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(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.

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  • $\begingroup$ In order to encourage further recommendations of textbooks in multivariable calculus meeting the criteria above, I have added a bounty of 50 reputation. (The recommendations thus far are fantastic and I thank the answerers; however, I would like to encourage further people to add their recommendations to this list.) $\endgroup$ Jun 14, 2011 at 6:13
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    $\begingroup$ @Javier: it's nice that you're so enthousiastic, but given that this question is very personal and subjective, your wording seems a bit stong. I don't understand claims like 'by far the best books are..' or 'by no means I would use any other titles', unless you have read every single existing book on earth about the subject. And 'no one will ever need any other book on the topic' almost sounds arrogant. $\endgroup$ Jun 16, 2011 at 17:36
  • $\begingroup$ @Javier: I understand that your answers are (meant to be read as) subjective. By the way, I also appreciate your passionatic book recommendations as seen in in other MS questions. Rather than cultural or educational, it is probably a personal thing: I tend to be overly modest and careful about what words to use. I don't like the wording 'no one will ever need...' because it suggests that anyone who thinks he does need another book, has not 'seen the light' or must be at a lower level or something like that. Anyway, let's keep at that, everone has his own styling of expressing himself. $\endgroup$ Jun 16, 2011 at 19:24
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    $\begingroup$ By the way, have you read/seen Dieudonné's Treatise on Analysis? While it is of course a series of books, as a whole it is mindblowingly comprehensive (as I just found out), and I think his style is very clear. I also like the combination of Lang's Undergraduate Analysis+Functional Analysis. I am not familiar with Zorich, by the way. $\endgroup$ Jun 16, 2011 at 19:28
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    $\begingroup$ @wildildildlife: I don't think there was anything offensive or arrogant in Javier's posts, it's just sheer enthusiasm. I always like to hear strong and passionate opinions about something I am evaluating or don't have an opinion of yet. And even if I do, and it's different from someone elses, it's still nice to hear a different view. You shouldn't think of it as offensive, it's not meant as "telling you what to do and think", it's just happiness that such a good book exists and an opinion that it's overlooked. Good math books are extremely rare in my experience... $\endgroup$
    – Leo
    Jun 17, 2011 at 0:35

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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.)

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    $\begingroup$ @wildildildlife This is an excellent recommendation! I had never heard about this book before but it does indeed look like a book satisfying my criteria. Thank you very much! $\endgroup$ Jun 10, 2011 at 10:57
  • $\begingroup$ @wildildildlife I really do have to emphasize that this must be one of the best multivariable calculus textbooks that I have seen. I am really pleased that you mentioned this book! I do have to read it in more detail but the choice of topics (in the table of contents) is exactly what a mathematics student needs plus a lot more. $\endgroup$ Jun 10, 2011 at 11:08
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    $\begingroup$ @wildildildlife Judging from its TOC, The Duistermaat and Kolk book looks really good. Nice reccomendation! $\endgroup$ Jun 10, 2011 at 12:06
  • $\begingroup$ This book is lying on my desk now. I took the course of Kolk using this book a couple of years ago. Back then I was still a physics student, and this might have pulled me over. It is indeed wonderful. $\endgroup$
    – Thomas Rot
    Jun 15, 2011 at 15:14
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    $\begingroup$ @Amitesh: You're welcome, and thanks for the bounty. It is indeed an amazing book :) Kolk en the late Duistermaat have done a remarkable job. [You might want to take a look at their other books on Lie groups and Distribution theory.] $\endgroup$ Jun 21, 2011 at 9:45
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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.

I was impressed to no end by his table of contents: enter image description here enter image description here

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    $\begingroup$ I was going to answer exactly those books!! I AGREE COMPLETELY. These titles are so amazingly good and complete, no one will ever need any other book on the topic. Callahan in particular is extremely well-done, filled with figures and graphics and geometric motivations. Zorich is more complete as it deals with one variable and some general Banach analysis and other more specialized topics. THEY ARE THE ULTIMATE REFERENCE FOR RIGOROUS THEORETICAL ADVANCED CALCULUS. $\endgroup$ Jun 14, 2011 at 13:39
  • $\begingroup$ +1 Thank you very much! This looks like a very good reference. $\endgroup$ Jun 15, 2011 at 5:47
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    $\begingroup$ I may have just seen the face of God in those tables of contents. +1. $\endgroup$ Jun 16, 2011 at 3:31
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    $\begingroup$ +1. I REALLY don't think any of us can improve on these recommendations.There are a few books just as good,but certainly not BETTER then these. $\endgroup$ Mar 19, 2012 at 17:23
  • $\begingroup$ please delete the expletive $\endgroup$
    – KingLogic
    May 27, 2018 at 22:01
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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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    $\begingroup$ Thanks Vishal! It's been a while since I asked this question (2 years!) but it's great to accumulate so many good references. I think this book is one of the best out of all those suggested by the answers! $\endgroup$ Sep 30, 2013 at 0:20
  • $\begingroup$ It is a lovely book indeed. I hope more and more users discover this book and benefit by it. $\endgroup$ Sep 30, 2013 at 4:19
  • $\begingroup$ @AmiteshDatta , then you should accept this answer instead of the other one. $\endgroup$
    – Julien__
    Apr 9, 2015 at 12:50
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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

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  • $\begingroup$ +1 Thank you very much! Could you please give a link to an online version of this book if possible? I cannot seem to view the book online. (The table of contents would also be useful.) $\endgroup$ Jun 10, 2011 at 11:27
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    $\begingroup$ @Shahab SHHHHHHHHHHHHHHHHH! THAT SITE DOES NOT EXIST! You want them to shut it down for copyright breach?!?LOL $\endgroup$ Nov 7, 2011 at 7:23
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    $\begingroup$ @Mathemagician1234 The site was shut down at some point in the past week ... $\endgroup$ Feb 19, 2012 at 1:39
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    $\begingroup$ @Amitesh Don't remind me. The corporate police state is encroaching further and further on the ability of poor students to have access to materials they can't afford. $\endgroup$ Mar 19, 2012 at 17:25
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    $\begingroup$ I personally don't like Wade... messy arrangement, ugly proofs, lack of important material, and bad writing... $\endgroup$
    – Eric
    Oct 26, 2017 at 16:25
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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.

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  • $\begingroup$ I will add that the notation for Derivatives used in this book has fallen out of favor, though I am using a first edition, does anyone know if the second edition is updated in this respect? $\endgroup$
    – user9352
    Jun 17, 2011 at 12:23
  • $\begingroup$ In the 2nd edition, at page 347, he write $\mathbf{T_c}:\mathbb{R}^n\to \mathbb{R}^n$ for the total derivative at $\mathbf{c}$ of $f:S\subset\mathbb{R}^n\to \mathbb{R}^n$. It's probably better to somehow use $f$ in the notation, like $\mathbf{T_c}f$. This would agree with what some people use for the 'tangent mapping' of a function between manifolds. But he does use $\mathbf{Df(c)}$ for the Jacobian matrix. $\endgroup$ Jun 18, 2011 at 11:08
  • $\begingroup$ If you can get an old hard copy that still has the library smell. Oh man, that is living. Also, in a similar vein, get a copy of Hans Sagan's advanced calculus text. These are both formidable texts from a simpler time with better students (in US schools anyway) $\endgroup$ May 30, 2018 at 4:53

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