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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.

Martin Sleziak
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user153025
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    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). – Jared May 24 '14 at 01:12
  • Another book that might help would be Applied Partial Differential Equations - Haberman (this was the book used in my PDE's class--again, I cannot really give a recommendation as I blew that course off). – Jared May 24 '14 at 01:20
  • Disclaimer: I have no idea what's on the Math GRE, but I would assume Stoke's Theorem and Green's Theorem would be on it...in which case either straight Vector Calculus or PDE's would probably be useful. – Jared May 24 '14 at 01:23
  • @Jared I think Stoke's theorem and Green's theorem are fair game on the exam. – user153025 May 24 '14 at 01:24
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    why are you guys offering suggestions if you don't know what the GRE covers? – symplectomorphic May 24 '14 at 01:26
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    Stokes' and Green's theorems are fair game, I had them on the subject test I took – Silynn May 24 '14 at 01:27
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    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. – KCd May 24 '14 at 01:30
  • @KCd I am really looking for a book that is aimed more towards upper level math majors, but if James Stewart is what the doctor prescribes, then that is what I will work through. Also, I removed that bit from the post. – user153025 May 24 '14 at 01:37
  • But your point is that you had not taken multivariable calculus, so your concern seems to be that you do not know the types of things learned specifically in multivariable calculus rather than upper level courses such as real analysis. I too did not take a standard multivariable calculus course in college, but I had read about it before then and I knew how to find double integrals in various ways and how to compute eigenvalues and eigenvectors of actual square matrices. If this is the type of material you never learned how to compute, get a book that gives you practice making computations. – KCd May 24 '14 at 01:44
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    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. – KCd May 24 '14 at 01:47
  • @symplectomorphic - The OP asked for multivariate calculus. I can guess what would be on the GRE based on that the questioner is asking about this topic. And my assumption appears to be correct about Stokes' and Green's theorem (since those are the essential concepts of Vector Calculus--and Green's theorem is a very important part of PDEs). Any introductory Calculus book will cover taking double, triple, and higher integrals. – Jared May 24 '14 at 01:47
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    Removing the part about Stewart's book from your question now makes my comment about it look slightly out of place. Be careful about editing questions after people respond to particular parts. – KCd May 24 '14 at 01:51
  • @Jared: Yep, that's what I used, still look at it now and then some 30 years later (ye gods!), always thought it was solid – MPW May 24 '14 at 01:51
  • @symplectomorphic: OP should learn the subject, not just target the exam. He may think a "crash course" is what he needs (and it may be what he needs for that exam), but think about how that will affect his professional life in the future. Would YOU want somebody like that teaching that course to you with such minimal understanding of the real content? Boo. – MPW May 24 '14 at 01:56
  • @MPW: did you see the books I recommended in my answer... ? I'm well aware he needs to learn it. I just didn't see the point of someone cluttering up this forum with "disclaimers" that they don't actually know whether their advice is useful. – symplectomorphic May 24 '14 at 02:04
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    @MPW I am hoping to learn multivariable calculus in the four weeks between when my REU ends and when school starts back up again. This is all I meant by "crash course". – user153025 May 24 '14 at 02:07

2 Answers2

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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.

Jesse Madnick
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    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? – user153025 May 24 '14 at 01:46
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    @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. – symplectomorphic May 24 '14 at 02:02
  • @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? – Christopher K May 24 '14 at 02:03
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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.

symplectomorphic
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  • Just as a note: I've seen the integration of differential forms on the GRE exam / practice tests. – Christopher K May 24 '14 at 02:05
  • @ChrisK: I haven't. point me to an example on an official practice test? – symplectomorphic May 24 '14 at 02:05
  • I also don't see where it would fit in the official list of topics. – symplectomorphic May 24 '14 at 02:08
  • How about Question 41 at https://www.ets.org/s/gre/pdf/practice_book_math.pdf? You have to parameterise and integrate. – Christopher K May 24 '14 at 02:09
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    @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. – symplectomorphic May 24 '14 at 02:11
  • But he does need to know how to parameterise differential forms. I guess this could be seen as the "substitution method", although I learned line integrals under the tent of "differential forms". – Christopher K May 24 '14 at 02:14
  • @ChrisK: you are confused. differential forms are not things you parametrize; embedded submanifolds of Euclidean space are. what you mean is that he needs to parametrize the unit circle. – symplectomorphic May 24 '14 at 02:14
  • The OP would need to pull back the 1-forms by means of the dual transformation to express $dx$ and $dy$ in terms of $dt$. I concede that the parameterisation is of the unit circle not of the differential forms themselves. – Christopher K May 24 '14 at 02:20
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    @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. – symplectomorphic May 24 '14 at 02:22
  • Fair enough. I learned the computation from a more theoretical approach. I thank you for correcting me on the point that we are parameterising the unit circle and not the differential forms. – Christopher K May 24 '14 at 02:23