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So I'm wanting a solid math book for Christmas. I have a solid background in Calculus and am currently working through baby Rudin. I really want a rigorous book dealing with multivariable calculus and linear algebra. How well does Apostol II do this? Would this be a good continuation book from my current study? If not, what book would you suggest?

EDIT: I really like Apostol's Multivariable Calculus and Linear Algebra book and its format. Will this book accomplish my purpose? Do you think it is a good book?

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This might help – Joe Johnson 126 Nov 6 '11 at 23:29
Or this one: – Jesse Madnick Nov 7 '11 at 1:33
up vote 9 down vote accepted

Ted Shifrin's book Multivariable Mathematics is actually one of the best books of this type while not being very well known. Unfortunately, it's very expensive, so unless you can find it in your library, I would choose something else.

Otherwise I would just recommend Spivak's Calculus on Manifolds together with some linear algebra book. For linear algebra I would recommend either Axler's Linear Algebra Done Right or Linear Algebra by Fiedberg, Insel and Spence

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how rigorous is Shifrin's book? – analysisj Nov 6 '11 at 23:53
Both of those linear algebra books are fantastic. – Brendan Cordy Nov 7 '11 at 2:46
Shifrin's book is very rigorous. As it's doing some pretty difficult differential geometry at a few points, it has to skip some steps. For the standard material like differential forms etc. it's not cutting corners. – pki Nov 7 '11 at 3:19

The best book on this for beginners is John and Barbara Hubbard's Vector Calculus,Linear Algebra And Differential Forms: A Unified Approach, 3rd edition. It is an incredibly rich treasure trove of results, all presented with full rigor and with a striking amount of originality. What I love best about this book is that it balances rigorous mathematics with applications better then any textbook I've ever seen. It is a bit pricey, but now as a gift for Christmas, it's perfect since it'll be a lifetime investment for a serious student of mathematics.

It's simply awesome, you're wasting your time buying anything else. (I'm not saying there's not any other good books on rigorous multivariable calculus-there certainly are. I just think any other book for a beginner is waste of money to buy. )

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I don't know how much improved the subsequent editions of the Hubbards' book is but the statement on p. 500 of the first edition "We can't draw you a picture of a form." was a bit of a let down. Much more ancient books like MTW Gravitation or Harold M. Edwards' "Advanced Calculus: A differential forms approach" do manage that. The latter also builds up the forms material at much more manageable pace for an undergraduate, although I found it too slow going. – Fizz Jan 17 at 3:46

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