Reddit mentions: The best calculus books

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.

1. Mathematics: A very Short Introduction : A very good book, but also very short book about mathematics by Timothy Gowers, a Field medalist and overall awesome guy, gives you a feelling for what math is all about.
   
   
2. Concepts of Modern Mathematics: A really interesting book by Ian Stewart, it has more topics than the last book, it is also bigger though less formal than Gower's book. A gem.
   
   
3. What is Mathematics?: A classic that has aged well, it's more textbook like compared to the others, which is good because the best way to learn mathematics is by doing it. Read it.
   
   
4. An Infinitely Large Napkin: This is the most modern book in this list, it delves into a huge number of areas in mathematics and I don't think it should be read as a standalone, rather it should guide you through your studies.
   
   
5. The Princeton Companion to Mathematics: A humongous book detailing many areas of mathematics, its history and some interesting essays. Another book that should be read through your life.
   
   
6. Mathematical Discussions: Gowers taking a look at many interesting points along some mathematical fields.
   
   
7. Technion Linear Algebra Course - The first 14 lectures: Gets you wet in a few branches of maths.
   
   Linear Algebra: An extremelly versatile branch of Mathematics that can be applied to almost anything, also the first "real math" class in most universities.
   
   
8. Linear Algebra Done Right: A pretty nice book to learn from, not as computational heavy as other Linear Algebra texts.
   
   
9. Linear Algebra: A book with a rather different approach compared to LADR, if you have time it would be interesting to use both. Also it delves into more topics than LADR.
   
   
10. Calculus Vol II : Apostols' beautiful book, deals with a lot of lin algebra and complements the other 2 books by having many exercises. Also it doubles as a advanced calculus book.
    
    
11. Khan Academy: Has a nice beginning LinAlg course.
    
    
12. Technion Linear Algebra Course: A really good linear algebra course, teaches it in a marvelous mathy way, instead of the engineering-driven things you find online.
    
    
13. 3Blue1Brown's Essence of Linear Algebra: Extra material, useful to get more intuition, beautifully done.
    
    Calculus: The first mathematics course in most Colleges, deals with how functions change and has many applications, besides it's a doorway to Analysis.
    
    
14. Calculus: Tom Apostol's Calculus is a rigor-heavy book with an unorthodox order of topics and many exercises, so it is a baptism by fire. Really worth it if you have the time and energy to finish. It covers single variable and some multi-variable.
    
    
15. Calculus: Spivak's Calculus is also rigor-heavy by Calculus books standards, also worth it.
    
    
16. Calculus Vol II : Apostols' beautiful book, deals with many topics, finishing up the multivariable part, teaching a bunch of linalg and adding probability to the mix in the end.
    
    
17. MIT OCW: Many good lectures, including one course on single variable and another in multivariable calculus.
    
    Real Analysis: More formalized calculus and math in general, one of the building blocks of modern mathematics.
    
    
18. Principle of Mathematical Analysis: Rudin's classic, still used by many. Has pretty much everything you will need to dive in.
    
    
19. Analysis I and Analysis II: Two marvelous books by Terence Tao, more problem-solving oriented.
    
    
20. Harvey Mudd's Analysis lectures: Some of the few lectures on Real Analysis you can find online.
    
    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.
    
    
21. Abstract Algebra: Dummit and Foote's book, recommended by many and used in lots of courses, is pretty much an encyclopedia, containing many facts and theorems about structures.
    
    
22. Harvard's Abstract Algebra Course: A great course on Abstract Algebra that uses D&F as its textbook, really worth your time.
    
    
23. Algebra: Chapter 0: I haven't used this book yet, though from what I gathered it is both a category theory book and an Algebra book, or rather it is a very different way of teaching Algebra. Many say it's worth it, others (half-jokingly I guess?) accuse it of being abstract nonsense. Probably better used after learning from the D&F and Harvard's course.
    
    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.

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.

Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:

 




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.

 




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

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

1. What's your current academic level? (Assuming, of course, you're still a student, rather than trying to learn mathematical proofs as an autodidact.)
   
   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 just beyond 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.
   
   
2. What's your current mathematical background?
   
   This may be especially true for things like logic and very elementary set theory.
   
   
3. What sort of access do you have to "formal" mathematical resources like textbooks, online materials, etc.?
   
   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.)
   
   
4. What resources are available to you for vetting your work?
   
   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 magnitude more effective when you have someone who can guide you.
   
   
5. Are you trying to learn the basics of mathematical proofs, or genuinely rigorous mathematical 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?
   
   
6. What experience have you already had with proofs in particular?
   
   Have you had at least, for example, a geometry class that's proof-based?
   
   
7. How would you characterize your general writing ability?
   
   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.
   
   

• The book How to Prove It: A Structured Approach by Daniel Velleman is a well-respected general introduction to ideas behind mathematical proof, as is How to Solve It: A New Aspect of Mathematical Method by George Pólya.
  
  
• Since you've already taken calculus, it would be worth reviewing the topic using a more abstract, proof-centric text like Calculus by Michael Spivak. This is a challenging textbook, but there's a reason people have been recommending its different editions over many decades.
  
  
• In order to learn how to write mathematically sound proofs, it helps to read as 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.
  
  
• It's like the old joke about how to get to Carnegie Hall: practice, practice, practice.
  
  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!

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 liberally
for 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 really need it for the above. If someone has
good 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.

****

> 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.

Math is going to be THE most important thing for him to conquer, and also probably the hardest after so many years. It's totally possible to do on your own, though, and I recommend brushing up before enrolling in classes. You could also take remedial math at a community college if he's the classroom type. Either way, the resources below should help.

I would highly recommend using this to go along with a calc textbook and/or class:


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


It helped me get an A in calc I and those fundamentals also helped me do well later on in Calc II. It explains problems in plain language, there are no confusing jumps and "what just happened" situations to overcome. My seat mate in Calc I liked the book so much that he asked me to use my Prime and order it off of amazon for him. It improved him from struggling to pass to at least +1 or +2 in letter grades.


First, though, he most absolutely known his algebra backwards and forwards and inside out. Calculus is all algebra. People don't fail calculus because of calculus, they fail it because their algebra sucks. This being the case, you can review all levels of algebra for free here:


http://www.wtamu.edu/academic/anns/mps/math/mathlab/

I'm not graduated yet, but I wouldn't care about the age of someone. In my experience engineers tend to be very practical minded. So if he's a good engineer, who cares what his age is? Good problem solving ability is the most important thing. I'm a nontrad myself, in my twenties. I think engineering draws a lot of second degree and non trad students, because it's so practical and a good way to support yourself.


Finance wise, 30K is something that should be easily repayable with an engineering degree. Our internships usually are in the $15-25 range during school, and after it starting salaries are in the 50-60K/year range. All of this depends a lot on talent and geographical location and job market and so on, but really, engineering's one of the best degrees to invest in. I have heard that civil engineering is a bit depressed right now, and that EE is probably doing the best. So it might be good to research if a particular engineering discipline might be not just more lucrative, but more interesting. (Does he have any interest in wiring, electricity, robots, computers, etc?)

http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918/ref=sr_1_1?ie=UTF8&qid=1342068971&sr=8-1&keywords=spivak%27s+calculus

This book starts with basic properties of numbers (associativity, commutativity, etc), then moves onto some proof concepts followed by a very good foundation (functions, vectors, polar coordinate). Be forewarned that the content is VERY challenging in this book, and will definitely require a determined effort, but it will certainly be good if you can get through it.

A more gentle introduction to Calculus is http://www.amazon.com/Thomas-Calculus-12th-George-B/dp/0321587995/ref=sr_1_1?s=books&ie=UTF8&qid=1342069166&sr=1-1&keywords=thomas%27+calculus and it is a much easier book, but you don't prove much in this one. Both of these can likely be found online for free. Also, if you want to get a decent understanding I recommend, http://www.amazon.com/How-Prove-Structured-Daniel-Velleman/dp/0521675995/ref=sr_1_1?s=books&ie=UTF8&qid=1342069253&sr=1-1&keywords=how+to+prove+it or http://www.people.vcu.edu/~rhammack/BookOfProof/index.html the latter is definitely free.

You may also need a more introductory text for trig and functions. I can't find the book my school used for precalc, hopefully someone else can offer a good recommendation.

Also, getting a dummies book to read alongside was pretty helpful for me, and Paul's online notes(website) is very nice.

There are a lot of good classics on /u/thebenson's list. I want to highlight the books that are good for what you'll be learning, and give you a sense of how the sequence works. And I'll add a few.


Calculus books: Thomas' Calculus, Calculus by James Stewart (not multivariable), and this cheap easy read by Morris Kline.

Have you learned calculus in the past? It sounds like you'll need it for at least one of those courses, but either way, it will definitely help you conceptually for the others. You should really try to get solid on this before you need to use it.



Intro physics books: Fundamentals of Physics (Halliday & Resnick), Physics for Scientists and Engineers (Serway & Jewett), Physics for Scientists and Engineers (Tipler & Mosca), University Physics (Young), and Physics for Scientists and Engineers (Knight) are all good. Gee, they get really unoriginal with the names, huh?

Each of these books assumes no background in physics, but you do need to use calculus. If you're going to take a class in basic mechanics that doesn't involve any calculus, you may find it more useful to get a book at that level. The only such book that I'm familiar with is Physics: Principles with Applications by Giancoli. I know there are many others, but I can't speak for them.



Mathematical methods: Greenberg is way more than you need here. I think you would find Engineering Mathematics by Stroud & Booth more useful as a reference, since it covers a lot of the less advanced stuff that you may need a refresher on.



Sequence: it's typical to start learning physics by learning about Newtonian mechanics, with or without calculus. After that, one often goes on to thermodynamics or to electricity and magnetism. It sounds like this is roughly how your program is going to work.

If you are learning mechanics with calculus, you can expect E&M to be even heavier on the calculus and thermodynamics to be less so. More calculus is not a bad thing. People often get scared of it, but it actually makes things easier to understand.

It is very typical that you will use only one book (from the intro books above) for all of these topics. You shouldn't need to get any books on specific topics.

**

The other books on /u/thebenson's list are all great textbooks, but I think you should avoid them for now. They generally assume a healthy background in basic physics, and they may not be very relevant to the physics you'll be studying.

But I do want to give some mention to Spacetime Physics* by Taylor and Wheeler, since I don't want to imply that this is a background-heavy book. On the contrary, this is one of the most beginner-friendly physics books ever written, and it is my favorite introduction to special relativity. Special relativity is probably not something you need to learn about right now, but if you have any interest, I seriously recommend finding an old used copy of this book—it's a fun read aside from any other uses!

I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.

Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.

This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:

• theoretical computer science (essentially a math text)
  
  
• set theory
  
  
• linear algebra
  
  
• algebra
  
  
• predicate calculus
  
  Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!
  
  (edit: can't count and thought five was four)

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 lower than 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 really knowing what it takes. Do you want to be that 1? If you don't want to do homework, fine; but you need to be doing work that allows you to reach your dreams.

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:

• http://kl1p.com/studytips a sort of summary, lots of the below info is covered, but not everything.
  
• Study less study smart overview -- this guy wrote out notes for
  
• http://www.lbcc.edu/LAR/studyskills.cfm (the videos) -- This guy is awesome. It's a full course on how to study. When I get a job after grad I'm definitely getting him a gift.
  
• http://altair.pw/pub/lib/How%20to%20Become%20a%20Straight-A%20Student.pdf
  
• http://www.cse.buffalo.edu/~rapaport/howtostudy.html -- note: has some repeated information from the first 2 links
  
• https://tomato-timer.com -- makes using the pomodoro technique a bit easier.
  
• http://khanacademy.org
  
• http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx -- apparently this guy's got really good, concise, notes. Also applies for MATH 152 and 153.
  
• How to Ace Calculus: The Streetwise Guide (recommended by the math profs, and I've seen tons of people say it helped a lot. It doesn't go in-depth, but gives you a simple overview of the concepts so you get a good idea of how and why they work)
  
  
  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)

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!

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.

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 means is 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 is alright, 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.

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 Astrophysics by 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.

The foundation of probability is measure theory, not nonstandard analysis (the topic that includes the hyperreals). So, when dealing with statements about probability, we deal with probability measures, which assign numbers in the real interval [0, 1] to subsets of the space of possible events. (Perhaps someone has studied a variant of measure theory that substitutes the hyperreals for the reals, but if so, it's sufficiently obscure that I've never heard of it.)

Also, nowhere in all this is anyone "raising a real number to the power of infinity". There are formal statements of the following sort:

>(*) The limit of 1/2^N as N approaches infinity is zero.

However, this is a statement about the limit of a sequence of real numbers, which is most definitely a real number, and is also formally defined in a way that makes no reference to "infinity". The expression "as N approaches infinity" is just a mildly informal (but much more readable) way of expressing that formal definition.

If you care to parse the formal statement, here it is:

>For any real number ε > 0, there exists a natural number N such that for all natural numbers n ≥ N, we have |1/2^(N) - 0| < ε.

This is how we precisely formalize statement (*).

For more information on limits of sequences, I recommend reading a book on mathematical analysis. Spivak's Calculus has a good chapter on this; it's an excellent book, so it's worth reading anyway.

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&qid=1331675661&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.

For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.

Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:

Discrete Mathematics with Applications by Susanna Epp

Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Numbers and Proofs by Allenby

Mathematics: A Discrete Introduction by Edward Scheinerman

How to Prove It: A Structured Approach by Daniel Velleman

Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi

Some special topics(elementary treatment):

Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby

A Friendly Introduction to Number Theory Joseph Silverman

Elements of Number Theory by John Stillwell

A Primer in Combinatorics by Kheyfits

Counting by Khee Meng Koh

Combinatorics: A Guided Tour by David Mazur


Just a nice bunch of related books great to have read:

generatingfunctionology by Herbert Wilf

The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule

A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger

If you wanna do graphics stuff, you wanna do some applied Linear Algebra:

Linear Algebra by Allenby

Linear Algebra Through Geometry by Thomas Banchoff, John Wermer

Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman

Best of Luck.

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 :-)

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:

1. Understanding Analysis by Steve Abbot
   
   
2. Yet Another Introduction to Analysis by Victor Bryant
   
   
3. Elementary Analysis: The Theory of Calculus by Kenneth Ross
   
   
4. Real Mathematical Analysis by Charles Pugh
   
   
5. A Primer of Real Functions by Ralph Boas
   
   
6. A Radical Approach to Real Analysis by David Bressoud
   
   
7. The Way of Analysis by Robert Strichartz
   
   
8. Foundations of Analysis by Edmund Landau
   
   
9. A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamzi
   
   
10. Calculus by Spivak
    
    
11. Real Analysis: A Constructive Approach by Mark Bridger
    
    
12. Differential and Integral Calculus by Richard Courant, Edward McShane, Sam Sloan and Marvin Greenberg
    
    
13. You can find tons more if you search the internet. There are more superstars of advanced Calculus like Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by Tom Apostol, Advanced Calculus by Shlomo Sternberg and Lynn Loomis... there are also more down to earth titles like Limits, Limits Everywhere:The Tools of Mathematical Analysis by david Appelbaum, Analysis: A Gateway to Understanding Mathematics by Sean Dineen...I just dont have time to list them all.
    
    Some user friendly books on Linear/Abstract Algebra:
    
    
14. A Book of Abstract Algebra by Charles Pinter
    
    
15. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl Meyer
    
    
16. Groups and Their Graphs by Israel Grossman and Wilhelm Magnus
    
    
17. Linear Algebra Done Wrong by Sergei Treil-FREE
    
    
18. Elements of Algebra: Geometry, Numbers, Equations by John Stilwell
    
    Topology(even high school students can manage the first two titles):
    
    
19. Intuitive Topology by V.V. Prasolov
    
    
20. First Concepts of Topology by William G. Chinn, N. E. Steenrod and George H. Buehler
    
    
21. Topology Without Tears by Sydney Morris- FREE
    
    
22. Elementary Topology by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and and V. M. Kharlamov
    
    Some transitional books:
    
    
23. Tools of the Trade by Paul Sally
    
    
24. A Concise Introduction to Pure Mathematics by Martin Liebeck
    
    
25. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston
    
    
26. Introductory Mathematics: Algebra and Analysis by Geoffrey Smith
    
    
27. Elements of Logic via Numbers and Sets by D.L Johnson
    
    Plus many more- just scour your local library and the internet.
    
    Good Luck, Dude/Dudette.

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!

It's really smart to be playing to your strengths: if you excel at language and writing, then read a book that talks about math in more detail. Textbooks are good for problems and for reference, but I find them very hard to read. They use equations where they should be using words.

Go to your local library, and look in the math section until you find something interesting. I found this book when I was struggling with calculus: How to Ace Calculus: The Streetwise guide. It was smart, funny, and really explained topics in ways I could relate to.

That's the kind of thing I would look for if I were you. Good luck! I hope you see post in all the ~430 comments!

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.

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.

I'm a physicist/mathematician, but I think it could be useful for you. Exterior algebra (differential forms) in particular is worth learning because it makes the theory of multivariable calculus much more elegant and simple. With exterior algebra you can see that the fundamental theorem of calculus, Green's theorem, and the divergence theorem are special cases of a generalized Stoke's theorem. Spivak's Calculus on Manifolds book (which is actually not a manifolds book despite the name) teaches calculus at an undergrad level using exterior algebra and differential forms if you're interested in learning this stuff.

Exterior algebra can be considered as part of geometric algebra, so you could continue on to learn geometric algebra if you enjoy exterior algebra.

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. :)

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.

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&ie=UTF8&qid=1413311074&sr=1-1&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.

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.

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.

First of all, I am glad to hear that your attitude towards math has changed so drastically! Welcome to the dark side. As for where to start relearning math, that probably depends on where you're starting from. Have you taken any classes on discrete math or linear algebra for your IT major? How about calculus?

If you want to work in mathematics, then you probably want a light introduction to proofs. Most undergraduates in math and computer science at my university use
Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games. I personally recommend How to Prove It by Velleman, which promotes a structured approach to proving that jives well with the step-wise refinement approach used by people in comp prog and comp sci.

If you have had some calculus, another good place to start is Calculus by Spivak. It will start you over from the basics, but with the rigor one should expect from a mathematics course.

Of course, I'm just making guesses about your background and your interests. What do you think you'd want to do with math or physics, or both? Perhaps you might enjoy scientific computing? If physics (or maybe even engineering) is your thing, then skip the first two books I recommended and try out Spivak. From there go on to books on linear algebra and differential equations -- the necessary math background -- while also checking out some physics books (I have a few to recommend, but that's off-topic. Let me know if that's something in which you're interested).

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 excellent text 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 Right and, 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.

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).

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).

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&s=books&qid=1239384587&sr=1-4

http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Applications/dp/0471000078/ref=sr_1_3?ie=UTF8&s=books&qid=1239384587&sr=1-3

Elements of the Theory of Functions and Functional Analysis, by Kolmogorov. One of my favorite pure math books (not that I've read many). Contrary to what others have said, I greatly enjoyed the introduction to important foundations of theory of sets, set topology and metric spaces. It has a great wealth of proofs that should aid your mathematical thinking in the future.

Also, anything by Kolmogorov is a joy to read.

Pretty cheap at $15

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.

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'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.

I know this isn't exactly what you're requesting (I assume you're requesting resources on the web for your consumption) but allow me to suggest the following book


A 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 why certain 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.

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:

1. Algebra Unplugged
   
2. Calculus for Cats
   
3. Calculus Made Easy
   
4. Trigonometry
   
   I wish you all the best!
   
   

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.

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.

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's Calculus might 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.

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'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&robot_redir=1

http://www.amazon.com/gp/aw/d/0716731606/ref=pd_aw_sims_3?pi=SL500_SY115&simLd=1

They're not exactly textbooks, but they appear to be good guides. Best of luck.

I'd say check with your professor first if it's for a class. You never know if you'll be missing a section. It helps to read what has changed in the newest edition. If it's minor cleanup or the addition of a single chapter you may be able to pass with the older version. However sometimes they change exercises and you'd be missing them for homework. Talk to the professor.

You can, however, check for Indian versions of books on Ebay or other places. These are usually paperbacks and are often in English, but they come at significantly reduced cost.

Otherwise, if this is for self-learning, I'd highly suggest looking at some Dover books. They pick up older classics or popular titles, often edit/update them a little, then publish in a cheap but nice looking and portable paperback.

E.g. Dover book on Infinitesimal Calculus for 4 bucks

There are hundreds of others. Many with good reviews, 4-5 star on Amazon. The presentation can be old-timey in some cases but the math is still relevant. I'm reading a book from the 1960's on "Information Theory" from Dover where you can see how this math motivated things like the internet and cell phones. It's based on Shannon's groundbreaking work in the 40s--much of it is still used to this day. They had the author (not Shannon) update it a bit for this new publication.

Textbooks (calculus): Fundamentals of Physics: http://www.amazon.com/Fundamentals-Physics-Extended-David-Halliday/dp/0470469080/ref=sr_1_4?ie=UTF8&qid=1398087387&sr=8-4&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&qid=1398087411&sr=8-2&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&qid=1398087498&sr=8-1&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&qid=1398087834&sr=8-1&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&qid=1398087878&sr=8-1&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&qid=1398087565&sr=8-1&keywords=irodov ,

Problem Books: Krotov; http://www.amazon.com/Science-Everyone-Aptitude-Problems-Physics/dp/8123904886/ref=sr_1_1?ie=UTF8&qid=1398087579&sr=8-1&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.

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.

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.

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.

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

:D
http://betterexplained.com/

http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Mathematics/dp/0486404536/ref=sr_1_1?ie=UTF8&qid=1422649729&sr=8-1&keywords=calculus+an+intuitive&pebp=1422649747330&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

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.

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

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.

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.

If you're looking for other texts, I would suggest Spivak's Calculus and Calculus on Manifolds. At first the text may seem terse, and the exercises difficult, but it will give you a huge advantage for later (intermediate-advanced) undergraduate college math.

It may be a bit obtuse to recommend you start with these texts, so maybe your regular calculus texts, supplemented with linear algebra and differential equations, should be approached first. When you start taking analysis and beyond, though, these books are probably the best way to return to basics.

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.

One of my favorite tutorials was a book called "How to Ace Calculus". It was a fun book to read and just so happened to go into great detail about calculus. I highly recommend it.

The exercises in Spivak’s Calculus (amzn) are the best part of the book.

• • 
    
    /u/WelpMathFanatic You’ll probably have a better (more efficient, more enjoyable) time if you take a course, or otherwise find someone to help you face to face. But if you’re studying by yourself you might want to look at a book about writing proofs, such as Velleman’s [How to Prove It](https://amzn.com/0521675995) or Hammack’s [Book of Proof*](https://amzn.com/0989472108). (Disclaimer: I haven’t read either of these.)
    
    

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.

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

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&sr=8-1&pi=SY200_QL40&keywords=humongous+book+of+calculus+problems&dpPl=1&dpID=515J89M2yTL&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.

My favorite two books for Calc 1,2, and 3 hands down:

How to Ace Calculus

How to Ace the Rest of Calculus

They're short, to the point, and pretty funny honestly.

For basic logic (first-order, classical) these are excellent textbooks...

• Goldrei, Propositional and Predicate Calculus
  
• Enderton, A Mathematical Introduction to Logic
  
• Smullyan, A Beginner's Guide to Mathematical Logic
  
  I should also mention that there are some free, open source textbooks on standard logic (good books, written by scholars in the field) such as the following...
  
  
• "Forall X" (Cambridge Edition)
  
  And if you want to go beyond classical logic, these are excellent textbooks...
  
  
• Priest, An Introduction to Non-Classical Logic
  
• Van Benthem, Modal Logic for Open Minds

> 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.

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.

> Second and third semester calculus

Is this vector calc? If so I enjoyed this book as it's very geometric, not at all rigorous and has lots of worked examples and exercises. Sorry it seems to be so expensive -- it wasn't when I bought it, and hopefully you can find it a lot cheaper if it's what you're looking for.

In general Stewart's big fat calculus book is a nice thing to have for autodidacts.

Obviously what you describe might include analysis, which these books won't help with.

>Formal logic theory (Think Kurt Godel)

I've heard Peter Smith's book on Godel is good, but haven't read it. Logic is a huge field and it depends a lot on what your background is and what you want to get out of it. You may need a primer on basic logic first; I like this one but again it's quite personal.

As a rising senior, I'll be attending a prestigious research program for 7 weeks to do some materials research, most likely biochemistry or biophysics.

Also, my school only goes as high as BC Calc for math. I took AB this year since BC didn't fit into my schedule, and the Assistant Superintendent was nice enough to set up a teacher to teach me Multivariable one-on-one next year, so long as I teach myself BC over the summer. Should be easy, and I might even start on Multivariable if I finish early.

Very excited! Should be a productive summer!

Edit: I'll be teaching myself from this book. It was recommended because it goes very in depth on proving various theorems that are usually just introduced without regard to why they work. I was told learning the theory behind calculus will help for when I take an Analysis class.

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.

If you are getting your degree in math or computer science, you will probably have to take a course on "Discrete math" (or maybe an "introduction to proofs") in your first year or two (it should be by your 3rd semester). Unfortunately, this will probably be the first time you will take a course that is more about the why than the how. (On the bright side, almost everything after this will focus on why instead of how.) Depending on how linear algebra is taught at your university, and the order you take classes in, linear algebra may be also be such a class.

If your degree is anything else, you may have no formal requirement to learn the why.

For the math you are learning right now, analysis is the "why". I'm not sure of a good analysis book, but there are two calculus books which treat the subject more like a gentle introduction to analysis-- Apostol's and Spivak's. Your library might have a copy you can check out. If not, you can probably find pdfs (which are probably[?] legal) online.

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.

Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.

Amazon search for Dover Books on mathematics

There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm

Pre-Calculus / Problem-Solving

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
  

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.

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.

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.

>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 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

Morris Kline's book. Not only is it the best calculus book for people in engineering and physics, I'd say it's the best of the calculus books still being published today, period.

once you get into partial differential equations, you'll be able to understand them. the basic ideas are pretty simple. there's just a bunch of computational overhead

this is a great book: https://www.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0813349109/ref=dp_ob_title_bk

it's informal and pretty easy to read. I don't remember it being so expensive though. i could've sworn i paid $20 for it

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

Strogatz is probably the best introductory book on the subject.

When studying nonlinear ODEs, analytical solutions are not always helpful and rarely necessary to understand the behavior of the dynamical system. If you absolutely need an answer (ie for a measured quantity) using RKF 4-5 (adaptive) for anything nonstiff is usually what you would do. There are no real good general tricks besides understanding system behavior without solving the ODE.

If you really want a close approximation, the only other option is to use perturbation theory (multiple scales, WKB, etc) to come up with an approximated solution. But it really isn't worth it in most cases (unless you have some eqution which is singularly perturbed). A good example of this is how to deal with the Schrodinger equation.

As for your example: it is separable, so separate and integrate. But if you have something remotely complicated you either won't get an analytical solution, or it will be such a pain that it isn't useful.

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/

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

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

I don't know of any video lectures that covers these topics, but I do know of a couple of good books that should be good resources to reference if you find Rudin a bit too terse in some places:

1. "Understanding Analysis" by Stephen Abbott - This should cover the first half of Rudin, plus the sequences/series of functions. I would really recommend, when you have the time, that you go back over Analysis with this book.
   
   
2. "Analysis on Manifolds" by James Munkres - Covers the stuff on Differential Forms. In fact, I would say that Rudin's main area of weakness in his Principles of Mathematical Analysis is precisely his coverage of differential forms, and so I would definitely pick up this book or the next.
   
   
3. "Calculus on Manifolds" by Spivak - This covers basically the same material as Munkres, but is more concise in the exposition. This is a classic, by the well-known differential geometer Michael Spivak. One warning, though: Spivak uses superscripts to index elements, so x = ( x^1 , x^2 , ... , x^n ) is how he writes points in R^n .
   
   I would recommend a combination of 2 and 3 for the differential forms and stuff from Rudin, and 1 for single variable real analysis.

Here's my radical idea that might feel over-the-top and some here might disagree but I feel strongly about it:

In order to be a grad student in any 'mathematical science', it's highly recommended (by me) that you have the mathematical maturity of a graduated math major. That also means you have to think of yourself as two people, a mathematician, and a mathematical-scientist (machine-learner in your case).

AFAICT, your weekends, winter break and next summer are jam-packed if you prefer self-study. Or if you prefer classes then you get things done in fall, and spring.

Step 0 (prereqs): You should be comfortable with high-school math, plus calculus. Keep a calculus text handy (Stewart, old edition okay, or Thomas-Finney 9th edition) and read it, and solve some problem sets, if you need to review.

Step 0b: when you're doing this, forget about machine learning, and don't rush through this stuff. If you get stuck, seek help/discussion instead of moving on (I mean move on, attempt other problems, but don't forget to get unstuck). As a reminder, math is learnt by doing, not just reading. Resources:

• math subreddit
  
• math.stackexchange.com
  

• math on irc.freenode.net
  

• the math department of your college (don't forget that!)
  
  
  Here are two possible routes, one minimal, one less-minimal:
  
  Minimal
  
  
• Get good with proofs/math-thinking. Texts: One of Velleman or Houston (followed by Polya if you get a chance).
  
• Elementary real analysis. Texts: One of Spivak (3rd edition is more popular), Ross, Burkill, Abbott. (If you're up for two texts, then Spivak plus one of the other three).
  
  
  Less-minimal:
  
  
• Two algebras (linear, abstract)
  
• Two analyses (real, complex)
  
• One or both of geometry, and topology.
  
  
  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.

Take a look at nonstandard analysis. I believe some studies in the 90s showed that students better understood these methods.

As for books, I can recommend Henle or the free book by Keisler at the high school level.

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.

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.

I've heard that, while Spivak's Calculus may be difficult because of proofs, it is good. However, his Manifolds is 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?

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.

The following are all really good and all very different. Check out the reviews and decide what fits you best. If I had to pick only one, I would pick one of the first three listed.

http://www.amazon.com/gp/product/0471530123

http://www.amazon.com/gp/product/1592577229

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

http://www.amazon.com/gp/product/0817636773

http://www.amazon.com/gp/product/0760706603

http://www.amazon.com/gp/product/B000ZEC19Y

http://www.amazon.com/gp/product/0070293317

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 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.

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.

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&qid=1466240189&sr=8-3&keywords=physics
Thank you for all your help so far (:

When I was (approximately) in 8th grade I read https://www.amazon.com/How-Ace-Calculus-Streetwise-Guide/dp/0716731606 and I loved it. :)

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 think I need to read up on dealing with infinite sets.

Your confusion was that you didn't distinguish between an existential quantifier and a universal quantifier. I don't think it's infinite sets themselves that are your issue. You might be better served by reading up on basic logic and proof techniques before reading up on basic set theory. This text is the one I used in my freshman introduction to proofs class and I thought it was pretty good.

This isn't an online resource, but this book is awesome for learning Calc 1.

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

Steven Strogatz is a great one too:

• Nonlinear Dynamics and Chaos
  
  
• Sync: The Emerging Science of Spontaneous Order

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 was or 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 needs a 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 learn programming, 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.

Perhaps non-standard calculus/analysis. There are some free texts here otherwise this one has good reviews.

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.

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

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.

Logic, Number theory, Graph Theory and Algebra are all too much for you to handle on your own without first learning the basics. In fact, most of those books will probably expect you to have some mathematical maturity (that is, reading and writing proofs).

I don't know how theoretical your CS program is going to be, but I would recommend working on your discrete math, basic set theory and logic.

This book will teach you how to write proofs, basic logic and set theory that you will need: http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995


I can't really recommend a good Discrete Math textbook as most of them are "meh", and "How to Prove It" does contain a lot of the material usually taught in a Discrete Math course. The extra topics you will find in discrete maths books is: basic probability, some graph theory, some number theory and combinatorics, and in some books even some basic algebra and algorithm analysis. If I were you I would focus mostly on the combinatorics and probability.


Anyway, here's a list of discrete math books. Pick the one you like the most judging from the reviews:

• http://www.amazon.com/Discrete-Mathematics-Applications-Susanna-Epp/dp/0495391328
  
• http://www.amazon.com/Discrete-Combinatorial-Mathematics-Applied-Introduction/dp/0201726343
  
• http://www.amazon.com/Discrete-Mathematics-Applications-Kenneth-Rosen/dp/0073383090
  
• http://www.amazon.com/Discrete-Mathematics-Computer-Science-Solutions/dp/053449501X
  
• http://www.amazon.com/Mathematics-Discrete-Introduction-Edward-Scheinerman/dp/0840049420 (read the reviews of the previous edition: http://www.amazon.com/Mathematics-Discrete-Introduction-Edward-Scheinerman/dp/0534398987)
  
  
  Don't bother trying to learn too much too soon, as you really do need to let time for the math to sink in.

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.

Definitely agree with the people recommending Calculus Made Easy by Silvanus P. Thompson. Often imitated, never equalled.

Another book similar to that is The Calculus for the Practical Man by J.E. Thompson. Besides its fame for being the book that Richard Feynman used to teach himself calculus, it has a completely nonstandard proof that the derivative of sin(x) is cos(x), using an argument based on arc length, which I haven't seen in any other book.

For more modern books I'd recommend Kline's book, which is underrated in my opinion. I'd avoid Spivak's book, which I feel is vastly overrated; it makes calculus even drier than the standard books do.

FYI, there's an answer book available for Spivak's Calculus that is very useful for self-study students.

For anyone interested in this topic, I can recommend two sources for newcomers.

Conversational, largely non-technical: Chaos: Making a New Science by James Gleick

Technical (requires knowledge of ordinary differential equations, but highly readable): Nonlinear Dynamics and Chaos by Steven H. Strogatz

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 just comes from the way we define sums of infinite sums, aka series. .999... is just shorthand for (.9+.09+.09+.009...), which is an infinite sum. We define the sum of a series to be equal to the limit of the partial sums. The limit is rigorously defined, and you can read the definition on wikipedia if you google "epsilon delta". The limit of an infinite sum, if it exists, is unique. For this infinite sum, that limit is exactly 1. By the way we define infinite sums, .999... is therefore exactly equal to 1.

It's not so bad when you remember that all real numbers have a representation as a non-terminating decimal. 0.5 can be written as 0.4999... and 1/3 can be written as 0.333... and pi can be written as 3.14159... for example.

And lastly, if .999... and 1 are different real numbers, then there must exist a number between them. This is because of an axiom we have called trichotomy: for any two real numbers a and b, exactly one of the following is true: a<b, a=b, a>b. If a=/=b, then there exists a real number between them, because the real numbers have a property called "dense". It is easy to prove that here is no such number between .999... and 1, real or otherwise. Therefore .999... is exactly equal to 1.

e: The sum (.9+.09+.009...) is bigger than every real number less than 1. You can check if you want. The smallest number that is greater than every real number less than 1 is 1 itself. We get this from an axiom called the "least upper bound property". Therefore .999... is at least 1. Using our rigorous definition of a limit, we find that it is exactly 1.

e2: Apostol's Calculus vol 1 is a fantastic place to start learning about rigorous math shit. Chapter one starts you out with axioms for real numbers, and about half way through chapter 1 you prove the whole thing about repeating decimals corresponding to rational numbers. It is slow and easy to follow. Other people recommend Spivak but I haven't seen it so idk.

Hey, so here's a list I have recommended lately.

For real science: Death by Black Hole

If you want some absurd (yet still real) science: What If?

If you like classic mysteries: A Study in Scarlet

And here's some things I have been looking to get, maybe you'll like one:

The Republic

and

The No Bullshit Guide to Math and Physics

Hope one of these sparks an idea!

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

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........!

'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.

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.

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.

Try Kolmogorov and Fomin's Elements of the Theory of Functions and Functional Analysis.

PROS:

• clear exposition
  
  
• short
  
  
• good, but not overly demanding exercises
  
  CONS:
  
  
• definitely not a reference text for the field
  
  
• no "applications section" to speak of
  
  
• old, and it shows in the nomenclature
  

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.

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&qid=1524052149&sr=8-1&pi=AC_SX236_SY340_FMwebp_QL65&keywords=quick+calculus was a perfect precursor for the class if you're pressed for time.

Fomin and Kolmogorov is a classic.

https://www.amazon.com/Elements-Functions-Functional-Analysis-Mathematics/dp/0486406830

Some of the terminology is out of date but it's a nice exposition.

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.. ._.

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.

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.

I found that the book Quick Calculus: A Self-Teaching Guide(Amazon link) was quite good.

Professor Landau would be proud of you: http://www.amazon.com/Foundations-Analysis-AMS-Chelsea-Publishing/dp/082182693X

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.