(Theoretical) Multivariable Calculus Textbooks [duplicate]

(Note that I have used bold text frequently simply to highlight the key points of my question for those who do not have the time to read through it thoroughly (it is not very long, however); I hope this is not considered offensive.)

There are many textbooks on multivariable calculus. However, some textbooks on multivariable calculus do not focus very much on the theoretical foundations of the subject. For example, a textbook might state a result along the lines of "the order of partial differentiation is immaterial" without proof and ask the student to use this rule to solve problems. Similarly, theorems such as those due to Green and Stokes are often not proved in their full generality.

Therefore, I ask the following question:

What are some good theoretical multivariable calculus textbooks?

Since "theoretical" is somewhat ambiguous, let me state the following criteria which I would like a "theoretical" textbook on multivariable calculus to satisfy:

• The textbook should be rigorous and it should not state a theorem without proof if the theorem is proved in at least one other multivariable calculus textbook. (Of course, the textbook may omit certain theorems; however, this criterion at least ensures that major theorems in multivariable calculus are not stated without proof and used purely for the sake of computations. Also, this criterion permits the textbook to state an interesting theorem if it is beyond the scope of all multivariable calculus textbooks.)
• The textbook should be primarily based on developing the theoretical foundations of multivariable calculus; therefore, applications such as learning how to compute the partial derivative of a function, learning how to solve extremum problems, learning how to compute etc. should be kept to a minimum. In particular, the textbook can assume that the reader has already seen at least an informal treatment of the subject where these aspects are emphasized.
• The textbook should have a rigorous treatment of differentiability in $n$-dimensional Euclidean space (e.g., the inverse and implicit function theorems should be proven), Riemann integration in $n$-space, and differential forms (e.g., Stokes theorem should be proven). It would also be a bonus if the book treated the general concept of a manifold.
• Textbooks with minimal prerequisites are preferred; however, please feel free to suggest books meeting the above criteria even if the prerequisites are quite demanding.
• Finally, it would also be preferable, but not essential, for the book to only treat multivariable calculus.

Examples of books meeting the above criteria: "Analysis on Manifolds" by James Munkres, "Principles of Mathematical Analysis" by Walter Rudin, and "Calculus on Manifolds" by Michael Spivak.

Although I have studied theoretical multivariable calculus already (four years ago), I could never find "the perfect book" (relative to myself, of course). Every book has its virtues; Rudin for its elegance, Munkres for its beautiful exposition, and Spivak for its "quick and dirty" approach. I am hoping that someone will be able to suggest a book that (relative to myself) is "perfect". Also, this question can be useful to other students who have not yet studied the subject and wish to learn it.

Thank you very much for all answers! Please do feel free to suggest as many books as you can think of so we can form a big list. Also, please try to explain why a particular book is good or at least why you think it is good. I suppose it is fine to suggest a book that is already suggested provided you have a different view as to why the book is good.

Solution 1:

A book fitting your description quite well is

Multidimensional Real Analysis by Duistermaat and Kolk, a 2-volume set: Differentiation and Integration.

It has rigorous, slick proofs, is highly theoretical, but with lots of (advanced) examples and many, many exercises. Much attention is given to the Inverse and Implicit Function theorem, and submanifolds of $\mathbb{R}^n$. The book is used in a second-year course at Utrecht University. I have to admit that it was quite hard to read for me when I took the course. But it is great as a reference, and years later I still consult it now and then.

Another nice book is Loomis & Sternberg - Advanced Calculus (freely available from Sternberg's website.)

Solution 2:

This is a lazy answer from a guy, who in his first and second year felt the need for an excellent exact rigorous and intuitive book in calculus, both one and several variables.

I haven't read any of the following books, but I have browsed through them.

• Mathematical Analysis I, Zorich, amazon, 578 pages
• Mathematical Analysis II, Zorich, amazon, 688 pages
• Advanced Calculus, Callahan

I was impressed to no end by his table of contents:

Solution 3:

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

Solution 4:

The second half of the book "An Introduction to Analysis" by William Wade supplies what you ask. (The first half is single variable.) There is even a section on elementary Fourier analysis.

Table of Contents (for multivariable part):

8 Euclidean Spaces

8.1: Algebraic Structure

8.2: Planes and Linear Transformations

8.3: Topology of $\mathbb{R}^n$

8.4: Interior, closure, and boundary

9 Convergence in $\mathbb{R}^n$

9.1: Limits of sequences

9.2: Limits of functions

9.3: Continuous functions

9.4: Compact sets

9.5: Applications

10 Metric Spaces

10.1: Introduction

10.2: Limits of functions

10.3: Interior, closure, boundary

10.4: Compact sets

10.5: Connected sets

10.6: Continuous functions

11 Differentiability in $\mathbb{R}^n$

11.1: Partial derivatives and partial integrals

11.2: Definition of differentiability

11.3: Derivatives, differentials, and tangent planes

11.4: Chain rule

11.5: Mean Value Theorem and Taylor's Formula

11.6: Inverse Function Theorem

11.7: Optimization (Lagrange Multipliers)

12 Integration on $\mathbb{R}^n$

12.1: Jordan regions

12.2: Riemann integration on Jordan regions

12.3: Iterated integrals

12.4: Change of variables

12.5: Partitions of unity

12.6: Gamma function and volume

13 Fundamental Theorem of Vector Calculus

13.1: Curves

13.2: Oriented curves

13.3: Surfaces

13.4: Oriented surfaces

13.5: Theorems of Green and Gauss

13.6: Stokes's Theorem

14 Fourier Series

14.1: Introduction

14.2: Summability of Fourier series

14.3: Growth of Fourier coefficients

14.4: Convergence of Fourier series

14.5: Uniqueness

15 Differentiable Manifolds

15.1: Differential forms on $\mathbb{R}^n$

15.2: Differentiable manifolds

15.3: Stokes's Theorem on manifolds

Solution 5:

I've been reading Mathematical Analysis by Tom Apostol to review much of this material. He seems to discuss and explain things more than Rudin. He even includes a few pictures, which are understandably primitive given the age of the book but I think they help. He also seems to have a lot of exercises.