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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.

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    $\begingroup$ My class used Vector Calculus by Marsden and Tromba. I can't necessarily recommend the book as the course wasn't all that great, but it was the book used by UT Austin math department (probably not anymore since this was over a decade ago). $\endgroup$ – Jared May 24 '14 at 1:12
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    $\begingroup$ why are you guys offering suggestions if you don't know what the GRE covers? $\endgroup$ – symplectomorphic May 24 '14 at 1:26
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    $\begingroup$ Stokes' and Green's theorems are fair game, I had them on the subject test I took $\endgroup$ – Silynn May 24 '14 at 1:27
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    $\begingroup$ Why do you consider Stewart's book as drivel for this exam? You do need to be able to compute quickly, say, double integrals. That is a skill you can pick up from computationally-oriented books on multivariable calculus. $\endgroup$ – KCd May 24 '14 at 1:30
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    $\begingroup$ I know someone who took only honors-level math classes in college, never saw how to calculate anything, and bombed the math subject GRE. If you deem books illustrating how to compute things as beneath you, be ready not to know how to answer computational questions on that test. The most important thing to know about that test is that you must work very quickly to get through all the questions. If you do not practice and keep track of time then you could easily find yourself running out of time with many questions left unanswered. $\endgroup$ – KCd May 24 '14 at 1:47
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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.


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.

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    $\begingroup$ I read and enjoyed Michael Spivak's Calculus before entering college. I now notice he has a text titled Calculus On Manifolds. Are you familar with this text? Is it relevant to the GRE? $\endgroup$ – user153025 May 24 '14 at 1:46
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    $\begingroup$ @user153025: manifolds aren't covered on the GRE. that book, though beautiful, is not the best for your needs. it is very compressed and not very computational. the beginning of Munkres's Analysis on Manifolds would be better, but you still don't need any of the manifold theory. $\endgroup$ – symplectomorphic May 24 '14 at 2:02
  • $\begingroup$ @user153025, not really. The motivation in Spivak is on proofs. Indeed, the last half of the book is devoted to chains and manifolds, which is completely irrelevant. I second that Munkres' Analysis on Manifolds is probably best. Or perhaps Folland's Advanced Calculus? $\endgroup$ – Chris K May 24 '14 at 2:03
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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.

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  • $\begingroup$ Just as a note: I've seen the integration of differential forms on the GRE exam / practice tests. $\endgroup$ – Chris K May 24 '14 at 2:05
  • $\begingroup$ @ChrisK: I haven't. point me to an example on an official practice test? $\endgroup$ – symplectomorphic May 24 '14 at 2:05
  • $\begingroup$ I also don't see where it would fit in the official list of topics. $\endgroup$ – symplectomorphic May 24 '14 at 2:08
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    $\begingroup$ @ChrisK: you must not know what a differential form is, or you're being excessively pedantic. nothing about that question requires knowing what a differential form is; the OP doesn't need to know about sections of the $k$-th exterior power of the cotangent bundle of a manifold. $\endgroup$ – symplectomorphic May 24 '14 at 2:11
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    $\begingroup$ @ChrisK: that is what I meant when I said you're being pedantic. you don't need to know about pullbacks to use a lower-level definition to compute a line integral given a parametrization. $\endgroup$ – symplectomorphic May 24 '14 at 2:22

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