I want to learn multivariable calculus and I need a book suitable for self-study.

I looked around on Amazon and found two books that seem to contain the right material:

Clark Bray: Multivariable Calculus


Larson/Edwards: Multivariable Calculus

Unfortunately, the first book is described as targeted at engineering majors (by the author himself on his website). And the second book seems to be too colorful and fancy so that I suspect it's also for engineering majors.

I major in pure maths.

Can I (a maths major) use either of these books (for engineering) to learn multivariable calculus?

And if not, which multivariable calculus book is suitable for a maths major?

I am looking for an undergraduate low-level text with lots of exercises and solutions. I want to practice the material as well as study proofs.


Why I am worried that these books might be no good to me:

One thing is that since they seem to be written for engineers I am worried that there are no or only few proofs. The other thing that I'm worried about is that they don't cover the same scope of material like a book targeted at maths majors.

Edit 2

After some more searching I found

Edwards: Advanced Calculus of several variables

It looks good topic-wise but its title suggests that it's advanced and I only know first year real analysis in one variable. This book contains a chapter about differential forms (isn't that rather advanced?) Also, it does have exercises but no solutions.

I am worried that this book is too advanced for me.

Does anyone have any experience with this book?

  • $\begingroup$ you might consider Susan Colley's Vector Calculus. I would say this is much more targeted towards math majors than the standard texts. Also it has a nice low-level differential forms chapter at the end. amazon.com/Vector-Calculus-Edition-Susan-Colley/dp/0131858742 I have free notes at supermath.info/CalculusIIIf2014.pdf you might find useful. $\endgroup$ – James S. Cook Jan 15 '15 at 1:25
  • $\begingroup$ @JamesS.Cook This book seems to have quite a few bad reviews. Thank you for the link to your notes but having exercises with solutions is essential to me and this is why I am looking for a book. I did notice that your notes contain some worked examples which I could use like an exercise with solution but I didn't see many of them... $\endgroup$ – user174981 Jan 15 '15 at 3:38
  • $\begingroup$ Bad reviews on Amazon don't mean much. Often you just see the slackerclass bemoaning the existence of a book that expects they think. That said, supermath.info/math231mission1f2013soln.pdf is a solution to those end of Chapter problems in my notes. Also, there is more at: supermath.info/MultivariateCalculus.html $\endgroup$ – James S. Cook Jan 15 '15 at 4:35
  • $\begingroup$ You could use Vol. 2 of Calculus by Apostol. Differential forms, in some sense, are a higher-dimension generalization of the part of calculus dealing with grad, div, curl, etc. Whether you should use a book that goes straight to dimension $n$, or one that focuses on dimension $\geq 3$ (leaving differential forms for a later course) will depend partly on what the expectations are in your university. Apostol's book has mostly complete proofs (except for particularly difficult points), but doesn't study manifolds or differential forms. $\endgroup$ – user208259 Jan 17 '15 at 6:33

I strongly recommend the book "Multivariable Mathematics" by our very own Ted Shifrin.

In terms of difficulty, it is suitable, in my opinion, as a first "proof based" math course. He does include the proofs of almost all of the important theorems in the text.

The book treats the needed linear (and multilinear!) algebra as needed for the development of multivariable calculus in the proper context, as well as an introduction to some topological ideas like compactness. A wide range of very nice problems, and beautiful exposition.

You will also learn the calculus of differential forms in a particularly down to Earth way.

I think this is exactly what you are looking for.

  • $\begingroup$ Thanks for the endorsement. Just before I retired, my students filmed the complete course I taught at UGA, so you can find accompanying lectures (and many different examples) on YouTube. $\endgroup$ – Ted Shifrin May 14 '20 at 23:00

Spivak "Calculus on Manifolds" is a favorite of mine. Do not be put off by the word "manifold"- they don't enter the picture till the last chapter. The presentation of Spivak is pretty elegant.

You can also have a look at Apostol's real analysis book. This book contains a lot of real analysis, and a few chapters on multivariable calculus. I personally find the treatment a bit less elegant than Spivak, but this book gives you a lot of examples.

(Also you might get copies of these books on the internet for free- but do not quote me on that ;-) )

  • $\begingroup$ Are you saying that both books I mention are unsuitable for a maths major? $\endgroup$ – user174981 Jan 15 '15 at 1:19
  • $\begingroup$ @user174981: I do not have any experience with the books you mentioned. The first one you say is for engineering students, so I guess it wouldn't be very proof based. If your question was regarding those two specific books, then I apologize. I can delete my answer, so as to not be misleading for others. $\endgroup$ – voldemort Jan 15 '15 at 1:21
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    $\begingroup$ I heatedly agree the books you mention are good to study, but, I think he is looking for something a bit lower level. These are the right books for him to read after he studies the basic multivariable material. For example, you probably wouldn't want to learn multivariate integration from these books $\endgroup$ – James S. Cook Jan 15 '15 at 1:33
  • $\begingroup$ @JamesS.Cook: You may be right about integration. I did learn integration from Spivak, but I had a lot of help from my teacher. Also, most (all?) of the time I integrated in multivariable calculus, I just used Fubini (after showing that we can apply Fubini), and then it was a case of single variable integration. $\endgroup$ – voldemort Jan 15 '15 at 1:36
  • $\begingroup$ I agree it's not a big deal, but, if he is mainly trying to learn multivariate computation then those books are heavily focused on theory. Maybe Marsden and Tromba would be good. Or, Salas and Hille, or Spivak's Calculus (which sadly I do not own)... $\endgroup$ – James S. Cook Jan 15 '15 at 4:37

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