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I'm getting into college next week, and here in my country we don't study calculus but we study what's called mathematical analysis, those are the topics covered in first year :

  • Real Number Properties

  • Functions & Limits (Including continuity)

  • Derivatives

  • Integrals and Riemann sums

  • Power series and Taylor expansion

  • Polynomial Approximation

  • Suites and Convergence theorem

  • ODE "Ordinary differential equations of first order"

  • Double integrals

  • Multi variable functions

And few other topics I don't remember, I don't know whether those are more related to calculus or mathematical analysis ?

I want to start studying my self this week but I can't decide whether I should use spivak calculus book (I've found that problems are very interesting and rigourous) Or MIT OCW lectures and problem sets for single variable calculus ?

Please help me choose one ?

Bernard
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Anis Souames
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  • I'm more concerned about the quality of the materials then their price . – Anis Souames Sep 05 '17 at 22:07
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    If you're interested in proofs and not just basic computations, Spivak is unquestionably the way to go. It won't get you multivariable calculus. Spivak is definitely an analysis course with calculus content. ... (And I wrote a multivariable sequel, in the spirit of Spivak's book, integrating linear algebra and multivariable calculus/analysis.) – Ted Shifrin Sep 05 '17 at 22:08
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    I had no idea you'd written a sequel, Ted. That's ANOTHER book to add to my to-read list. :) To OP: If you find you can do the problems in Spivak, then that's a great route to follow. Many are so challenging that they can be dis-spiriting, and if you find that happening, consider engaging with some of the OCW content to get you going again. – John Hughes Sep 05 '17 at 23:11

1 Answers1

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I think the best answer to this question is that if you can afford it, you should use both: while many of us don't follow our own advice due to time constraints, it is much better to use more than one resource to learn things. Advantages include

  • exposure to different style and notation,
  • different proofs,
  • different motivations,
  • more exercises,
  • and best of all, if you really can't understand one source's version of a particular result, looking in another really clarifies things (For example, I really didn't understand manifolds the first time, but found a source that had the chart functions the other way round, and suddenly everything made sense. And you wouldn't believe how long it took to find an explanation of Lie derivatives that I was satisfied with.).

But Spivak's Calculus an excellent textbook: has plenty of exercises that actually tell you interesting things, as well as being good tests of knowledge, motivates most things carefully before plunging in with epsilons and deltas, and is overall one of the more readable books out there. (Nice big margins to make notes in if you're using the dead tree edition and that way inclined, too.)

(It doesn't cover anything about ODEs or multivariable functions, however. For those you need things that are normally labelled "A Second Course in Analysis" or similar, like Körner's A Companion to Analysis, or Rudin's Principles of Mathematical Analysis, for example.)

Chappers
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