as their nominal goal preparing for the Putnam, which can often feel like a VERY ambitious way to learn proofs, akin to learning to swim by being thrown into a lake.

As a general rule, I'd say that working on proof-based contest questions that are

justbeyond the scope of what you think you can solve is probably a good initial source of problems. You don't want something so difficult that it's simply discouraging. Further, contest questions typically have solutions available, either in printed books or available somewhere online.This may be especially true for things like logic and

veryelementary set theory.Some recommendations will make a lot more sense if, for example, you have access to a quality university-level library, since you won't have to spend lots of money out-of-pocket to get copies of certain textbooks. (I'm limiting my recommendations to legally-obtained copies of textbooks and such.)

Imagine trying to learn a foreign language without being able to practice it with a fluent speaker, and without being able to get any feedback on how to improve things. You may well be able to learn how to do proofs on your own, but it's

orders of magnitudemore effective when you have someone who can guide you.rigorousmathematical proofs?Put differently, is your current goal to be able to produce a proof that will satisfy yourself, or to produce a proof that will satisfy someone

else?Have you had at least, for example, a geometry class that's proof-based?

Proofs are all about

communicating ideas. If you struggle with writing in complete, grammatically-correct sentences, then that will definitely be a bottleneck to your ability to make progress.---

With those caveats out of the way, let me make a few suggestions given what I think I can infer about where you in particular are right now.

How to Prove It: A Structured Approachby Daniel Velleman is a well-respected general introduction to ideas behind mathematical proof, as isHow to Solve It: A New Aspect of Mathematical Methodby George Pólya.Calculusby Michael Spivak. This is a challenging textbook, but there's a reason people have been recommending its different editions over many decades.writemathematically sound proofs, it helps toreadas many as you can find (at a level appropriate for your background and such). You can find plenty of examples in certain textbooks and other resources, and being able to work from templates of "good" proofs will help you immeasurably.Learning proofs is in many ways a skill that requires cultivation. Accordingly, you'll need to be patient and persistent, because proof-writing isn't a skill one typically can acquire passively.

---

How to improve at proofs is a big question beyond the scope of what I can answer in a single reddit comment. Nonetheless, I hope this helps point you in some useful directions. Good luck!

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

Apostol and Spivak are the best calculus texts I know; paperback versions of each exist.

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 really believe that Michael Kelly's "Humongous Book of" series are the best resources for getting through all math classes up to Calculus II. These books contain every single type of problem you will ever encounter in Algebra I & II, Geometry, Trig, and Calc I & II,

all solved in great detail.They are like Schaums Outlines but much more reliable.https://www.amazon.com/Humongous-Basic-Pre-Algebra-Problems-Books/dp/1615640835

https://www.amazon.com/Humongous-Book-Algebra-Problems-Books/dp/1592577229

https://www.amazon.com/Humongous-Book-Geometry-Problems-Books/dp/1592578640

https://www.amazon.com/Humongous-Book-Trigonometry-Problems-Comprehensive/dp/1615641823

https://www.amazon.com/Humongous-Book-Calculus-Problems-Books/dp/1592575129

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

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.

Thomas' CalculusCalculus books:

,Calculusby 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), andPhysics 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 Applicationsby Giancoli. I know there are many others, but I can't speak for them.

Engineering MathematicsMathematical methods: Greenberg is way more than you need here. I think you would find

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.

**

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!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

Business calculus. I struggled until I bought a copy of Forgotten Calculus. The book made sense to me and I did well in class after using it. My professor also thought the book was terrific, so I gave it to him after the final.

Every Thing Must Go: James Ladyman & Don RossThe Philosophy of Complex Systems: Anthology edited by Cliff HookerNatural Born Cyborgs: Andy ClarkCognitive Surplus: Clay ShirkyQuantum Mechanics and ExperienceandTime & Chance: David AlbertTime's Arrow and Archimedes' Point: Huw PriceScience in the Age of Computer Simulation: Eric WinsbergAnt Encounters: Interactions Networks and Colony Behavior: Deborah GordonThe Construction of Social Reality: John SearleUnsimple Truths: Science, Complexity, and Policy: Sandra MitchellThe Devil in the Details: Asymptotic Reasoning in Explanation, Reduction, and Emergence: Robert BattermanComplexity: A Guided Tour: Melanie MitchellScientific Metaphysics: Anthology edited James LadymanNonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering: Stephen StrogatzThe Ethical Project: Philip KitcherFoundations of Complex Systems Theories: Sunny AuyangA Vast Machine: Computer Models, Climate Data, and the Politics of Global Warming: Paul Edwardsonce 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.

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

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.

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 Easyby 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.

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.

http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&amp;qid=1344481564&amp;sr=8-1&amp;keywords=spivaks+calculus

I haven't read all of it, but even the bit I did read was very challenging and it is generally recommended around here for a rigorous introduction to calculus. Be warned, it is pretty challenging, especially if you aren't comfortable with proofs.

Look at worked-out problems. I highly recommend books in this series: http://www.amazon.com/Humongous-Book-Calculus-Problems-People/dp/1592575129/ref=pd_sim_b_2

Beyond that, slog through practice problems. Math is a language. You can know a mind-blowing concept, but you won't develop an intuition for it without repeated exposure. This includes the stuff you might look at and think there's no reason for you to know that cuz software will handle it - if you're looking at a proof that makes zero sense without knowing what happens when you divide logarithms, you're going to be lost.

https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918

From my experience, Calculus in America is taught in 2 different ways: rigorous/mathematical in nature like Calculus by Spivak and applied/simplified like the one by Larson.

Looking at the link, I dont think you need to know sets and math induction unless you are about to start learning Rigorous Calculus or Real Analysis. Also, real numbers are usually introduced in Real Analysis that comes after one's exposure to Applied/Non-Rigorous Calculus. Complex numbers are, I assume, needed in Complex Analysis that follows Real Analysis, so I wouldn't worry about sets, real/complex numbers beyond the very basics. Math induction is not needed in non-proof based/regular/non-rigorous Calculus.

From the link:

For Calc 1(applied)- again, in my experience, this is the bulk of what's usually tested in Calculus placement exams:Solving inequalities and equations

Properties of functions

Composite functions

Polynomial functions

Rational functions

Trigonometry

Trigonometric functions and their inverses

Trigonometric identities

Conic sections

Exponential functions

Logarithmic functions

For Calc 2(applied)- I think some Calc placement exams dont even contain problems related to the concepts below, but to be sure, you, probably, should know something about them:Sequences and series

Binomial theorem

In Calc 2(leading up to multivariate Calculus (Calc 3)). You can pick these topics up while studying pre-calc, but they are typically re-introduced in Calc 2 again:Vectors

Parametric equations

Polar coordinates

Matrices and determinants

As for limits, I dont think they are terribly important in pre-calc. I think, some pre-calc books include them just for good measure.

Let me recommend Spivak,

Calculus.Might not be the type of ‘refresher’ you are looking for though.

I have to profoundly disagree. You're not "bam, you're writing code". That's like saying if I throw you into the deep end of a pool "bam, you're swimming". No, you're flailing for dear life.

There's a reason that Python is a better first language than Java. In Java everything must be in a class. You can't teach "Hello, world" without invoking the concepts of classes, objects, methods and variables. Generally this means the instructor will say something along the lines of "Type this in and just ignore all of this other stuff for now." This leaves the student feeling like I did when I attempted to learn Calculus with a bunch of math geeks in college when my mind is not wired for math: you lose confidence. Even if your program compiles/you get the right math answer, you say: "I just wrote down a bunch of gibberish and I have no idea what it means or how it worked. I wonder if I ever will." I passed Calc I (with a D) yet at the end of the course I still didn't know what calculus

wasor why one moved their x's here or their y's there. I had no understanding, and you can imagine how that set me up for Calc II (two tries, two F's). Contrast this approach with Ken Ahmdahl's Calculus For Cats which is mostly words and not a single exercise.The beginner to programming

needsa 45-page intro trying to introduce them to the concepts of computer programming. Otherwise they're just memorizing keywords and actions they don't understand. I remember what I went through with Calculus so I can relate (even though I'm too old to really remember how easy/hard it was to learn programming). Maybe other people don't remember what it was like to learn their first computer language. No concept of variables, local/global scope, flow control, types, etc. I can't believe that throwing a complete newbie into the deep end ever produces good results.This poster doesn't need to learn

Python; they need to learnprogramming, and that's something else entirely. Python can be a good tool to do that, but one does not approach doing that like one does one's fifth computer language.He really should get this for Christmas

[Calculus Made Easy] ( http://www.amazon.com/gp/aw/d/1456531980/ref=redir_mdp_mobile?pc_redir=T1) for calculus. This book should be the standard textbook.

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.

If it's a review I suggest a book called Quick Calculus, also look over Paul's math notes trig and algebra review.

I had the misfortune of teaching calculus from the Anton/Bivens/Davis book. That book is filled with so much needless fluff - not to mention contrived "applications" and convoluted explanations - that I will never use it again. I much prefer Calculus: An Intuitive and Physical Approach by Morris Kline. I think it's the best of the calculus texts still being published, and gives a much better feel for the subject than the standard books.

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

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.

You might look at Michael Spivak's Calculus ( http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098896 ). In the preface to the second edition, Spivak writes:

>I have often been told that the title of this book should really be something like "Introduction to Analysis", because the book is usually used in courses where the students have already learned the mechanical aspects of calculus--such courses are standard in Europe.

The book starts by developing the real and complex number systems and later goes into proofs that pi is irrational, e is transcendental, etc.

Please note that I'm not a math major and have only just started working through the Spivak book myself, so I'm far from an authority on the subject. But it's the book I stumbled onto when I was looking for a similarly non-numeric perspective on calculus and basic analysis and so far I've been pleased with it.

If it helps, here are some free books to go through:

Linear Algebra Done Wrong

Paul's Online Math Notes (fantastic for Calc 1, 2, and 3)

Basic Analysis

Basic Analysis is pretty basic, so I'd recommend going through Rudin's book afterwards, as it's generally considered to be among the best analysis books ever written. If the price tag is too high, you can get the same book much cheaper, although with crappier paper and softcover via methods of questionable legality. Also because Rudin is so popular, you can find solutions online.

If you want something better than online notes for univariate Calculus, get Spivak's Calculus, as it'll walk you through single-variable Calculus using more theory than a standard math class. If you're able to get through that and Rudin, you should be good to go once you get good at linear algebra.

I know this isn't

exactlywhat you're requesting (I assume you're requesting resources on the web for your consumption) but allow me to suggest the following bookA large part of truly understanding mathematics is built upon the foundation of understanding and being able to correctly write proofs. The book above will introduce you to proofwriting and do so while teaching you

whycertain things you learned in college-level calculus I and II are correct; this may prove more rewarding of an experience than simply crunching answers based on theorems that the book tells you are true.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.

There's a lot of fun and interesting physics and astronomy that can be understood with little more than solid algebra skills. Add a little bit of introductory calculus, and there's a lot to keep you busy. If you're brave enough to dive into calc, I recommend this book.

Since you expressed particular interest in Astronomy, I would suggest using that as an anchor point. Get a good Astrophysics text like

An Introduction to Modern Astrophysicsby Carroll and start there. Inevitably, you will come upon concepts that you're shaky on-- luckily this is the age of the internet! I find HyperPhysics is a great resource (which appears to be down at the moment).If you find that Newtonian physics is tripping you up, I recommend Basic Physics: A Self-Teaching Guide to fill in the gaps.

'Quick Calculus' is based around a self-tutoring approach, and is great for brushing up. You can probably find a PDF of it around somewhere.

I found that the book

Quick Calculus: A Self-Teaching Guide(Amazon link) was quite good.This was offered as an optional text for one of my beginning calculus courses: Quick Calculus

It's kind of like choose your own adventure calculus--if you get the correct answer to a problem, you go to one page, but if you get an incorrect answer, it sends you to the dungeon to work through the concept again.

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

secondtime, 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.

Presumably: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order

Look on page 80-81 of this book: http://www.amazon.com/dp/0486450015 . It's available in the book preview.

Can't beat Dover books. The reviews are quite good for this one, and the price is right (less than $10).

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

Yeah I agree with analmouthwash, holy shit. That's pretty high end for a calc 2 class.. anyway, i would definitely recommend How to Ace the Rest of Calculus . It's not a go-to reference book but it really makes the topics fun/easy to understand and gives one or two good, easy-to-follow examples for each topic.

I was looking for the classic

Calculus Refresherby A. Albert Klaf, when I found this brief refresher for statistics majors and this longer one for a course; I also found a similar book calledForgotten Calculus.It also seems like certain computer science classes send out refreshers on linear algebra, including this one that focuses on matrix operations; I also found this lovely set of slides for an actual refresher course intended for people who took Linear Algebra a while ago.

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

For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.

I guess I also just disagree with a lot of people about the

"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's

excellenttext Topology of Surfaces.If you're interested in a sequence of books...keep reading.

If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent

Linear Algebra Done Rightand, maybe, one of those big, dumb algebra books like Dummit and Foote.Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.

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 Scienceby James GleickTechnical (requires knowledge of ordinary differential equations, but highly readable):

Nonlinear Dynamics and Chaosby Steven H. StrogatzYou 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

I'll be working through Spivak's calculus for fun. Wish me luck!

For getting more intuition on proofs I would suggest the following book

http://www.amazon.com/Nuts-Bolts-Proofs-Third-Introduction/dp/0120885093/ref=sr_1_1?ie=UTF8&amp;qid=1311007015&amp;sr=8-1

I think Rudin might be really tricky at your level, you can keep with it if you want, but I think Calculus by Michael Spivak would be much more approachable for you.

http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1311007057&amp;sr=1-1

i personally prefer Yurope! Hillary's Invasion. Very insightful reading.

I didn't mean to make it sound so serious :) However, stress, drinking, and insomnia can all have some unexpectedly large effects, so it may be worth dropping into a counseling session if your university has one.

In regards to math education and intuition, something I found very useful was to read some books that start from scratch, like Burn Math Class, or Spivak's calculus for a real challenge. You're at a point in your education where you have the sophistication to understand the foundations of math, so you can start to rebuild intuition about a lot of things that will make university-level math much more sensible.

Start with 3 Blue 1 Brown's Essence of Calculus Series - https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

and follow the following books -

Calculus by Spivak - https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918

Calculus Made Easy - http://calculusmadeeasy.org/

Follow all the concepts and solve the examples and exercises.

Feel free to ask the questions here or in mathsoverflow.

Last but not the least, PRACTICE, PRACTICE, PRACTICE........!

I am surprised no one has mentioned M. Spivak's very well known text Calculus. I thought this book was a pleasure to read. His writing was very fun and lighthearted and the book certainly teaches the material very well. In my opinion this is the best introductory calculus text there is.

When I first started learning math on my own, I started learning calculus from something like this. Though I enjoyed it, it didn't really show me what 'real math' was like. For learning something closer to higher math, a more rigorous version would be something like this. It's all preference, though.

If you don't know much about calculus at all, start with the first one, and then work your way up to Spivak.

If you want to learn serious mathematics, start with a theoretical approach to calculus, then go into some analysis. Introductory Real Analysis by Kolmogorov is pretty good.

As far as how to think about these things, group theory is a strong start. "The real numbers are the unique linearly-ordered field with least upper bound property." Once you understand that sentence and can explain it in the context of group theory and the order topology, then you are in a good place to think about infinity, limits, etc.

Edit: For calc, Spivak is one of the textbooks I have heard is more common, but I have never used it so I can't comment on it. I've heard good things, though.

A harder analysis book for self-study would be Principles of Mathematical Analysis by Rudin. He is very terse in his proofs, so they can be hard to get through.

If you have a chance, I recommend checking out some textbooks on real analysis, which will guide you through the derivations and proofs of many theorems in calculus that you've thus far been expected to take for granted.

Some would recommend starting with Rudin's Principles of Mathematical Analysis, and it's certainly a text that I plan to read at some point. For your purposes, I might recommend Spivak's Calculus since it expects you to rigorously derive some of the most important results in calculus through proof-writing exercises. This was my first introduction to calculus during high-school. While it was overwhelming at first, it prepared me for some of my more advanced undergraduate courses (including real analysis and topology), and it seems to be best described as an advanced calculus textbook.

Linear algebra is about is about linear functions and is typically taken in the first or second year of college. College algebra normally refers to a remedial class that covers what most people do in high school. I highly recommend watching this series of videos for getting an intuitive idea of linear algebra no matter what book you go with. The book you should use depends on how comfortable you are with proofs and what your goal is. If you just want to know how to calculate and apply it, I've heard Strang's book with the accompanying MIT opencourseware course is good. This book also looks good if you're mostly interested in programming applications. A more abstract book like Linear Algebra Done Right or Linear Algebra Done Wrong would probably be more useful if you were familiar with mathematical proofs beforehand. How to Prove it is a good choice for learning this.

I haven't seen boolean algebra used to refer to an entire course, but if you want to learn logic and some proof techniques you could look at How to Prove it.

Most calculus books cover both differential and integral calculus. Differential calculus refers to taking derivatives. A derivative essentially tells you how rapidly a function changes at a certain point. Integral calculus covers finding areas under curves(aka definite integrals) and their relationship with derivatives. This series gives some excellent explanations for most of the ideas in calculus.

Analysis is more advanced, and is typically only done by math majors. You can think of it as calculus with complete proofs for everything and more abstraction. I would not recommend trying to learn this without having a strong understanding of calculus first. Spivak's Calculus is a good compromise between full on analysis and a standard calculus class. It's possible to use this as a first exposure to calculus, but it would be difficult.

You do realize that there is guesswork but the extremes of the confidence interval are strictly positive right? In other words, no one is certain but what we are certain about is that optimum homework amount is

positive. Maybe it's 4 hours, maybe it's 50 hours. But it's definitely not 0.I don't like homework either when I was young. I dreaded it, and I skipped so many assignments, and I regularly skipped school. I hated school. In my senior year I had such severe senioritis that after I got accepted my grades basically crashed to D-ish levels. (By the way this isn't a good thing. It makes you lazy and trying to jumpstart again in your undergrad freshman year will feel like a huge, huge chore)

Now that I'm older I clearly see the benefits of homework. My advice to you is not to agree with me that homework is useful. My advice is to pursue your dreams, but when doing so be keenly aware of the pragmatical considerations. Theoretical physics demands a high level of understanding of theoretical mathematics: Lie groups, manifolds and differential algebraic topology, grad-level analysis, and so on. So get your arse and start studying math; you don't have to like your math homework, but you'd better be reading Spivak if you're truly serious about becoming a theoretical physicist. It's not easy. Life isn't easy. You want to be a theoretical physicist? Guess what, top PhD graduate programs often have acceptance rates

lowerthan Harvard, Yale, Stanford etc. You want to stand out? Well everyone wants to stand out. But for every 100 wannabe 15-year-old theoretical physicists out there, only 1 has actually started on that route, started studying first year theoretical mathematics (analysis, vector space), started reading research papers, started reallyknowingwhat it takes. Do you want to be that 1? If you don't want to do homework, fine; but you need to be doingworkthat allows you to reach your dreams.I have yet to read it myself, but the classic text for Calculus is Spivak's

Calculus. It is very highly recommended.Hey y'all! I'm 16, and am about to finish Spivak's Calculus. Assuming that I know everything up to Algebra II, AP Statistics, Trigonometry, a bit of linear algebra (please specify if the subject requires extensive knowledge here), and have thoroughly gone through Spivak's Calculus, what should be the next thing I study? And what textbook(s) would you recommend for learning that subject?

Right now I'm leaning towards Real Analysis, or Multi Variable Calculus, or maybe Topology, or... case in point, I am very undecided and am in need of recommendations.

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.

I am doing this very thing. I found some fantastic books that might help get you (re)started. They certainly helped me get back into math in my 30s. Be warned, a couple of these books are "cute-ish", but sometimes a little sugar helps the medicine go down:

I wish you all the best!

Hi great story, how do you manage a fulltime job and a parttime physics degree? Do you have any children? Here in the Netherlands there are no partime physics studies. Where do you study? I’m getting smooth at algebra right now using

No bullshit guide to math and physics: https://www.amazon.com/No-bullshit-guide-math-physics/dp/0992001005 hard to follow at times since Savov not always explains very much. Overal a good book.

Okay then I'll make some recommendations!

If you're into science: [Death by Black Hole] (https://smile.amazon.com/Death-Black-Hole-Cosmic-Quandaries/dp/039335038X/ref=sr_1_1?ie=UTF8&amp;qid=1491764342&amp;sr=8-1&amp;keywords=death+by+black+hole)

If you like classic mysteries: [A Study in Scarlet] (https://smile.amazon.com/Study-Scarlet-Arthur-Conan-Doyle/dp/1514698854/ref=sr_1_1?ie=UTF8&amp;qid=1491764421&amp;sr=8-1&amp;keywords=a+study+in+scarlet)

If you want something absurd: [What If?] (https://smile.amazon.com/What-If-Scientific-Hypothetical-Questions/dp/0544272994/ref=sr_1_1?ie=UTF8&amp;qid=1491764514&amp;sr=8-1&amp;keywords=what+if)

And here's some things I have been looking to get: [The Republic] (https://smile.amazon.com/dp/0141442433/ref=wl_it_dp_o_pC_nS_ttl?_encoding=UTF8&amp;colid=226JO509390Q8&amp;coliid=I2FVTI8SC5VCYS)

and

[The No Bullshit Guide to Math and Physics] (https://smile.amazon.com/dp/0992001005/ref=wl_it_dp_o_pd_nS_ttl?_encoding=UTF8&amp;colid=226JO509390Q8&amp;coliid=I1P7W8OMFLADHK)

Hope one of these sparks an idea!

Hi Micromeds, I hope you like the NO BS guide to MATH & PHYS. It's clear you have the right attitude—the best way to learn math is by solving lots of practice problems.

Sometime in January I'll be releasing the NO BS guide to LA, so if you like the first book you should check out the sequel. Extended preview for anyone interested: https://minireference.com/static/excerpts/noBSguide2LA_preview.pdf

BTW, for fellow Canadians, there's a crazy rebate on the MATH & PHYS book on amazon.ca today:

https://www.amazon.ca/dp/0992001005/

This book (available on Amazon as well http://www.amazon.com/Calculus-Made-Silvanus-Phillips-Thompson/dp/1456531980) helped me understand a lot of the whys of calculus http://www.gutenberg.org/files/33283/33283-pdf.pdf

ja os espertos usam isso

Good luck! It's some fun stuff. I'd also recommend this book if you don't already have it:

http://www.amazon.com/The-Humongous-Book-Calculus-Problems/dp/1592575129

Sorry, my post wasn’t very clear. Those were actually specific titles.

Practical Algebra:

https://www.barnesandnoble.com/w/practical-algebra-peter-h-selby/1114284979

Geometry and Trigonometry for Calculus:

https://www.barnesandnoble.com/p/geometry-and-trigonometry-for-calculus-peter-h-selby/1114965492/2676067143387

Those are both very good. My calculus recommendation is a little unconventional, so maybe it’s not for you, but I’d get Calculus: An Intuitive and Physical approach.

https://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics-ebook/dp/B00CB2MK6C

That book is far more wordy than your average calculus text, but I think that makes it great for self teaching. If you pick up something like 3000 Solved Calculus problems to go along with it you should be in great shape.

I know that’s not exactly cheap, but you should be able to pick up all of those for less than $100. Good luck!

Edit: all of the statistics texts in the last paragraph of my original post are available freely (legally, I believe) online.

You could try "Precalculus" by Stitz & Zeager. Chapters 10 and onwards is their Trigonometry book. This should be a very smooth book to work through.

Have you already picked out a Calculus textbook? Also, what are her plans as an MIT student? If she's going into engineering and the like, I would say Larson's "Calculus" (solutions manuals vol 1, 2) would be good enough.

If she plans on being a math student, though, I would say give her a a few months with Velleman's "How To Prove It". Afterwards, I can't recommend Spivak's Calculus (Answer Book) and Jim Hefferon's Linear Algebra (solutions manual on same page) enough. This is a good time to introduce mathematical rigor as a normal thing in mathematics because, really, this is what math is about.

No, he wrote a book on single-variable calculus, too: https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918

None of the questions you asked is “silly” or “simple.” There’s a whole lot going on in calculus, most of which is typically explained in a real analysis course. Rigorous proofs of things like the mean value theorem or various forms of integration are challenging, but they will provide the clarity you’re looking for.

I recommend that you check out something like Spivak’s Calculus, which is going to give a more rigorous intro to the subject. Alternately, you can just find a good analysis or intro to proofs class somewhere. It’s a fascinating subject, so good luck!

I would highly advise going with the 31/37 route. As both of the above courses are proof based, they will be play an integral role in upper year courses. Please be warned that they are extremely challenging but worthwhile courses. I would highly recommend you start preparing for the above two courses. For A37, I would suggest starting with Spivak:

https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918

You can start with Calculus by Spivak. If you're going to buy it then wait until after the Fall semester begins; the price is inflated right now because students need it for school.

This is a PDF of the third edition of the above book.

This is an excellent introduction to logic and proofs. You will want a strong understanding of how mathematicians communicate via proof and that book will really help.

The math subreddit is primarily undergrads talking about various topics. Make a point of just hanging out and reading stuff. If you don't understand something just tell us and they'll do their best to help out.

Hang out on the math stack exchange and ask questions about things you do not understand while trying to help with things you do understand.

Hope that helps!

You'll remember and forget formulae as you use them. It's the using them that makes things concrete in your head.

Once you're comfortable with algebra, trig. I'm assuming you've had geometry, since you were taking algebra 2; if not, geometry as well.

Once you're comfortable with those topics, you'll have enough of the basics to start branching out. Calculus is one obvious direction; a lot people have recommended Spivak's book for that. Introductory statistics is another (far too few people are even basically statistically literate.) Discrete math is yet another possibility. You can also start playing with "problem math", like the Green Book or Red Book. Algebraic structures is yet another possibility (I found Herstein's abstract algebra book pretty easy to read when we used it in school).

Edit: added Amazon links.

Question about Spivak's Calculus and Ross' Elementary Classical Analysis:

Are they books treating mathematics on the same level? Do they treat the rigorous theoretical foundation and computational techniques equally well? Can each one be an alternative to the other? Could someone please give brief comparative reviews/comments on them?

This question is also on r/learnmath: HERE.

This is good advice. Source: I flunked a private engineering school at age 17, in spite of of being 99th percentile in the ACT. Reason? Besides socialization issues, poor mathematics and academic preparation at my rural high school, where few went to college, let alone out-of-state.

I'm a strong believer in self-education (and self-employment) and am currently rectifying the above-stated issues.

Came here to plug Spivak's

Calculus. It's a bit harder and more detailed than most calculus texts used today, but that's because he actually explains all the tricky bits, rather than just using hand-waving to finish those tricky bits. (It was the hand-waving that always left me confused in classroom teaching.) Spivak'sCalculusmight not be the place to start, but it's where you want to end up, so I want you to know about it.Peace out, bro, and keep working. We'll make it. ME/EE is a great combo, btw. ME is the first branch of engineering, though it was called something else, when "engines of war", catapaults and whatnot, was the only game in town. But, all machines need sensors, controls, and power, which is the EE bit. Put it together, and you get mechatronics, which is part of the future.

One piece of added advice: stick to one of the main-line branches of engineering: mechanical, electrical, chemical, maybe civil, instead of one of the new, hybrid branches, like biomedical, etc. The jobs are more plentiful, you'll get a sounder foundation in engineering principles, and specializing is still possible.

Ed: Do you already know about MIT's Open Course Ware site? Most MIT courses are online with videoed lectures, recommended textbooks, homework and tests. It's a great resource. They also have edX, a co-operative venture with a bunch of fancy schools.

Calculus - Michael Spivak

Hey, I have found a nice text book called no bullshit guide with math and physics.

https://www.amazon.ca/No-bullshit-guide-math-physics/dp/0992001005

No bullshit guide to math and physics: Ivan Savov ... - Amazon.ca

So far it's the best I've ever read. The author made another one only for linear algebra as well.

https://www.amazon.ca/No-bullshit-guide-linear-algebra/dp/0992001021

No bullshit guide to linear algebra: Ivan Savov ... - Amazon.ca

Hello again! I have a question about the young and fredman physics textbook. Which version should I purchase and should I also purchase the solutions manual? Also would that textbook be better than say this https://www.amazon.com/No-bullshit-guide-math-physics/dp/0992001005/ref=sr_1_3?ie=UTF8&amp;qid=1466240189&amp;sr=8-3&amp;keywords=physics

Thank you for all your help so far (:

First, to get a sense as to the world of math and what it encompasses, and what different sub-subjects are about, watch this: https://www.youtube.com/watch?v=OmJ-4B-mS-Y

Ok, now that's out of the way -- I'd recommend doing some grunt work, and have a basic working knowledge of algebra + calculus. My wife found this book useful to do just that after having been out of university for a while: https://www.amazon.com/No-bullshit-guide-math-physics/dp/0992001005

At this point, you can tackle most subjects brought up from first video without issue -- just find a good introductory book! One that I recommend that is more on computer science end of things is a discrete math

book.

https://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025

And understanding proofs is important: https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472108

http://www.amazon.com/No-bullshit-guide-math-physics/dp/0992001005

http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics-ebook/dp/B00CB2MK6C

Hi, this book really helped me.

https://www.amazon.com/No-bullshit-guide-math-physics/dp/0992001005

You're welcome! I love helping people and want to teach in the future, so "thanks!" is probably my #1 favorite thing!

One thing I didn't mention in the above post that I'm starting to realize as I go through more higher level Math classes (Linear Algebra this semester) --

find MULTIPLE explanations for a concept and find MANY worked out, annotated problems.This is coming up because I'm having a lot of difficulty with Linear Algebra. The calculations are simple (it's basically just solving systems of equations from pre-Calc), but understanding what everything

meansis a whole different story.I spoke with some of the instructors in my colleges math department this semester and they all agreed on those two points. My Linear Instructor isn't

bad, he just doesn't teach in a style I learn well in. Our textbook isalright, but doesn't have many examples worked out (maybe 3-5 per section, but each one is so fundamentally different nothing it's hard to understand what's being illustrated and why).The Shaum's Outline series came highly recommended. The head of the department specifically cited Shaum's Linear Algebra for the reason she passed linear (she could never attend class and was teaching herself out of the textbook at the time, so the more help the better). While I can't comment on the Calculus version, I'm loving what I've seen so far in the Linear version. So if you ever need to see more problems worked out and Paul's math notes isn't doing it for you, try getting a Shaum's book. They're pretty inexpensive at around $10-20 on Amazon -- I picked up a used copy for $8.

I also picked up the No Bullshit Guide to Linear Algebra after a tutor friend recommended it. The author has a Calculus/Physics integrated version. It's pricy, but if you're having a difficult time understanding your Calculus book or just need a legitimately no-bullshit explanation of a concept, it's a great option. Again, I haven't seen the Math/Physics version, but if it's anything like the Linear one it'd be mad helpful.

Lastly, if you still need more examples or explanations, the book my college is using for Calculus is available here on reddit.

UK Amazon

US Amazon

"Humongous Book of Calculus" explains in english without treating you like a dummy or a 5 year old in need of a story.

Don't feel badly. Calc II favors rote memorization, which a lot of people have to work at. You just have to practice.

Some stuff I did to get it down:

-Write the formulas on your bathroom mirror with dry erase. Every time you go to wash hands or pass it, try to review it a bit.

-Write the new formula 30+ times. It sucks. You are going to hate it, but damn if it doesn't work. As you're writing, try to review which variables mean what.

-Practice problems while waiting in line, commuting, etc. I liked this book (The Humongous Book of Calculus Problems) for some great explanations and practice problems: https://www.google.com/url?sa=t&amp;source=web&amp;rct=j&amp;url=https://www.amazon.com/Humongous-Book-Calculus-Problems-Books/dp/1592575129&amp;ved=2ahUKEwimlKb11f7jAhXUqZ4KHdXBC0QQFjAAegQIARAB&amp;usg=AOvVaw38Qmi3pxSjppZwJW6CBno8

I need either this or this. I'm taking Calculus II this semester for the second time. I'm aiming to be a math major, but I had difficulty last time. I'm already off to a better start this semester, but I want as much practice as possible. I'm aiming for a Masters in Math. I'm lucky that I have high grades and the F from last semester only dropped me down to a 3.2 GPA. I can't afford to have it drop any lower. I can't afford to spend any more time at this level. I have a Calculus workbook that my mom bought me, but it only covers Calc I and about two chapters of Calc II.

Actually.. Anything from my School Stuff WL is stuff I feel I need in order to do well at school. I really need to get organized with my school work and papers.. ._.

Most of the trig and precal you need will be built in to calculus problems. I would recommend just jumping in and doing lots of problems. The Humongous Book of Calculus Problems starts with trig and precal and moves into calculus, with everything explained. http://www.amazon.com/gp/aw/d/1592575129/ref=mp_s_a_1_1?qid=1452092788&amp;sr=8-1&amp;pi=SY200_QL40&amp;keywords=humongous+book+of+calculus+problems&amp;dpPl=1&amp;dpID=515J89M2yTL&amp;ref=plSrch.

It is also cheap. They also make one for algebra and trig but you probably don't need it. There is also an awesome free calculus book here:

https://www.math.wisc.edu/~keisler/calc.html

Along the way if you get stuck on something specific and a written explanation won't suffice, check khan Academy or YouTube for it.

Also if you plan on studying mathematics or anything closely related, you will likely need an analysis course, in which case Spivak's "Calculus" provides an excellent bridge.

books that have helped me( i keep them as reference)

https://www.amazon.com/Forgotten-Algebra-Barbara-Lee-Bleau/dp/0764120085/ref=sr_1_2?ie=UTF8&amp;qid=1500377758&amp;sr=8-2&amp;keywords=forgotten+algebra

https://www.amazon.com/Forgotten-Calculus-Barbara-Bleau-Ph-D/dp/0764119982/ref=sr_1_1?ie=UTF8&amp;qid=1500379330&amp;sr=8-1&amp;keywords=forgotten+calculus

https://www.amazon.com/Pre-calculus-Demystified-Second-Rhonda-Huettenmueller/dp/0071778497/ref=sr_1_1?ie=UTF8&amp;qid=1500379354&amp;sr=8-1&amp;keywords=precalculus+demystified

https://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics-ebook/dp/B00CB2MK6C/ref=sr_1_3?ie=UTF8&amp;qid=1500379374&amp;sr=8-3&amp;keywords=calculus

https://www.amazon.com/Calculus-Idiots-Guides-Michael-Kelley-ebook/dp/B01E6H5C5A/ref=sr_1_2?ie=UTF8&amp;qid=1500379523&amp;sr=8-2&amp;keywords=calculus+idiots+guide

I know a few people who highly recommend How to Prove It by Velleman. I've never read it so I can't say for sure. The first book I used to learn mathematical logic was Lay's Analysis with an Intro to Proof. I can't recommend that book enough. The first quarter of the book or so is a pretty gentle introduction to mathematical logic, sets, functions, and proof techniques. I imagine it will get you where you need to be pretty quickly.

Haha, studying for the GRE, I know that now, but I was never aware of how important it was. The most topology I dealt with was in my complex analysis class and in my multivariable class where we used this book. That class initiated my masochistic addiction to math.

I say masochistic because I also studied biology to some depth. I was always rushing to catch up with one major or the other. So point-set topology probably got lost in crossfire of a laundry list of other classes I had to take. But I don't regret bio: I want to do applied math focusing on biological problems, i.e. dynamical systems, high dimensional networks, and other problems motivated by bioinformatics, computational biology, and biophysics. Ideally, I can get into Duke or UCLA's biomath programs; they seem pretty well established from the research I've done. However, they're definitely "reach" schools. Not putting all of my eggs in those baskets.

Linear Algebra (preferably proof based, theoretical) Introduction to Proofs (usually a perquisite for Linear Algebra) and Multivariable Calculus

Here’s a pdf of the textbook: http://alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

Note that this pdf does not cover differential forms but continue to read if you want to know more about the topic!

Now, if you want to learn more about differential forms, read chapter 8 of this textbook: https://www.amazon.com/Multivariable-Mathematics-Algebra-Calculus-Manifolds/dp/047152638X

A more economical way to learning differential form is to watch MATH 3510 videos on YouTube, particularly on differential forms. MATH 3500-3510 is a rigorous year sequence that covers many topics including differential geometry. These lectures go by the book I listed above. The prerequisite for this course is Calculus II but as a caveat, this course is not easy to self-study.

Gilbert Strang wrote one of the standard textbook in linear algebra and teaches out of it in his class on MIT OpenCourseware.

I preferred Shifrin's

Multivariable Mathematicsand there also videos of him teaching the class. But the books have different sensibilities and I thought one worked well as a back up and different perspective to the other.Plus, in Shifrin's text, multivariable calculus and linear algebra are treated at the same time, which made a lot of sense at the time. It makes a lot about the two subjects make more sense.

I haven't read it personally, but some agree on Quick Calculus being an approachable book for covering both the techniques and the concepts applied in calculus. The "why" behind the techniques often gets hidden away from non-maths majors, so this book supposedly works as a good self-supplement.

I hadn't taken a math class in over 5 years when I enrolled in Calc I. This book https://www.amazon.com/gp/aw/d/0471827223/ref=mp_s_a_1_1?ie=UTF8&amp;qid=1524052149&amp;sr=8-1&amp;pi=AC_SX236_SY340_FMwebp_QL65&amp;keywords=quick+calculus was a perfect precursor for the class if you're pressed for time.

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've heard that, while Spivak's

Calculusmay be difficult because of proofs, it is good. However, hisManifoldsis basically a graduate level reference book, and isn't the best multivariable calculus book for rebuilding/reteaching the basics of it. I've read that this is good in that regard.I'd also hope to find a book that goes into the physics side. I've heard this is good for that.

Have you heard anything on these? Have other suggestions?

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.

Would something like this be up your alley? Or would you prefer an analysis book? Edit: or of course something less rigourous.

It's really smart to be playing to your strengths: if you excel at language and writing, then read a book that

talksabout 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).

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

Parts will come back a LOT so you want to be familiar with that if at all possible. I think that is the technique I use most and unlike Trig Sub you cannot just use a table.

When I was struggling in the calc series I found the How To Ace Calculus books to be very helpful. They are good at translating the math into verbal explanations of concepts so I could connect the computations to a bigger picture. You might see if your library has them, if not, they are very cheap on Amazon. The 2nd one has about 25 pages on Series and then the rest is stuff that you see through Calc 2 and Calc 3.

What part of series is messing you up? Just a general foggy confusion or is there something more specific?

Forgotten Algebra and Forgotten Calculus have good reviews on Amazon. I've been meaning to get them myself.

Forgotten Algebra https://www.amazon.com/dp/1438001509/ref=cm_sw_r_cp_apa_i_SjJBCb4Z0CC0T

Forgotten Calculus https://www.amazon.com/dp/0764119982/ref=cm_sw_r_cp_apa_i_ClJBCbXZE0CYS

I never used these books but they are designed for people who already taken these classes and need a refresher.

Reviewing old material from previous math course is part of the struggle when learning higher levels of math. Reviewing them is part of the course. Also, use the calculator as much as you can. It may cut down on mistakes when doing the simplier math. Everyone have this issue. Its not the calculus per say that will mess you up but the simplier math especially when doing multistep problems. for example, there is a polynominal equation function and num solver on my ti 36x pro. I used to do it by hand or type it out the slow way in the calculator using parenthesis and division. It only hurt me in the long run. My calculator also have a fraction button which is something I use more often this semester. its faster, less accident prone, and It will show what I typed in without having to scroll. Become a calculator guru. Find out what calculator that the higher classes allow and read the manual and learn how to make the most out of it. At my college, it is the ti 36x pro.

I am have taken calc 1 to 3 and i am currently taking differential equations. I always have to go baxk and review my trig identites and integral and derivatives of trig functions. Algebra in calculus is something all student will have a hard time remember. So review will be essential. As bad as algebra maybe, Trig is far more easily and more common to forget that I find in myself and my classmates.

Find the zeroes mean where (points) does the line (function) go through the x and y axis.

http://www.amazon.com/Forgotten-Calculus-Barbara-Bleau-Ph-D/dp/0764119982

I wouldn't bother with Apostol's Calculus. For analysis, you should really look at the first two volumes of Stein and Shakarchi's Princeton Lectures in Analysis.

Vol I: Fourier Analysis

Vol II: Complex Analysis

Then, you should pick up:

Munkres, Analysis on Manifolds or something similar, you could try Spivak's book but it's a bit terse. (on a personal note, I tried doing Spivak's book when I was a freshman. It was a big mistake).

In truth, most introductory undergrad analysis texts are actually more invested in trying to teach you the rigorous language of modern analysis than in expositing on ideas and theorems of analysis. For example, Rudin's Principles is basically to acquaint you with the language of modern analysis -- it has no substantial mathematical result. This is where the Stein Shakarchi books really shines. The first book really goes into some actual mathematics (fourier analysis even on finite abelian groups and it even builds enough math to prove Dirichlet's famous theorem in Number Theory), assuming only Riemann Integration (the integration theory taught in Spivak).

For Algebra, I'd suggest you look into Artin's Algebra. This is truly a fantastic textbook by one of the great modern algebraic geometers (Artin was Grothendieck's student and he set up the foundations of etale cohomology).

This should hold you up till you become a sophomore. At that point, talk to someone in the math department.

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.

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!

You can find Michael Spivak's Calculus, which everyone tells me ought to be titled "Introduction to Analysis" on libgen.

Calculus

Spivak

http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098896

I've only used it briefly but Spivak's Calculus is pretty popular around here.

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."

You could try Principles of mathematical analysis by Rudin. This is too much for me, so be warned.

I find Spivak's Calculus to be a lot more palatable, but I've read less of it than Rudin.

If your Calculus is rusty before Rudin read Spivak Calculus it is great intro to analysis and you will get your calculus in order. Rudin is going to be overkill for you. Also before trying to do proofs read How to prove it It is a great crash course to naive set theory and proof strategies. And i promise i won't bore you with math any more.:D

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.

Spivak's Calculus is a great resource that I used for a real analysis class. The first exercise is something on par with proving that 1+1=2 and it goes on to build all of Calculus from there.