Find a Rigorous Calculus of Multiple Variables Book for Advanced Learners

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mstud
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Hi! I am looking for a very rigorous book on some of the topics covered in Calculus of Multiple Variables.

My University uses the last part of Adams "Calculus: a complete course" and I found the presentation therein more fit for people needing to know enough to perform the calculations than for those who want to understand the concepts...

At best it could also cover implicit functions and vector fields In addition to vector calculus ( multiple integrals and the other content I am currently satisfied with what I know so far, so it's no need for it to be included)

So you'll know a bit more about my level: Books I've studied so far include baby Rudin, Carothers "Real Analysis", parts of the (first 7 chapters) "Big Rudin", the "complex variables"- book of Brown & Churchill. I also have studied much of the Fraleigh's "Introduction to Abstract Algebra", but that's a book I don't understand to well (especially after the first 10 chapters or so).

However I want to go back to the Calculus III content and study it in a far more rigorous way than the treatment in Adams' book.

I have read some of the other threads and I feel many are asking for a good book to introduce themselves to the subject. Thus I am not sure which book will be the best for my level, what would you recommend?

Any suggestions would be greatly appreciated!
 

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I only know non-rigourous maths, but one book which was quite readable for non-experts is Spivak's "Calculus on Manifolds", which is apparent also a rigourous book. It covers Stokes's theorem, which is the central bit of multivariable calculus used in electrodynamics.
 
Rigor does not necessarily always equal understanding the concepts. So, personally, I'd say if you want to understand vector calculus, you should learn electricity and magnetism from a physics or engineering book. I suspect Div, Curl, Grad, and All That is a good book, from what I've heard, but I've never read it. Another more rigorous, but not completely rigorous book that deals with Stokes theorem and differential forms is Mathematical Methods of Classical Mechanics. The way to really understand vector calculus is through physical intuition, rather than mathematical rigor. Another book to look at might be A Geometrical Approach to Differential Forms.
 
Maybe Apostol? I've never seen the second volume though.
 
EDIT: Wow I didn't check the date of this thread until I had posted. Old!

homeomorphic said:
Rigor does not necessarily always equal understanding the concepts.

I agree. Read Div, Curl, Grad, and All That by Schey (little TAI) then Generalized Vector and Dyadic Analysis by Tai (BIG TAI). After that you will probably have a good understanding of the motivation for Calculus of Multivariables. Not sure about a more rigorous text.