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Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.

After reading the wiki article "Second partial derivative test", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.

Motivated by this, I'd like to put the question here:

What are the usual references for the theoretical treatment for the Multivariable calculus?

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up vote 21 down vote accepted

I usually think of multivariable calculus as being divided into four parts:

  • (Partial) Differentiation
  • (Multiple) Integration
  • Curves and Surfaces in $\mathbb{R}^3$
  • Vector Calculus (Green's Theorem, Stokes' Theorem, Divergence Theorem)

For differentiation, you can use Principles of Mathematical Analysis by Rudin (Chapter 9). Actually, this text also discusses integration and vector calculus (Chapter 10), but I personally found Rudin's treatment of such hard to follow when I was first learning the subject.

For differentiation, integration, and vector calculus you can use Calculus on Manifolds by Spivak, or Analysis on Manifolds by Munkres.

For curves and surfaces, you can use basically any book on elementary differential geometry. One of the most widely-used is Differential Geometry of Curves and Surfaces by do Carmo, though I highly recommend Elementary Differential Geometry by Pressley.

A few remarks as to where these topics end up going, with a slant towards differential geometry:

Differentiation of functions $f\colon \mathbb{R}^n \to \mathbb{R}^m$ can be generalized to differentials (a.k.a. pushforwards) of maps $f\colon M \to N$ between differentiable manifolds.

All of the references I mentioned above treat multiple Riemann integration of functions $f\colon \mathbb{R}^n \to \mathbb{R}$. This can be generalized to multiple Lebesgue integration via consideration of product measures.

The theory of curves and surfaces leads naturally towards Riemannian geometry, though certainly other branches of geometry also generalize this subject.

In vector calculus, one discusses line integrals and surface integrals of both functions and (co)vector fields. To my mind, the intricacies of such processes are not fully realized until one studies the integration of differential forms on differentiable manifolds.

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There are many great books that cover multivariable calculus/analysis, but I'm not sure a "standard" really exists.

That being said, here are a few that I like in no particular order:

Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach by Hubbard and Hubbard. This text includes proofs of the major theorems of vector calculus and, as a great benefit to the self-learner, a solutions manual for many of the problems so that you can check your work. It should be approachable by anyone with a good background in basic calculus.

Wendell Fleming's Functions of Several Variables is a somewhat more sophisticated treatment of the subject but still elementary and approachable. The author works up to proving Stoke's Theorem on manifolds. It does include measure theory but really only enough for integration in a general context to make sense.

If you're really interested in the geometric aspects of advanced calculus, take a look at Callahan's recently released Advanced Calculus: A geometric View. This treatment is far from standard but is quite rigorous and the author works very hard to convey the geometric content of the material. The book has tons of illustrations and uses numerical computer experiments to aid intuition.

Another book that also focuses on geometric content is Baxandall and Liebeck's Vector Calculus. The name of this book might remind you of very elementary texts that focus on plug-and-chug, but this impression would not be accurate. This is a very solid text, one that I wish had been taught to me in "Calculus III".

A text that is more sophisticated than all of the above but extremely well-written and approachable is Shroeder's Mathematical Analysis: A Concise Introduction. This has a much more "analytical" flavor to it, more along the lines of, Say, Rudin, but was written to be self-contained and require few prerequistites. The text starts out easy and slow but builds rapidly to cover the major theorems of analysis in general contexts (i.e., Banach, Hilbert spaces).

I'll also tip my hat to a text that was mentioned in another answer: Loomis & Sternberg's Advanced Calculus. This is a wonderful book and it is a shame that it is no longer in print.

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A GREAT list,3Sphere-one I give a hearty thumbs up to; especially Hubbard and Hubbard. This book should be in every serious mathematics students' library. – Mathemagician1234 Sep 30 '11 at 9:32
Loomis & Sternberg is back in print now. – Nate C-K May 19 '15 at 15:40

Since I agree with some of the extremely good recommendations given in the post (Theoretical) Multivariable Calculus Textbooks I shall mention my picks here.

My personal advise is the two volumes by Zorich - "Mathematical Analysis vol. 1" and "Mathematical Analysis vol. 2". Take a careful look at the table of contents of both since they deal with all rigorous calculus needed from real numbers and functions of one variable to multivariable calculus and vector analysis, curves and surfaces, differential forms, series and asymptotic methods. Most proofs are included: from the usual easy rules and techniques of differentiation and indefinite integration in one variable up to very important results in multivariable calculus like the general Stoke's theorem and the change of variables formula inside of a multiple Riemann integral (a fundamental result not proved in many other books as rigorously beyond heuristic justifications for double or triple integrals).

A book focusing on multivariable calculus only with tremendous visual insight (filled with figures) and motivation is Callahan's - "Advanced Calculus: A Geometric View". It is a very beautifully written book which is kind of less tough than Zorich but also less ambitious in scope.

I really think that they are a wonderful complete reference for the real analysis needed just before entering general measure theory (Lebesgue integration) and other more advanced topics such as operators and Hilbert spaces (as a great complement to Zorich, dealing with this adv. subjects a great book is Kantorivitz - "Introduction to Modern Analysis")

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I'm very partial to "Russian" origin texts and completely agree about Zorich-it's a wonderful book that also includes many physical applications! – Mathemagician1234 Sep 30 '11 at 9:40

Edwards - Advanced Calculus of Several Variables

Loomis & Sternberg - Advanced Calculus (freely available from Sternberg's website.)

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Loomis and Sternberg is INSANELY hard for undergraduates. I know it was designed for the superhuman honor students at Harvard in the 1960's,but it borders on the absurd in terms of it's level of abstraction and compactness. It's a really good book for more advanced students,though-like those with a year of real analysis a la Rudin and a strong linear algebra course. Hubbard and Hubbard,though,is a LOT more pleasant to read and covers a lot of important material that can't be found anywhere else. I'd start with that book and then move on to Loomis/Sternberg. – Mathemagician1234 Sep 30 '11 at 9:37

Spivak's Calculus on Manifolds is a great book. It's available from Dover Westview Press and quite cheap not expensive. It hits on everything in @Jesse Madnick 's list.

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Actually, Calculus on Manifolds is offered by Westview Press, not Dover, and at over $40 for 160 pages, I really wouldn't consider it "cheap"... – ItsNotObvious Jun 5 '11 at 21:43
Actually I initially mentioned this book in my answer, but then removed it because I am pretty sure it does not talk about things such as the second partial derivative test. Spivak has a clear goal, to get to Stokes theorem as fast as possible, so as a comprehensive reference on multivariable analysis it is probably not so good. – wildildildlife Jun 5 '11 at 22:05
I edited to correct the publisher. Also, I find it to be a good reference and easy to read. It may not be as complete as some of the other books listed above, but I feel Spivak's book deserves mentioning. – wckronholm Jun 5 '11 at 23:10

Wade's "Introduction to Analysis" is a great down to earth book that covers single variable and multivariate analysis.

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There is a recent book Functions of Several Real Variables. It has lot of good examples and exercises and is certainly theoretical.

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After looking far and wide for such a textbook, I finally settled on Serge Lang's Calculus of Several Variables.

It is rigorous, easy to read, and perhaps one of the best textbooks I have ever come across. Do give it a look.

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