(Part 2) Top products from r/learnmath

>my first venture into proofs?

Have you had no prior experience with rigorous proofs, other than some elements of your linear algebra class? Not even something like a discrete math class? I'd worry that as an already-busy grad student, this might be biting off more than you can chew.

One additional question: is "grad analysis" a graduate-level class in analysis beyond an undergraduate-level class also offered at your school? I ask because typically, such a graduate-level class would assume considerable familiarity with undergrad-level analysis as a prerequisite. If you're in a situation where understanding the rigorous ε-δ definition of limit isn't something you've already internalized intuitively, then you'll likely find a grad-level introduction to something like measure theory to have a very steep learning curve.

---

I second /u/Gwinbar's recommendation above of Stephen Abbott's Understanding Analysis as a textbook for self-directed learning. But even that might be premature if you don't first develop sufficient background in the basics of set theory and mathematical logic. In particular, lots of concepts in analysis involve logical quantifiers, meaning that you'll need to be comfortable with both the meaning of a statement like

• For all ε>0, there exists a δ>0 such that if 0<|x-a|<δ, then |f(x)-L|<ε
  
  and how you would take the logical negation of the above statement. If none of this is familiar or transparently clear to you, then you might be better served by taking an undergraduate class in real analysis. Another option, of course, would be to audit a class, though that would be less advantageous in the context of buttressing your CV.
  
  ---
  
  I think the best advice I can give you at this point would be to talk to someone at your school. Someone in the economics department would have the best sense of how valuable having a graduate-level analysis class could be for your pursuit of a doctorate—as well as how damaging flaming out from such a class might be. I'd recommend talking to someone at your school's math department, too, since the best way to evaluate your background would be through a conversation by someone who's familiar with your school's analysis curriculum. They're in the best position to make the recommendation that best fits your current background level in mathematics, given what your school's academic standards are for such analysis classes. They can also provide final exams from past iterations of the undergrad- and grad-level analysis courses, respectively. That might give you some additional data to illuminate what such classes entail.
  
  I hope you can find more concrete information that's more custom-tailored to your specific circumstances. Good luck, whatever you decide!

> To be honest, I do still think that step 2 is a bit suspect. The inverse of [;AA;]is [;(AA)^{-1};] . Saying that it's [;A^{-1}A^{-1};] seems to be skipping over something.

I realized how right you are when you say this after I reread the chapter on Inverse Matrices in my book. I am using Introduction to Linear Algebra by Gilbert Strang btw. I'm following his course on MIT OCW.

The book saids: If [;A;] and [;B;] are invertible then so is [;AB;]. The inverse of a product [;AB;] is [;(AB)^{-1}=B^{-1}A^{-1};].

So, before I went through with step two, I would have to have proved that [;A;] is indeed invertible.

>Their proof is basically complete. You could add the step from A2B to (AA)B which is equivalent to A(AB) due to the associativity from matrix multiplication and then refer to the definition of invertibility to say that A(AB) = I means that AB is the inverse of A. So you can make it a bit more wordy (and perhaps more clear), but the basic ingredients are all there.

I will write up the new proof right here, in its entirety. Please let me know what you think and what I need to fix and/or add.

Theorem: if [;B;] is the inverse of [;A^2;], then [;AB;] is the inverse of A.

Proof: Assume [;B;] is the inverse of [;A^2;]

1. Since [;B;] is the inverse of [;A^2;], we can say that [;A^2B=I;]
   
   
2. We can write [;A^2B=I;] as [;(AA)B=I;]
   
   
3. We can rewrite [;(AA)B=I;] as [;A(AB)=I;] because of the associative property of matrix multiplication.
   
   
4. Therefore, by the definition of matrix invertibility, since [;A(AB)=I;], [;AB;]is indeed the inverse of [;A;].
   
   Q.E.D.
   
   Do I have to include anything about the proof being correct for a right-inverse and a left-inverse?
   
   > That's a great initiative! Probably means you're already ahead of the curve. Even if you get a step (arguably) wrong, you're still practicing with writing up proofs, which is good. Your write-up looks good to me, except for the questionability of step 2. In step 3 (and possibly others) you might also want to mention what you are doing exactly. You say "therefore", but it might be slightly clearer if you explicitly mention that you're using your assumption. You can also number everything (including the assumption), and then put "combining statement 0 and 2" to the right (where you can also go into a bit more detail: e.g. "using associativity of multiplication on statement 4").
   
   I haven't began my studies at university yet, but I sure am glad that I exposed myself to proofs before taking an actual discrete math class. I think that very few people get exposed to proof writing in the U.S. public school system. I've completed all of the Khan Academy math courses, and the MIT OCW Math for CS course is still very difficult. I basically want to develop a very strong foundation in proof writing, and all the core courses I will take as a CS major now, and then I will hopefully have an easier time with my schoolwork once I begin in the fall. Hopefully this prior knowledge will keep my GPA high too. I really appreciate all the constructive criticism about my proof. I will try to make them as detailed as possible from now on.
   

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:

1. Here's Linear Algebra described in the sequence above: I'll just leave it blank because I hate pointing fingers.
   
   
2. Here's a more serious intro to 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:
   
   
3. Here's non-serious Calculus described in the sequence above: I won't name names, but I assume a lot of people are familiar with these expensive door-stops from their freshman year.
   
   
4. Here's an intro to proper, rigorous Calculus:
   
   Calulus by Spivak
   
   3. Full-blown undergrad level Analysis(proof-based):
   
   Analysis by Rudin
   
   4. More advanced Calculus for advance undergrads and grad students:
   
   Advanced Calculus by Sternberg and Loomis
   
   The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
   
   Here's how you start studying real math NOW:
   
   Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
   
   Discrete Math by Susanna Epp
   
   How To prove It by Velleman
   
   Intro To Category Theory by Lawvere and Schnauel
   
   There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
   
   If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
   
   Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
   
   I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
   
   Good Luck, buddyroo.

Yes it is definitely possible! I was in high school 12 years ago and I failed grade 9 math 3 times and now I have a degree with a minor in math.

I started university late and so not only did I do pretty poorly in high school math (I didn't have a conceptual understanding of anything), but it had been about 5 or 6 years since I'd used any of it, in other words I was in way worse shape than you are now.

If you are interested in science as opposed to math in itself, then you will be taking a calculus course, a linear algebra course and a stats course in university most likely, and fortunately it is surprisingly easy to get caught up!

Over the course of one month in the summer I worked through a book like http://www.amazon.com/Algebra-Trigonometry-Transcendentals-Calculus-Edition/dp/0321671031 and made sure I understood the concepts (not just memorized the answers). Anytime I didn't understand something I would look it up online, ask someone for help, or watch a khan video. The most important thing is working hard to practice and understand pre-calculus: doing a lot of exercises, and asking yourself "why does this work", whenever you can.

Trust me, if I can go from failing grade 9 math three times to doing pretty well in courses like calculus and proof based classes like set theory, anyone can.

Since you're into programming and think very logically, I recommend as the stickied post on this sub 'How to Learn Math', specifically the sources from Art of Problem Solving AoPS.

If you don't mind spending a little then I think they're a great source for building up to pre-calculus and calculus. I started with their books and then build up enough confidence to read works from Serge Lang (like 'Basic Mathematics' and 'Geometry: a High School Course') and there's something so compelling in Lang's writing that make his reading enjoyable (mind you the basic math books from Lang and not the more advanced books that some people might consider a mess).

You can check also the free videos from AoPS on youtube Link, the way Richard Rusczyk teaches math makes me want to learn math everyday. Mind you, that although AoPS advertises itself for math competition students or gifted students, older students (including adults) can also benefit greatly from this.
Their approach is to let the student think and try to come up with a solution before teaching you the theory behind, I find their teaching ideal for people with interest in programming (like their problem solving books).

Since you mentioned pre-algebra using Khan Academy, if you're also interested in some examples from AoPS pre-algebra book just message me.
I know sometimes using Khan can also be confusing, I had those feelings at first too, before diversifying into more sources to learn from.

Edit: Also don't worry and cheer up! Whether you have dyscalculia or not, I don't think it changes the fact that you want to understand math and by asking here you're also proving that fact. When I was younger, dumber and more immature I also thought I had dyscalculia, no matter how I tried to understand and do math, I just couldn't do it. Eventually finding sources like AoPS, gave me the little push I needed, giving me another perspective and made me realize how terrible my fundamentals were. Don't be afraid to start from the most basics, also Barbara's Oakley book is great too link, math is a lot about practice, having your fundamentals well set and looking from multiple perspectives.

Awesome stuff! Let me volunteer my time; if you ever want to ask a question of someone (who you will soon outstrip in mathematical ability if not interest), PM me.

One thing that might help you is to go through a history of math. This way you're learning math the way the world has been learning math. It may give you a better understanding of why things are done the way they are. Mathematics for the Non-mathematician is something I'd like to work my way through (anyone interested in doing an online reading group this summer?) that may help you. Johnson/Mowry: Mathematics: A Practical Odyssey is a text I worked through that discussed some of the history and went in something resembling historical order (I had the third edition). As far as getting texts for self study goes, keep an eye out for older editions; you'll save a lot on a subject that doesn't change over time.

Good luck! I'm rooting for you!

wildberryskittles recommended the classics but teaching methods have improved since then in my opinion.

You should revisit algebra, geometry, and trigonometry before tackling a book like Calculus Made Easy. For algebra, Practical Algebra: A Self-Teaching Guide seems like a great place to start. After that, head on to geometry with something like Geometry and Trigonometry for Calculus. The book Precalculus Mathematics in a Nutshell might also be helpful.

You should spend some lovely evenings with my friend, Stewart. If you find my friend Stewart too hard on you, take some exercises from my little friend Thomas! And if you want even more fun, my friend Piskunov has some lovely exercises for you!

And ask your questions here :-)

How long ago did you do it? I work with calculus and statistics a lot and I often go back to earlier concepts to make sure my foundations are still strong.

I would recommend looking at this book and just quickly running through the exercises. That will give you a good idea about what you need to focus on. If you feel comfortable with those, something like this might be good to check out since it's made for self-teaching as opposed to being used in conjunction with a class.

Edited to add: math is like any language, in that the more you practice and manipulate numbers the better you'll be at it!

I only skimmed your post but I'd advise against beginning at arithmetics. It'll be boring as fuck, you'll mostly find material intended for children and you're probably gonna lose interest. Also there really isn't much to it.

One book that paints a bigger picture is this:

https://www.amazon.com/Mathematics-Nonmathematician-Morris-Kline/dp/0486248232/

Though old, it's a pretty interesting and well written book and it covers the basics of many topics. It has countless real applications of mathematics and even a lot of history. You can find it on Library Genesis but the physical copy is 8 bones right now so I'd just go for that tbh.

From there you might want to dive deeper into whatever topic interested you most, if that's Calculus you might want to get some kind of precalculus book and then "Calculus - A physical approach" which was written by Morris Kline as well. I personally really enjoy this guy's style, can recommend his stuff, but there are a lot of other good textbooks out there. Spivak and Rudin might be suitable alternatives.

Here are some books I'd recommend.

General Books

These are general books that are more focused on proving things per se. They'll use examples from basic set theory, geometry, and so on.

1. How to Prove It: A Structured Approach by Daniel Velleman
   
2. How to Solve It: A New Aspect of Mathematical Method by George Pólya
   
   Topical Books
   
   For learning topically, I'd suggest starting with a topic you're already familiar with or can become easily familiar with, and try to develop more rigor around it. For example, discrete math is a nice playground to learn about proving things because the topic is both deep and approachable by a beginning math student. Similarly, if you've taken AP or IB-level calculus then you'll get a lot of out a more rigorous treatment of calculus.
   
   

• An Invitation to Discrete Mathematics by Jiří Matoušek and Jaroslav Nešetřil
  
• Discrete Mathematics: Elementary and Beyond by László Lovász and Jaroslav Pelikan
  
• Proofs from THE BOOK by Martin Aigner and Günter Ziegler
  
• Calculus by Michael Spivak
  
  I have a special place in my hear for Spivak's Calculus, which I think is probably the best introduction out there to math-as-she-is-spoke. I used it for my first-year undergraduate calculus course and realized within the first week that the "math" I learned in high school — which I found tedious and rote — was not really math at all. The folks over at /r/calculusstudygroup are slowly working their way through it if you want to work alongside similarly motivated people.
  
  General Advice
  
  One way to get accustomed to "proof" is to go back to, say, your Algebra II course in high school. Let's take something I'm sure you've memorized inside and out like the quadratic formula. Can you prove it?
  
  I don't even mean derive it, necessarily. It's easy to check that the quadratic formula gives you two roots for the polynomial, but how do you know there aren't other roots? You're told that a quadratic polynomial has at most two distinct roots, a cubic polynomial has a most three, a quartic as most four, and perhaps even told that in general an n^(th) degree polynomial has at most n distinct roots.
  
  But how do you know? How do you know there's not a third root lurking out there somewhere?
  
  To answer this you'll have to develop a deeper understanding of what polynomials really are, how you can manipulate them, how different properties of polynomials are affected by those manipulations, and so on.
  
  Anyways, you can revisit pretty much any topic you want from high school and ask yourself, "But how do I really know?" That way rigor (and proofs) lie. :)

One thing I like to remind people, is that Linear Algebra is really cool and though it tends to come "after" calculus for some reason, it really has no explicit calc prerequisite.

I highly recommend Dr. Gilbert Strang's lectures on it, available on youtube and ocw.mit.edu (which has problems, solutions, etc, also)

I think it's a great topic for right around late HS, early college. And he stresses intuition and imo has the right balance of application and theory.

I'd also say that contrary to most peoples' perceptions, a student's understanding of a math topic will vary greatly depending on the teacher. And some teachers will be better for some students, others for others. That's just my opinion, but I firmly believe it. So if you find yourself struggling with a topic, find another teacher/resource and perhaps it will be more clear. Of course this shouldn't diminish the effort needed on your part, learning math isn't a passive activity, one really has to do problems and work with the material.

And finally, proofs are of course the backbone of mathematics. Here is an intro text I like on that.

Oh okay, one more thing, physics is a great companion to math. I highly recommend "Classical Mechanics" by Taylor, in that regard. It will be challenging right now, but it will provide some great accompaniment to what you'll learn in upcoming years.

Well there are a lot of useful links in this /r/math post (check the comments, too).

In addition to Khan Academy, there's MIT OCW, Paul's Math Notes, and PatrickJMT. There's also the Art of Problem Solving books.

But really, you don't need to watch calculus videos if you're going to take classes this summer. Your time might be better spent doing the exercises on Khan Academy to make sure there aren't gaps in your knowledge.

I highly recommend books by James Gleick, specifically Chaos, Genius, Isaac Newton, and The Information. Also, Polya's How to Solve It, GEB (join us in /r/geb!), and GH Hardy's A Mathematician's Apology. Here are some lists of popular math books.

You might find this collection of links on efficient study habits helpful.

> don't think that there is a logical progression to approaching mathematics

Well, this might be true of the field as a whole, but def not true when it comes to learning basic undergrad level math after calc 1, as the OP asked about. There are optimized paths to gaining mathematical maturity and sufficient background knowledge to read papers and more advanced texts.

> Go to the mathematics section of a library, yank any book off the shelf, and go to town.

I would definitely NOT do this, unless you have a lot of time to kill. I would, based on recommendations, pick good texts on linear algebra and differential equations and focus on those. I mean focus because it is easy in mathematics to gloss over difficulties.

My recommendation, since you are self-studying, is to pick up Gil Strang's linear algebra book (go for an older edition) and look up his video lectures on linear algebra. That's a solid place to start. I'd say that course could be done, with hard work, in a summer. For a differential equations book, I'm not exactly sure. I would seek out something with some solid applications in it, like maybe this: http://amzn.com/0387978941

That is more than a summer's worth of work.

Sorry, agelobear, to be such a contrarian.

You could consider starting with a book like Velleman's How to Prove It. It doesn't have to be that book, there are also free options online, but learning some logic and set theory from a book like that is a good way to figure out how to work with the other subjects you're working on.

Then, you could find a rigorous treatment of the subjects you want to learn. Something like Axler's Linear Algebra Done Right or Spivak's Calculus.

Learning math from textbooks like this is harder, but you end up with a better understanding of the math.

If you want to do statistics in a rigorous way you should start with calculus and linear algebra.

For calculus I recommend Paul's notes -> http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx
They are really clearly written with good examples and provide good intuition.
As supplement go through 3blue1borwn Essence of calculus. I think it's an excellent resource for providing the right intuition.

For linear algebra - linear algebra - Linear algebra done right as already recommended. Additionally, again 3blue1brown series on linear algebra are top notch addition for providing visual intuition and understanding for what is going on and what it's all about.

Finally, for statistics - I would recommend starting with probability calculus - that way you'll be able to do mathematical statistics and will have a solid understanding of what is going on. Mathematical statistics with applications is self-contained with probability calculus included. https://www.amazon.com/Mathematical-Statistics-Applications-Dennis-Wackerly/dp/0495110817

I'm 2 years into a part time physics degree, I'm in my 40s, dropped out of schooling earlier in life.

As I'm doing this for fun whilst I also have a full time job, I thought I would list what I'm did to supplement my study preparation.

I started working through these videos - Essence of Calculus as a start over the summer study whilst I had some down time. https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

Ive bought the following books in preparation for my journey and to start working through some of these during the summer prior to start

Elements of Style - A nice small cheap reference to improve my writing skills
https://www.amazon.co.uk/gp/product/020530902X/ref=oh_aui_detailpage_o02_s00?ie=UTF8&psc=1

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

Calculus: An Intuitive and Physical Approach
https://www.amazon.co.uk/gp/product/0486404536/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1

Trigonometry Essentials Practice Workbook
https://www.amazon.co.uk/gp/product/1477497781/ref=oh_aui_detailpage_o05_s00?ie=UTF8&psc=1

Systems of Equations: Substitution, Simultaneous, Cramer's Rule
https://www.amazon.co.uk/gp/product/1941691048/ref=oh_aui_detailpage_o05_s00?ie=UTF8&psc=1

Feynman's Tips on Physics
https://www.amazon.co.uk/gp/product/0465027970/ref=oh_aui_detailpage_o07_s00?ie=UTF8&psc=1

Exercises for the Feynman Lectures on Physics
https://www.amazon.co.uk/gp/product/0465060714/ref=oh_aui_detailpage_o08_s00?ie=UTF8&psc=1

Calculus for the Practical Man
https://www.amazon.co.uk/gp/product/1406756725/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1

The Feynman Lectures on Physics (all volumes)
https://www.amazon.co.uk/gp/product/0465024939/ref=oh_aui_detailpage_o09_s00?ie=UTF8&psc=1

I found PatrickJMT helpful, more so than Khan academy, not saying is better, just that you have to find the person and resource that best suits the way your brain works.

Now I'm deep in calculus and quantum mechanics, I would say the important things are:

Algebra - practice practice practice, get good, make it smooth.

Trig - again, practice practice practice.

Try not to learn by rote, try understand the why, play with things, draw triangles and get to know the unit circle well.

Good luck, it's going to cause frustrating moments, times of doubt, long nights and early mornings, confusion, sweat and tears, but power through, keep on trucking, and you will start to see that calculus and trig are some of the most beautiful things in the world.

If you're currently at the pre-calc level, you could probably get away with learning from khan academy for a little while. After that (and building some familiarity with proof writing), you'd be ready for some of the pure math classes like abstract algebra and real analysis. For those courses, you'll probably want to check out some Open Courseware. You'd want to treat it like a real class; watch the lectures online and read from the textbooks, while working on problem sets.

While you're working your way through the khan academy stuff, you might want to check out Stewart's calculus book; it's pretty solid for making your way through the calculus sequence.
I'd ask around for a good linear algebra book, since I haven't encountered a decent one that's at that level.

A lot of people recommend Khan Academy, but you cannot really learn from the Khan Academy; there is just too much material to cover. I recommend either going into an algebra class at your local community college, and/or get some good algebra/maths books. This one gets a lot of praise on Amazon.com:

http://www.amazon.com/Practical-Algebra-Self-Teaching-Guide-Second/dp/0471530123/ref=sr_1_fkmr0_1?ie=UTF8&qid=1288684060&sr=8-1-fkmr0

and, this one is the one nobel laureate Richard Feynman taught himself with:

http://www.amazon.com/Algebra-practical-Mathematics-self-study/dp/B0007DZPT6

Well, if you want something light and accessible and suitable for the layperson, I'm quite fond of Jan Gullberg's Mathematics from the Birth of Numbers. It goes over basically everything you would typically learn in primary and secondary school, and it presents everything with historical background. But it doesn't go into tremendous detail on each topic, and it doesn't provide the most rigorous development. It's more of a high-level overview.

But if you really want to learn some mathematics, on a deep and serious level, be prepared to read and study a lot. It's a rewarding journey, and we can give you book recommendations for specific topics, but it does take a lot of discipline and a lot of time. If you want to go that route, I would recommend starting with an intro to proofs book. I like Peter J. Eccles's An Introduction to Mathematical Reasoning, but there are many other popular books along the same line. And you can supplement it with a book on the history of mathematics (or just read Gullberg alongside the more serious texts).

Mathematical Proofs: A Transition to Advanced Mathematics was the book for my college proofs class. I found it to be a good resource and easy to follow. It covers some introductory set theory as well. Just be prepared to work through the proof exercises if you really want a good intuition on the topic.

Your question is pretty vague because studying "mathematics" could mean a lot of things. And yes, your observation is correct: "There are a lot of Mathematical problems which are extremely difficult". In fact, that's true for a lot of people as well. So I suggest that you choose a certain field and delve into that.

For proof based subjects, the most basic to start with is Real Analysis. I recommend Stephen Abbott's Understanding Analysis as it is a pretty well-explained book.

Man, I feel like I'm pimping this book all over the place lately, but seriously everyone struggling in math (or science) should check out "A Mind For Numbers" by Barbara Oakley. She addresses the common reasons people fail at math and also teaches how to use your diffuse and focused modes of thinking (very similar, imo, to the comment /u/The_White_Baron made about math requiring both creative and critical thinking). Diffuse mode thinking is where your brain takes the ideas you've taken in during focused mode and makes the intuitive connections to other topics and areas of knowledge, setting up more diverse connections to that concept, which makes retrieval easier. I implemented the ideas presented in the book in my statistics course and went from a 79% (just below class average) on the first exam to a 97.5% (highest in the class) on the next exam. It is absolutely one of the most valuable books I've read in years.

​

Link for anyone who is interested: https://www.amazon.com/Mind-Numbers-Science-Flunked-Algebra-ebook/dp/B00G3L19ZU

Parametric and vector functions will be very useful to have a basic grasp of. If you're in engineering, you'll be doing a lot of work with vector calculus in that class, and any familiarity you can get with it will help you immensely.

Get a copy of Div, Grad, Curl and All That. http://www.amazon.com/Div-Grad-Curl-All-That/dp/0393925161

It's a great book to read along with your textbook. Wouldn't hurt to start reading it now, actually, while you have the time to go back and refresh yourself on any topics the book assumes knowledge on that are rusty for you.

Do you understand what unknown variables are and why you are solving for them? Do you know why you are asked to move variables from one side to another?

Regarding problem solving...

If you are dead serious and really want to learn problem solving as a general skill, and are willing to read something that has a few examples a bit over your head but is extremely helpful in general, then may I suggest George Polya's How to Solve It. It is written at probably a high school geometry level but many of his discussions are generic enough that they should give you some insight into the problem solving process.

Essentially Polya wrote a book (maybe the book) on problem solving patterns i.e. when faced with a problem ask this set of questions and try strategy A or B, etc. He has I think 12 core questions to always ask. I found it very helpful myself. The first third or so of the book is a narrative of him showing how an ideal teacher would apply his teaching methods to guide students to discover concepts on their own.

A PDF of his original 1945 edition is available here: https://notendur.hi.is/hei2/teaching/Polya_HowToSolveIt.pdf

But a new edition paperback is on Amazon for $14, I have it and have made tons of pencil notes in the margins.

BTW if you do try to read it, you only need to know a few things to have the first part make sense. A "rectangular parallelipiped" (horrible name) is just a rectangular prism, so imagine it is your classroom's four walls floor and ceiling, etc. If you know how to find the diagonal length of a square or rectangle (the length of a line between two opposite corners) you probably know enough to basically follow along since that is the core of his example. If not, here's the trick, just divide the square (or rectangle) into two triangles and apply the pythagorean theorem. A huge part of his problem solving method revolves around asking yourself if you know of a similar problem with a similar pattern that you can adapt to solve your current problem. It's like being asked to find the area of a half circle, you don't know the formula, but you know the formula for the area of a circle, so you can use that as a base and adapt it to the problem of the half circle.

BTW 2: Math is hard. For everybody. People who are good at math paid for it in blood sweat and tears.

I learned from Wackerly which is decent, though I think Devore's presentation is better, but not as deep. Both have plenty of exercises to work with.

Casella and Berger is the modern classic, which is pretty much standard in most graduate stats programs, and I've heard good things about Stat Labs, which uses hands-on projects to illuminate the topics.

To piggy back off of danielsmw's answer...

> Fourier analysis is used in pretty much every single branch of physics ever, seriously.

I would phrase this as, "partial differential equations (PDE) are used in pretty much every single branch of physics," and Fourier analysis helps solve and analyze PDEs. For instance, it explains how the heat equation works by damping higher frequencies more quickly than the lower frequencies in the temperature profile. In fact Fourier invented his techniques for exactly this reason. It also explains the uncertainty principle in quantum mechanics. I would say that the subject is most developed in this area (but maybe that's because I know most about this area). Any basic PDE book will describe how to use Fourier analysis to solve linear constant coefficient problems on the real line or an interval. In fact many calculus textbooks have a chapter on this topic. Or you could Google "fourier analysis PDE". An undergraduate level PDE course may use Strauss' textbook whereas for an introductory graduate course I used Folland's book which covers Sobolev spaces.

If you wanted to study Fourier analysis without applying it to PDEs, I would suggest Stein and Shakarchi or Grafakos' two volume set. Stein's book is approachable, though you may want to read his real analysis text simultaneously. The second book is more heavy-duty. Stein shows a lot of the connections to complex analysis, i.e. the Paley-Wiener theorems.

A field not covered by danielsmw is that of electrical engineering/signal processing. Whereas in PDEs we're attempting to solve an equation using Fourier analysis, here the focus is on modifying a signal. Think about the equalizer on a stereo. How does your computer take the stream of numbers representing the sound and remove or dampen high frequencies? Digital signal processing tells us how to decompose the sound using Fourier analysis, modify the frequencies and re-synthesize the result. These techniques can be applied to images or, with a change of perspective, can be used in data analysis. We're on a computer so we want to do things quickly which leads to the Fast Fourier Transform. You can understand this topic without knowing any calculus/analysis but simply through linear algebra. You can find an approachable treatment in Strang's textbook.

If you know some abstract algebra, topology and analysis, you can study Pontryagin duality as danielsmw notes. Sometimes this field is called abstract harmonic analysis, where the word abstract means we're no longer discussing the real line or an interval but any locally compact abelian group. An introductory reference here would be Katznelson. If you drop the word abelian, this leads to representation theory. To understand this, you really need to learn your abstract/linear algebra.

Random links which may spark your interest:

• Fourier analysis also finds applications in number theory, for example in the Hardy-Littlewood circle method.
  
• Solving cubic equation using discrete Fourier transform
  
• The Wavelet transform
  
• Fourier optics
  
• Monstrous moonshine
  
• Fractional Laplacian/calculus

Try to find entry points that interest you personally, and from there the next steps will be natural. Most books that get into the nitty-gritty assume you're in school for it and not directly motivated, at least up to early university level, so this is harder than it should be. But a few suggestions aimed at the self-motivated: Lockhart Measurement, Gelfand Algebra, 3blue1brown's videos, Calculus Made Easy, Courant & Robbins What Is Mathematics?. (I guess the last one's a bit tougher to get into.)

For physics, Thinking Physics seems great, based on the first quarter or so (as far as I've read).

>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.

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.

http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?s=books&ie=UTF8&qid=1405668438&sr=1-1&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.

I've heard (more than once) that you go to Calculus to finally fail Algebra or Trig. The latter is proving true for me. I've got a copy of this book (my calc class is taught out of Early Transcendentals ... so it pairs up perfectly - check for something similar for your text if you can find it) and it's a good resource for when I can't remember how to do something.

Aside from that, figure out where your weaknesses are. Don't spend valuable time further working on stuff you're already rock solid on - target weak areas and work on them.

Khan Academy, Professor Leonard are both great resources. Wolfram Alpha subscriptions (only a few bucks a month) also has a button that shows step-by-step how they get to the answer, which is frequently useful to me.




Edit: OH! Also, don't go it alone. You've obviously found this place, which is great. Your school probably has a math tutoring center, or math lab, or your professor's office hours, or a TA you can bother or whatever. Those things are all "free" - but in reality they're not. They're included in the price of your tuition. You've already paid for them: USE EM!

I found Calculus from Larson and Edwards pretty good.

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&cursor=1

I personally think you should brush up on frequentist statistics as well as linear models before heading to Bayesian Statistics. A list of recommendations directed at your background:

• For Frequentist Statistics and Probability: Introduction to Probability, Mathematical Statistics With Applications or since you are more interested in applications of Stats to Machine Learning, All of Statistics.
  
  
• For Bayesian Statistics, I would start with Statistical Rethinking and Introduction to Bayesian Statistics for an intro.
  
  

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.

For vector calculus, you might enjoy the less formal British text Div, Grad, Curl, and All That by H. M. Schey; for group theory in brief, consider the free textbook Elements of Abstract and Linear Algebra by Edwin H. Connell.

Alternatives to Schey's book include the much more formal Calculus on Manifolds by Michael Spivak, which does have more exercises than Schey but uses most of them to develop the theory, rather than as the mindless drills that fill an ordinary textbook; Michael E. Corral's free textbook Vector Calculus isn't huge but is written closer to an ordinary textbook.

My professor wrote this this book. It is excellent if you already have a memory of PDE's. It is also inexpensive.

So I keep pointing students to this one book, and they keep telling me it saved their asses. So I'll point you to it too: Just in Time Algebra and Trig for Early Transcendentals Calculus

It has everything you need, while remaining short and inexpensive. I know of no better preparatory book for calculus, and that includes Stewart, Sullivan, and Blitzer-- all the big names.

I used this book for my PDE class. It's meant to be supplemented with notes, but it does a decent job explaining material on it's own. That and it's super cheap for a math book.

Note that it has several errors in the answer key.

https://www.amazon.com/Calculus-James-Stewart/dp/1285740629/ref=sr_1_2?ie=UTF8&qid=1466199140&sr=8-2&keywords=james+stewart+calculus+8th+edition

It covers Calculus 1-3. I have a downloaded version from a friend of the whole book.

I find Gilbert Strang's Introduction to Linear Algebra quite accessible, and seems to be aimed towards the practical (numerical) side of things. His video lectures are also quite good, IMHO.

Probability and Random Processes by Grimmett is a good introduction to probability.

Mathematical Statistics by Wackerly is a comprehensive introduction to basic statistics.

Probability and Statistical Inference by Nitis goes into the statistical theory from heavier probability background.

The first two are fairly basic and the last is more involved but probably contains very few applied techniques.

It's useful for Electromagnetic physics. Surface integrals are used for finding the flux through a Gaussian surfaces so you can use Gauss' Law on non-symmetrical surfaces. Line integrals are used with Ampere's Law to find the magnetic flux. Once you learn the mechanics of working with multivariable calculus, you should read "Div, Grad, Curl and All That"

I'm pretty sure everyone struggles with that stuff to some degree. I know I did. Time and practice got me past it.

I've been reading A Mind For Numbers and it has some solid advice on how to study math. Nothing earth-shattering (focus for a while, take a break; don't procrastinate; be active in learning, etc), and nothing everyone here is telling you, but it's conveniently packaged.

How to Read and Do Proofs by Daniel Solow this book saved my life in Abstract Algebra.

I can't really give a better testimonial other than I read through this book and applied a couple of the concepts and did very well in the course.

One thing to remember, you can always reverse your steps, if you are stuck at some point, then work backwards from the end and you can sometimes meet up to the point you were stuck at.

Also, How to Prove It by Daniel J Velleman is another classic book that can help.

This is what my school requires of us. https://www.amazon.com/gp/aw/d/1285740629/ref=mp_s_a_1_1?ie=UTF8&qid=1501294155&sr=8-1&pi=AC_SX236_SY340_FMwebp_QL65&keywords=calculus+stewart+8th+edition

Don't forget the $130 Web assign!

A lot of early math tends to come down to how often you do problems, and computation classes can generally be seen as rote learning. I'd suggest you start doing some proofs, they force you to understand what you are doing, rather then just doing what you've seen. Pick up http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521446635

or, http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?s=books&ie=UTF8&qid=1345011596&sr=1-1&keywords=spivaks+calculus

Not knowing random operations as you listed is fine, with time you will get quicker, but don't worry if you need to consider it for a moment.

Axler's Linear Algebra Done Right is something you might enjoy looking at; since his basic point of view is that linear algebra is generally done wrong.

http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/0387982582

Also endorse this book as a primer on mathematical thinking. No background necessary: http://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069111966X

"What is Mathematics ? An elementary approach to Ideas and methods ( 2nd Edition) by Richard Courant (Author), Herbert Robbins (Author), Ian Stewart (Editor) "

+1 for Professor Leonard on YouTube - Slow and Steady pace, lots of detail, lots of repetition.

Also, buy this book and implement what it teaches you: https://www.amazon.co.uk/Mind-Numbers-Science-Flunked-Algebra-ebook/dp/B00G3L19ZU

They also have a Coursera course: https://www.coursera.org/learn/learning-how-to-learn

There is actually a book called An Introduction to Mathematical Reasoning.

Understanding Analysis by Stephen Abbott https://www.amazon.com/dp/1493927116/

Topics in Algebra by I.N. Herstein https://www.goodreads.com/book/show/1264762.Topics_in_Algebra

The Feynman lectures on physics http://www.feynmanlectures.caltech.edu/

I've got nothing for Economics, but the above would be my personal recommendations for self-study and just general reading.

https://www.amazon.com/Calculus-James-Stewart/dp/1285740629/ref=sr_1_1?ie=UTF8&qid=1484794699&sr=8-1&keywords=calculus+8th+edition

For history:https://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg/dp/039304002X/ref=sr_1_1?s=books&ie=UTF8&qid=1538960578&sr=1-1

This book is decent: https://www.amazon.com/Mathematics-Nonmathematician-Dover-Books/dp/0486248232

Stewart, Larson, Rogawski

Try picking up a book. I recommend this one. You can also use Rudin but it will be more difficult.

If you are using notes and online research, it may be that the exercises you've been working on are coming from many different areas and aren't really focused on one topic in particular. This may be the reason that every problem seems to require a new trick.

While it's certainly not the best or broadest advice, I've always found that, whenever a problem starts to get excessively complicated, the mean value theorem always seems to be the why-didn't-I-think-of-that trick that solves it.