I start UChicago in the fall and would like to test into their Honors Analysis sequence (the test being in late September). I’m about midway through Spivak and I really enjoy it. I hope to finish the book in the next few weeks but I'm conflicted about what to go on to next. I would like to do Lang’s Linear Algebra (and hopefully start Herstein’s Topics in Algebra if I have time) but I’m not sure if Lang’s book on multivariable calculus would be better preparation for the test. I’ve heard Spivak’s Calculus on Manifolds is good as well. With these considerations in mind, what is likely to be the best/wisest course of action (if you have experience with Chicago’s Honors Analysis, all the better).

Also, I hope this question doesn't get flagged. There are many students in similar situations (making hard decisions about what areas to study after calculus) for whom this question would likely serve well and provide guidance.


closed as off-topic by José Carlos Santos, cmk, Cesareo, Widawensen, DMcMor Jul 9 at 16:36

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    $\begingroup$ An off-topic remark: I think Spivak's Calculus on Manifolds is a bit controversial. I agree with an Amazon reviewer's comment: "every budding mathematician should be aware of Spivak's treatment of Stokes' theorem". It's a bit like Bak and Newman's Complex Analysis, which contains a proof of the prime number theorem using complex analysis. Both books have some ground breaking works, but that doesn't mean that they are necessarily good for private study. $\endgroup$ – user1551 Jul 9 at 11:37
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    $\begingroup$ I would recommend a simultaneous study. I mean, knowing Linear Algebra is an essential thing to do in Mathematics, once you are well formed on Vector Spaces you should be able to land (better) the concepts seen in Multivariable Calculus. I'm not familiar with the test but also I would recommend two more books: The first volume of Calculus by Apostol and Friedberg's Linear Algebra, just because sometimes two books are not enough and personally I think that Spivak's on Manifolds gives a lot of things by granted. $\endgroup$ – Eric Toporek Jul 9 at 14:46
  • $\begingroup$ You should learn both subjects together, because in the beginning, they complement each other very well. You said you liked Spivak's calculus, so I'd suggest you take a look at Loomis and Sternberg's Advanced Calculus (available for free online). In the preface to chapter 1, they say the following: "The calculus of functions of more than one variable unites the calculus of one variable, which the reader presumably knows, with the theory of vector spaces, and the adequacy of its treatment depends directly on the extent to which vector space theory really is used. (cont) $\endgroup$ – peek-a-boo Jul 9 at 22:14
  • $\begingroup$ "The theories of differential equations and differential geometry are similarly based on a mixture of calculus and vector space theory. Such "vector calculus" and its applications constitute the subject matter of this book, and in order for our treatment to be completely satisfactory, we shall have to spend considerable time at the beginning studying vector spaces themselves." As you can see, linear algebra is a huge part of multivariable calculus. So if you really had to choose only one, I'd say go for linear algebra, because it's a prerequisite. (I benifitted greatly from such an approach) $\endgroup$ – peek-a-boo Jul 9 at 22:16