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This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like:

-Differentiability. -Open mapping theorem. -Implicit function theorem. -Lagrange multipliers. Submanifolds. -Integrals. -Integration on surfaces. -Stokes theorem, Gauss theorem.

I need to know if anyone of you guys know good textbooks that contain practice problems with full solutions or hints that can be used to understand the material. Most of the textbooks I found are covering only the material with few examples.

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This may be helpful math.stackexchange.com/questions/46482/… –  user38268 Jan 7 '12 at 23:53
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5 Answers 5

up vote 3 down vote accepted

I like C.H. Edwards, "Advanced Calculus of Several Variables." It's cheap and contains many exercises and examples.

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http://www.math.harvard.edu/~shlomo/

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That's too hard for the average undergraduate, Adam. Hell,it was too hard for some of the honor students at Harvard that Sternberg and Loomis wrote the original notes for in Math 55 there! –  Mathemagician1234 Jan 8 '12 at 3:32
    
Nothing is too hard - only too fast. ;) –  Adam Jan 8 '12 at 5:08
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Spivak has a good book in multivariable calculus. Harvard, I think, uses Hubbard's Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Another book which is used is Shifrin's Multivariable Mathematics, which I believe Harvard uses as well.. For Shifrin's book at least, you can buy a solution manual online.

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+1. I think these are the 2 best options for good students currently for an advanced calculus/vector analysis course. Hubbard and Hubbard is especially wonderful for it's many physical applications,historical notes and "God-I-Wish-Someone-Had-Pointed-That-Out-To-Me-When-I-Was-First-Learning-This" observations peppered throughout. –  Mathemagician1234 Jan 8 '12 at 3:36
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Here are some free options:

http://www.whitman.edu/mathematics/multivariable/

http://synechism.org/drupal/cfsv/

http://www.mecmath.net/

I maintain a catalog of free books at http://theassayer.org , and you could poke around there for further possibilities.

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A couple of references that come to mind that satisfy your criteria for exercise solutions are:

(1) Multivariable Analyis by Shirali and Vasudeva. Most problems are provided with complete solutions.

(2) Analysis in Vector Spaces by Akcoglu, et al. This comes with a student solutions manual that contains solutions to all of the odd-numbered exercises. The biggest drawback to this text is, unfortunately, the price - which has almost doubled since I purchased it last year (!)

In both cases, the problems are very good and are at a level commensurate with the material. Both of these texts cover the inverse/implicit function theorems and integration including Stokes, the divergence theorem, etc. Also, both use the language of differential forms for the development of integration theory.

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