I'm currently enrolled in Calculus II (Integral Calculus), and I am slowly working my way through Spivak's Calculus.

I intend to take Multivariable Calculus next semester. The course description is as follows:

I was wondering if there were any clearly recommended books, similar to Spivak, for the above-mentioned class/content. I've got a couple of calculus textbooks that include multivariable content, but I'm looking for something that takes a more rigorous approach as a supplement.

Also, since I'm here already, I'll be taking a Discrete Mathematics course with the following course description:

When I studied Number Theory, back in the 80s, we used the text "An Introduction to the Theory of Numbers" by Niven and Zuckerman https://www.amazon.com/Introduction-Theory-Numbers-Ivan-Niven/dp/0471625469 --can you believe it, I still have the book. I can't tell you whether number theory and discrete math are exactly the same thing, but they are certainly related. I fell asleep in that class a lot. Basically, back then number theory had one application: Making soundproof rooms. However it is said that number theory is the queen of mathematics.

From my reading of Wikipedia, it looks like Discrete Mathematics covers a lot of ground of which, Number Theory is a small piece which also interacts with Continuous Mathematics. Also I see that number theory has application in cryptography (factoring of large numbers). Do you know what you'll be studying in your class? That might give us a better idea of which books to recommend. Perhaps a better recommendation would be Godel Escher Bach: An Eternal Golden Braidhttp://www.google.com/url?sa=t&sour...o_28BA&usg=AFQjCNF5uApwP33mpSyy7w5YpA5UT2FnBw by Douglas Hofstadter. EDIT: Ignore the book on Number Theory, I just read your quoted section. Definitely get Godel Escher Bach (one of the less expensive books you'll be buying).

Well, Spivak's Calculus on Manifolds is the likely suggestion.

However, the theory behind calculus in R^n is usually written in differential forms and a bunch of other fancy math. This is how the aforementioned book is, how Munkres' Analysis on Manifolds is, how chapters 9 & 10 in baby Rudin are, etc.

They'll cover the same thing as your calc III class will, but in a very different way.

Wendell Fleming's "Functions of Several Variables" is pretty decent, but nowhere near as fun to read as Spivak. The exercises are kind of blah, too. Still worth a look to see if the style agrees with you.

If that one seems too elementary and you're feeling more ambitious, check out Loomis and Sternberg, "Advanced Calculus." That one will put hair on your chest. I think it's out of print, but the author has made it available as a PDF for free: