Recommendations for Rigorous Multivariable/Vector Calculus Books

[avec_holl]

Does anyone have some suggestions for a good multi-variable/vector calculus book? I have a fairly reasonable math background - managed to self-teach myself calculus through Micahael Spivak's text Calculus pretty successfully - and I'm looking for something that's fairly rigorous. One of my math teachers recommended either Tom Apostol's text Calculus: Vol 2 or Spivak's text Calculus on Manifolds but I'm not sure if these texts are going to be beyond me (mathematically speaking). Price isn't really an issue but a cheaper text would be nicer. So, does anyone have some suggestions?

 

[Daverz]

Spivak is extremely dense. You might take a look at C. H. Edwards, Jr. Advanced Calculus of Several Variables, which is friendlier.

 

[slider142]

Spivak's "Calculus on Manifolds" is an excellent text, although it is better complemented if you learn some vector calculus first. I like Hubbard/Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms" myself.

 

[avec_holl]

I just looked into Vector Calculus, Linear Algebra, and Differential Forms and it looks like an excellent textbook - very well reviewed. Thanks for the suggestion!

 

[Landau]

You might also want to take a look at Vector Calculus by Marsden and Tromba, although lots of proofs of theorems are omitted (or only accessible on a website).
Do you know any linear algebra? Apostol's Vol.2 covers that. Keep in mind that Apostol's Vol.2 and Spivak's Calculus on Manifolds cover completely different subjects, so I wonder why your math teacher recommended one of those (as if they were interchangeable). If you think Spivak is too dense, there is this book Analysis on Manifolds by Munkres, which essentially is an expanded (less dense) version of Spivak.