Theoretical Multivariable Calculus books

[bacte2013]

Dear Physics Forum advisers,

Could you recommend books that treat the multivariable calculus from a theoretical aspect (and applications too, if possible)? I have been reading Rudin's PMA and Apostol's Mathematical Analysis, but their treatment of vector calculus is very confusing and not really motivating at all. I unfortunately did not take a Calculus III course, so I am not familiar with the elementary presentation of the calculus of several variables. However, I am very determined to learn it.

I know books like Kaplan, Buck, Loomis/Sternberg, Hubbard/Hubbard, Apostol's Calculus, and Zorich, but I personally never had a chance to look through them. All of them were checked out. Is any of them good for learning the multivariable calculus? By the way, I see many names for "calculus of several variables" in the Tags column: multivariate calculus, multivariable calculus, vector calculus, calculus on manifolds, manifolds analysis, etc. Several of the books I mentioned have title called "Advanced Calculus". Are they same description for studying the multivariable calculus?

 

[axmls]

Spivak's Calculus on Manifolds looks good from what I've seen. If it's anything like his regular Calculus, it should be great.

However, I'd avoid really focusing on the theoretical aspects until you've at least gone over the basic mechanics of vector calculus that one would learn in a calculus 3 course.

 

[micromass]

If it's anything like his regular Calculus, it should be great.

It's actually nothing like his regular calculus book, sadly enough. Calculus on manifolds feels rushed and it lacks motivation. But since the OP is mainly looking for a theoretical source and is already quite familiar with analysis, I think Spivak would be a great book for him.

 

[bacte2013]

Spivak's Calculus on Manifolds looks good from what I've seen. If it's anything like his regular Calculus, it should be great.

However, I'd avoid really focusing on the theoretical aspects until you've at least gone over the basic mechanics of vector calculus that one would learn in a calculus 3 course.

Thank you for the advice. Why is it important to learn the basic mechanics of vector calculus before studying the theoretical aspect? I am very curious about that. I did not have a problem jumping to the analysis without strong foot on the Calculus I-II. Do you have any recommendation on a book or source where I can learn the basics of vector calculus?

 

[bacte2013]

It's actually nothing like his regular calculus book, sadly enough. Calculus on manifolds feels rushed and it lacks motivation. But since the OP is mainly looking for a theoretical source and is already quite familiar with analysis, I think Spivak would be a great book for him.

I am quite surprised about the thinness of Spivak, however, I do not like the presentation. I was looking for one that treats the multivariate calculus in a comprehensive, detailed level, and one that also connects the ideas to different mathematical topics like geometry and diff. equations. I just briefly went through Loomis/Sternberg and Hubbard/Hubbard, and those books seem good to me.

 

[Xiuh]

I always recommend these 3:
-Edwards. Advanced Calculus of Several Variables
https://www.amazon.com/dp/0486683362/?tag=pfamazon01-20
-Fleming. Functions of Several Variables
https://www.amazon.com/dp/0387902066/?tag=pfamazon01-20
-Munkres. Analysis on Manifolds
https://www.amazon.com/dp/0201315963/?tag=pfamazon01-20

 

[George Jones]

Do you have any recommendation on a book or source where I can learn the basics of vector calculus?

I like "Calculus of Several Variables" by Lang,

https://www.amazon.com/dp/0387964053/?tag=pfamazon01-20

 

[verty]

Bacte2013 has asked about vector calculus before. He wanted two books on every subject, his question was what two books to use for vector calculus. He had settled on Lang and possibly Hubbard/Hubbard. I thought they wouldn't compliment each other very well, there would be a lot of overlap. I suggested, since he had settled on Lang (he said wanted to use it), that he might like to have supplementary problems. I was thinking that is surely why he wants two books (as a policy), to have ample problems. And I was confident that Lang would explain everything perfectly well. I therefore suggested Marsden & Tromba for excellent problem coverage. A used copy was not too expensive and I believe he did buy those two books.

 

[feyn_stein17]

I would recommend Colley's Vector Calculus. At JHU it is used for the honors calc iii course, which treats multivariable calculus from a rigorous theoretical standpoint, but still covers all of the basic applications for the math as well.
Amazon link:
https://www.amazon.com/dp/0321780655/?tag=pfamazon01-20