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Question/Advice regarding multivariable calculus

  • #1
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Main Question or Discussion Point

Hi,

I want to re-learn multivariable calculus, after I have learned it, not in the best possible way... and feel bad about it.

I have seen the recommendations here about Hubbard/Shifrin/Fleming/Edwards. I have also seen the books by Munkres/Spivak/Apostol.
I didn't really like Hubbard's book because it is way too verbose for me. I like(and think it is necessary) to read some motivation, but in my opinion it is too much.

From the books above, I like the most Munkres',Apostol's and Fleming's books. But I am not sure if those are "the right" books to learn the subjects from. Regarding Fleming's book, unfortunately it contains a lot of new things that I didn't learn, and I am not sure if I need to learn them or not, especially if more advance topics are based on those topics(mostly chapter 2, from section 2.6- topological spaces- onwards).

My primary goal is to strength my understanding in the "classical" vector calculus, and if time allows then go into differential forms.

Thank you.
 

Answers and Replies

  • #2
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A really good way to strengthen your "div, grad, curl, and all that" vector calculus would be to study an application which uses a lot of it, such as classical electrodynamics. Given your reaction to advanced calculus textbooks, I think you might like Wangsness' text.

The extra topics in Fleming come in very handy later on in both mathematics and physics.

Topology is useful for improving ones understanding of the mathematical structures used in both general relativity and quantum field theory, for example. The candidate theories for quantum gravity rely even more heavily on topological concepts.

Learning Lebesgue integration helps cement one's understanding of point-set topology, and also provides a better framework for thinking about integrating over general sets rather than simple intervals or curves.

That being said, given your attitude towards Hubbard, you'd probably prefer Spivak's Calculus on Manifolds for the coverage of differential forms, though Fleming's presentation is closer to Spivak than Hubbard, in my opinion. I think Fleming's text is probably a better value for someone in your situation.
 
  • #3
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A really good way to strengthen your "div, grad, curl, and all that" vector calculus would be to study an application which uses a lot of it, such as classical electrodynamics. Given your reaction to advanced calculus textbooks, I think you might like Wangsness' text.

The extra topics in Fleming come in very handy later on in both mathematics and physics.

Topology is useful for improving ones understanding of the mathematical structures used in both general relativity and quantum field theory, for example. The candidate theories for quantum gravity rely even more heavily on topological concepts.

Learning Lebesgue integration helps cement one's understanding of point-set topology, and also provides a better framework for thinking about integrating over general sets rather than simple intervals or curves.

That being said, given your attitude towards Hubbard, you'd probably prefer Spivak's Calculus on Manifolds for the coverage of differential forms, though Fleming's presentation is closer to Spivak than Hubbard, in my opinion. I think Fleming's text is probably a better value for someone in your situation.
Thank you for the answer.

A problem with your first suggestion is that I am really bad at physics. I am a math major.

Another thing, when I asked about the sections in Fleming's book which I didn't learn and don't know if I should(right now), it's because I am not sure if the proofs in the rest of the book use those parts.

Thank you.
 
  • #4
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Another thing, when I asked about the sections in Fleming's book which I didn't learn and don't know if I should(right now), it's because I am not sure if the proofs in the rest of the book use those parts.
I would say that the sections on Euclidean spaces and topology are more practice for learning the style in which Fleming uses set theoretic expressions alongside his exposition. You should be able to easily get away with jumping straight to chapter 3 and using the first two chapters as reference material, and for extra problems when you want practice dealing with sets. If you run into notation that isn't defined in chapters 3+, you can flip through chapters 1-2 quickly to see where it's first used, and work through that example.

If you're familiar with set theory, and understand basic topological ideas from analysis like compactness, countable and finite covers, and such, you should be fine going straight to chapter 3.
 
  • #5
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I would say that the sections on Euclidean spaces and topology are more practice for learning the style in which Fleming uses set theoretic expressions alongside his exposition. You should be able to easily get away with jumping straight to chapter 3 and using the first two chapters as reference material, and for extra problems when you want practice dealing with sets. If you run into notation that isn't defined in chapters 3+, you can flip through chapters 1-2 quickly to see where it's first used, and work through that example.

If you're familiar with set theory, and understand basic topological ideas from analysis like compactness, countable and finite covers, and such, you should be fine going straight to chapter 3.
Thank you. I appreciate your help!
 
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