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 thetheoretical treatment forMultivariable calculus?