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My course outline:

Differentiation of functions of several variables: partial derivatives, differential, differentiability, inverse function theorem, implicit function theorem, free extremum problems, constrained extremum problem, method of Lagrange multipliers

Integration in $R^n$:measure zero and content zero sets, integrability, Fubini's Theorem, partition of unity, change of variables

Integration on chains: tensors, alternating tensors, vector fields, differential forms, Poincare Lemma, Stokes' Theorem

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The classical source for most of these topics is Rudin : Principles of Mathematical Analysis. – Euler....IS_ALIVE Jan 20 '13 at 8:42
(reference-request) should not be used as a standalone tag; see tag-wiki and meta. – Martin Sleziak Aug 2 '13 at 6:29

The book by Moskowitz and Paliogiannis: Functions of Several Real Variables, discusses almost all the above topics in detail, and I strongly recommend it. It is rigorous with many good Examples and Problems with Hints and full Solutions.

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