(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.
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 Rn. 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.