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I would like to buy a book to study multivariable calculus. Currently, the texts I have in mind are:

  1. Vector Calculus, Linear Algebra, and Differential Forms A Unified Approach by Hubbard & Hubbard
  2. Multivariable Calculus with Applications by Lax & Terrell
  3. Functions of Several Real Variables by Moskowitz & Paliogiannis

I want a book that has a clear expositions of the subjects of multivariable calculus. Also, I would like a book that avoids leaving proofs as excercises to the reader, or at least that does not do it most of the time. If possible, a book that also contains multiple examples/exercises with (fully) detailed explanations/solutions to at least some of the examples/exercises. I do not mind a rigorous approach to the subject as long as the content is explained with detail.

Which of the books mentioned above fits best the description? Also, if you have other books in mind, feel free to recommend them as well. Thanks in advance!

Note: I have taken two proof-based calculus classes and one proof-based linear algebra class.

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    $\begingroup$ I can suggest an addition: Advanced Calculus by Loomis and Sternberg is a classic and freely available $\endgroup$
    – balddraz
    Commented Jun 12, 2023 at 21:13
  • $\begingroup$ @athalhaids I've hear about that text, but I've also hear that it can be too advanced. Is this true? $\endgroup$
    – user926356
    Commented Jun 12, 2023 at 21:49
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    $\begingroup$ it is advanced, yes, in the sense that they do things very generally (differential calculus in Banach spaces, they treat ODEs very generally etc), but generality in and of itself is nothing to be feared, especially since you say you’ve had proof-based courses before. Also, it is very well explained (my favorite for learning the material (I used Loomis, Spivak and Munkres essentially simultaneously)). If you like, you can accompany it with Bamberg and Sternberg, which is a toned-down version, but has many interesting physical applications, and also more pictures. $\endgroup$
    – peek-a-boo
    Commented Jun 12, 2023 at 21:52
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    $\begingroup$ @user926356 I suggested it since you already took proof-based calculus and linear algebra classes. Also for a book that advanced, it is not terse (which is sadly a common feature at higher levels e.g. Spivak's Calculus on Manifolds): it is very detailed and has many well-developed examples. $\endgroup$
    – balddraz
    Commented Jun 12, 2023 at 22:00
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    $\begingroup$ Be careful with Hubbard and Hubbard. It is very idiosyncratic and, at least in the early editions, exercises are uneven and not plentiful enough. I started teaching a course out of it and in short order wrote my own text. (There are also lectures from my course, following my book, linked in my profile.) Both texts are somewhat unique in that they integrate linear algebra and multivariable calculus/analysis. Since you’ve already studied linear algebra, you might skip tgat or review it lightly. My approach emphasizes the geometry in linear algebra; that is not true of many books and courses. $\endgroup$
    – Ted Shifrin
    Commented Jun 12, 2023 at 22:24

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How about Multivariable Calculus, by Don Shimamoto? It is a rigorous text, free, suitable for proof-based multivariable calculus, and has solutions to odd-numbered problems in the back.

I just took a quick browse through the book. It's true that he does leave some proofs of important theorems to the exercises, but it looks like he also leaves a number of hints. Still, I think it's a good choice.

When I took Multivariable Calculus, we used the book by Colley, but I think I would have preferred learning from Shimamoto's book.

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    $\begingroup$ Thanks! I'll take a look at it $\endgroup$
    – user926356
    Commented Jun 12, 2023 at 22:40

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