Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.
After reading the wiki article "Second partial derivative test", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.
Motivated by this, I'd like to put the question here:
What are the usual references for the
theoretical treatment for
Multivariable calculus?
Best Answer
I usually think of multivariable calculus as being divided into four parts:
For differentiation, you can use Principles of Mathematical Analysis by Rudin (Chapter 9). Actually, this text also discusses integration and vector calculus (Chapter 10), but I personally found Rudin's treatment of such hard to follow when I was first learning the subject.
For differentiation, integration, and vector calculus you can use Calculus on Manifolds by Spivak, or Analysis on Manifolds by Munkres.
For curves and surfaces, you can use basically any book on elementary differential geometry. One of the most widely-used is Differential Geometry of Curves and Surfaces by do Carmo, though I highly recommend Elementary Differential Geometry by Pressley.
A few remarks as to where these topics end up going, with a slant towards differential geometry:
Differentiation of functions $f\colon \mathbb{R}^n \to \mathbb{R}^m$ can be generalized to differentials (a.k.a. pushforwards) of maps $f\colon M \to N$ between differentiable manifolds.
All of the references I mentioned above treat multiple Riemann integration of functions $f\colon \mathbb{R}^n \to \mathbb{R}$. This can be generalized to multiple Lebesgue integration via consideration of product measures.
The theory of curves and surfaces leads naturally towards Riemannian geometry, though certainly other branches of geometry also generalize this subject.
In vector calculus, one discusses line integrals and surface integrals of both functions and (co)vector fields. To my mind, the intricacies of such processes are not fully realized until one studies the integration of differential forms on differentiable manifolds.