Multivariable Calculus for GRE

Question

This is going to sound strange, but I am a third year math major who never took multivariable calculus (despite having taken courses on Galois and Lebesgue theory, etc). I plan to take the GRE next year and need to learn multivariable calculus (and analysis) over the summer.

What are some good textbooks for a quick crash course on multivariable calculus that would be germane to the GRE Subject Exam?

Edit: How about this book, for example? Regarding its reviews

Edit 2: I have a pretty solid grasp of undergraduate linear algebra (having taken two courses in linear algebra and TAing the lower level course of the two). As such, the book may assume linear algebra as a prerequisite.

Answer 1

Apostol is a nice reference. The incredibly informative book by Hubbard, which uses much more modern and conventional notation than Apostol, integrates multivariable calculus with linear algebra, but it also discusses differential forms and manifolds, which you don't really need to know for the GRE. (Hubbard's book goes just a little more in depth than the book by Ted Shifrin, who frequently posts in this forum. But his book also includes differential forms.)

You might also find the 18.02 material at MIT OpenCourseWare useful. The course isn't theoretical; it focuses on computational fluency.

Answer 2

I learned multivariable calculus from Paul's Online Math Notes.

If you want a physical textbook, I second Jared's recommendation of Marsden & Tromba's Vector Calculus. It has a somewhat more theoretical flavor to it than James Stewart's books.

Another standard text is Edwards & Penney, which I've used to tutor students. However, it's essentially on the same plane as Stewart.

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Now for a few comments.

First of all, if you're studying for the GRE, then you might not want a textbook that emphasizes theory. First and foremost, you need to be able to solve basic problems and calculate things, so in that sense a book like Stewart's might actually be the most appropriate.

Speaking of Stewart, not everyone holds his books in such disregard. I don't love his textbooks personally, but I do understand and appreciate why they're the standard.

Finally, I'd like to take a second and exude some enthusiasm for the subject. Multivariable calculus is one of my favorite areas of math, and was crucial in helping me develop intuition for (and interest in) differential geometry. In my (admittedly limited) experience, undergraduates skipping multivariable calculus and ordinary differential equations is not too atypical. However, I would hope that all serious math students eventually go back and learn both subjects, appreciating them for their inherent beauty.