Can anyone recommend such a textbook (Vector Calc)

[nicksauce]

I would like an advanced calculus that covers the following
-A bit on vectors
-A bit on functions of several variables, Chain Rule
-Vector valued functions and things like Curvature, and Binormal vector etc.
-Multi-var Taylor's Theorem
-If possible, A bit on differentiation under the integral (Leibniz rule)
-Extrema of MV functions (Lagrange multipliers, second derivative test, etc)
-Multiple integrals
-Line integrals and surface integrals
-Stoke's theorem, Divergence theorem, Green's Theorem
-If possible a bit on Differential Forms and Manifolds
And the text should ideally have a good level of rigor, meaning that there is more of a focus on proving theorems and propositions than on examples.

Any suggestions?

Cheers, Nick

[axeae]

I'm not sure if it's exactly what you're looking for, but it's a great book and might be worth looking into---Spivak's Calculus on Manifolds.

[quasar987]

FYI, Calc on Manifolds have everything in your list except:

-Curvature, and Binormal vector etc.
-Multi-var Taylor's Theorem
-Extrema of MV functions (second derivative test, etc)

Leibniz's rule and Lagrange multipliers are treated in an exercise.

[las3rjock]

I have not yet had the opportunity to work through it myself, but by all accounts, Spivak's Calculus on Manifolds is the classic text. It is very brief, though, so it might be a good idea to supplement it with a more detailed text. One inexpensive option is C. H. Edwards, Jr.'s Advanced Calculus of Several Variables. It's available in a Dover edition, and it's been used as the text for the second course in multivariable calculus at the University of Michigan.

[nicksauce]

Well I found "Vector Calculus" by Colley at my library, and it does have everything I would like, although doesn't seem entirely rigorous, so perhaps I will use it along with Spivak. Thanks for the comments!

[mathwonk]

i also like fleming's functions of several variables, and the very high level book by loomis and sternberg is free at sternberg's website.

another classic is williamson crowell and trotter.