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I'm looking for a tetxbook that covers Multivariable Calculus and/or Vector Calculus theoretically.

I have done Analysis (single-variable) at the level of Introduction to Real Analysis by Bartle and Sherbet and Principles of Mathematical Analysis by Walter Rudin. Except for the theory of integration. I haven't done a lot of exercises from "Baby Rudin" but I have done a course on Topology so some of the exercises will be approachable now, though I may have to go and revise Topology to refresh my concepts/recall the theorems.

Anyway, I'm looking for a textbook that extends the material in the aforementioned books to Multivariable and/or Vector Calculus. Any suggestions?

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  • $\begingroup$ Related. $\endgroup$ – user137731 Jul 9 '16 at 19:01
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Check out Mathematical Analysis I (for single variable) and Mathematical Analysis II (for mulitivariable calculus) by Vladimir A. Zorich

https://www.amazon.com/Mathematical-Analysis-Universitext-Vladimir-Zorich/dp/3540874518

Eastern European style. Typical reference in U.S. is Michael Spivak Calculus on Manifolds and more comprehensive lesser known Analysis on Manifolds by Munkres (the same Mukres who wrote the First Course in Topology).

Any serious understanding of Multivariable Calculus will have to involve concurrent course in Electromagnetic field theory (Classical Electrodynamics) as well as understanding of Hodge star operator and at least a rudiments of De Rham-Hodge Theory.

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  • $\begingroup$ when does one usually learn about the Hodge star operator/De-Rahn Hodge Theory? I believe I'm still a long way from learning them since I'm a student of physics with a mathematical leaning so i have double the amount of material to juggle. Also, how would a couese on Electromagnetic field theory help? I have an introductory graduate level course on electrodynamics next term. $\endgroup$ – Junaid Aftab Jul 9 '16 at 19:17
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    $\begingroup$ Typically never unless you attend some serious place like MIT, University of Michigan etc. Even worse once you learn about it in some kind second graduate course in Algebraic Topology the physical intuition is completely removed. Unfortunately in U.S. and Europe (not including Russia) we have decoupled so much our math courses from physics courses that it hurts me to speak about it as somebody who knows little bit about Math Physics $\endgroup$ – Predrag Punosevac Jul 9 '16 at 19:31
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Honestly I would recommend that you use a University Calculus text, practice doing problems and read the proofs there. You will see that even the extremely non-rigorous proofs are hard to follow. The problem with tackling an intense book off hand is that you may go through the whole text without being able to do simple computations.

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  • $\begingroup$ Most standard university calculus textbooks don't seem to offer even non rigorous proofs. Any suggestions for books on this front. $\endgroup$ – Junaid Aftab Jul 11 '16 at 17:44
  • $\begingroup$ Look at Rogawski 2nd and Lang Calc of Several Variables. $\endgroup$ – Faraad Armwood Jul 11 '16 at 17:52
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Introduction to Analysis by Wade. In my opinion it contains very readable, rigorous, and user-friendly proofs of the vector calculus theorems (for $\mathbb R^3$). I think it is the most clearly written analysis text I have seen where the reader is required to do fewer "side calculations".

Spivak's Calculus on Manifolds is concise and a bit harder to read, but contains more general versions of the theorems for any dimension using differential forms.

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