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Please give me advice on the following goals I have for studying calculus.

1 A more rigorous understanding

I've studied single-variable and vector calculus and some ode, and have become familiar with the methods used. However, I have trouble when I try to prove theorems (when I can switch the order of integration etc), or assessing when I can or cannot integrate a function. Also, I'm not comfortable with the notion of "area element" or switching dxdydz with dV.

I think my understanding of the concepts such as continuity and limits is poor.

What are some good online resources to improve my situation? Do you think studying proofwiki's proofs on calculus is a good idea? If so, what are the key theorems that I should start with?

2 Making a connection with physics

Also, I want to study calculus in relation to physics, so as to be more familiar the above mentioned area elements that I often saw arise in relation to physics problems. What are good textbooks/online resources to study calculus in the context of physics?

3 PDEs

What are good textbooks to study PDEs? I am studying on my own, so my requirements are

  • thorough explanation
  • a lot of exercises.

Advice on self-studying mathematics in general is also greatly appreciated.

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  • $\begingroup$ Take a look at Spivak's Calculus. It's rigorous and is a good introduction to more abstract mathematics. Another one is Understanding Analysis by Abbott. You could take a look at Rudin's Principles of Mathematical Analysis but its at a higher level than both of these. You can find these online.. $\endgroup$ – user70962 Jun 9 '13 at 9:06
  • $\begingroup$ This might be helpful: math.stackexchange.com/questions/44522/… $\endgroup$ – N.U. Jun 9 '13 at 9:13
  • $\begingroup$ For a more thorough introduction single variable calculus, I second Spivak's Calculus. Its quite rigorous and contains many thought provoking exercises. You can pick up a copy of the thrid edition from Barnes and Noble (bn.com) for about $50. $\endgroup$ – Gamma Function Jun 9 '13 at 9:30
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I would recommend a notepad and taking some time to imagine yourself sitting in the MIT classroom when viewing their online videos on physics and calculus. They have loads of additional resources available (notes, books, exercises, assignment, past papers, etc).

If you want some more rigor, I would recommend that you master the concepts of logic, naive set theory, functions, cardinality, natural numbers and the real numbers. Most books on analysis brush over these ideas before moving on to limits and continuity. There is a video series on youtube hosted by Harvey Mudd College which offers a course based on Rudin's Principles of Analysis. It is very good. Also, the book 'Analysis by Steven Lay' has an excellent introduction to formal logic. You can not move on to theorems and proofs until you have mastered the basics of formal logic.

For PDE's, I found Csordas and Haberman to be very good.

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