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I've been trying to educate myself in various areas of mathematics. I have been out of any formal math education for quite some time and so I brushed up on some basic (really, really basic) stuff including algebra and so forth. I'm working my way through some calculus and I'm using Michael Spivak's Calculus (4th ed.), which is fantastic, but I'm finding it to be a bit difficult at times. Do you have any recommendations for a good book or any other path that might make sense for someone without a very strong, formal background in math for learning calculus?

I should add that a major motivation in my renewed interest in math is related to my interest in applying it to computer science. I know that many will respond saying that calculus is, therefore, not the best place to spend my energies (discrete math might be better) and I'm sure that's a reasonable point, but I'm also interested in calculus per se. I guess there are a lot of areas in computer science where I feel like I'm lacking because I'm not competent with functions, etc, for example in algorithm analysis and big-O notation.

I realize this was a bit rambly and some of this might be totally off -- if it is, it's from pure ignorance -- and for that I apologize. I guess I'm just sort of confused as to where my energies would best be spent given my goals.

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The 'obvious' recommendation would be Stewart's book, but it may be too far in the opposite direction. Are you interested in rigorous proof, or mostly grokking the main ideas and applications? – Robert Mastragostino Jul 16 '12 at 3:04 – Artem Jul 16 '12 at 3:05
@RobertMastragostino, thanks for your response. See my edit above. – LuxuryMode Jul 16 '12 at 3:05
I second Stewart, even though it is polar opposite of Spivak's style as noted by Robert. – Joe Jul 16 '12 at 3:05
Everyone has to start somewhere when getting into formal mathematics. Spivak is a good book for learning formal proof-based mathematics. Rudin's Principle of Mathematic Analysis is another good book for learning formal rigorous calculus. – William Jul 16 '12 at 3:05
up vote 2 down vote accepted


MIT's OpenCourseWare initiative is a excellent place for any self-learner. See

Prof. Jerison is very clear, if a bit slow. The best part about these is the lectures and recitation videos. In these videos, you get to have someone explain the concepts to you - which is great when you get burned out from the textbook. Moreover, you feel like part of the crowd (if only till you realize you are in fact not in a lecture hall)!

Another great resource is the Khan Academy. See I find it is best not to take a course in Khan Academy, but rather to look at videos in areas in which you have trouble. This is just personal preference - many people love taking full courses from Khan. The best thing about Khan Academy is that Salman Khan is a great explainer. He goes slow enough for the concepts to sink in, but doesn't dumb it down just to make is easier.

I don't have any good textbook recommendations. From what I've heard Spivak is one of the best though.

I would really recommend that you do as many exercises as possible. Find problems that force you to find new and clever implementations of the things you learn; don't use the same process over and over again for each problem. As a high schooler who spends his summers self-learning, this is the most important thing that I have learned and it is the most important advice I can give you.

EDIT: Another MIT course, Introduction to Algorithms, I think, does math for computer science. It teaches asymptotic notation, and as someone who has read quite a bit of the textbook for the course, I can vouch for its excellence.

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thanks Eric these are excellent recommendations. I was watching the introduction to algorithms course, but found myself lost pretty quickly when they started to go through asymptotic notation pretty fast – LuxuryMode Jul 16 '12 at 3:14
I didn't watch many lectures. What I do remember is that the book contained far more information than the lecture. The book is a beast (~800 pages of small text) but it is highly informative. I will admit that I was still quite confused at times reading it. – Eric Thoma Jul 16 '12 at 3:20

Gilbert Strang's Calculus. The best all purpose, general level calculus text bar none. Tons of applications, carefully and beautifully written and containing many applications and insights you simply won't find anywhere else.

And best of all,the first edition's available online for free.

That's the best one if you're not interested in a specific aspect of the subject.

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started reading the book this morning, so far a very enjoyable read and super clear. thanks for the recommendation! – LuxuryMode Jul 16 '12 at 14:10

Calculus is the basic language used in science, so no matter where your interest lies it should be useful. You may check this book:

I have never read it and it is recommended to me by a friend when I ask what book to suggest for my students to read for calculus as I am going to be calculus TA next term. For serious calculus books, I would recommend Zorich's Analysis I.

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MIT videos are COOL!

If you would like to have some paperbook mayby this one will be helpful? The most basic things in mathematics are covered too. Really good for self studing.

Maths: A Student's Survival Guide: A Self-Help Workbook for Science and Engineering Students.

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