While I learned multivariable calculus a few years ago, I have never felt I understand it well enough. Now I have time to go back and correct this. Since I have been through subjects like real analysis, I want this understanding to be more rigorous than the more intuition based learning that I initially encountered.
That lead me to consider differential form. Is it any good idea to learn things like line integrals, Lagrange multipliers and the like through differential forms?
Things that bother me the most about my understanding of multivariable calculus is, for instance, while I do know the purpose of the Jacobian in changing variables, I cannot prove it. In fact, beyond an intuitive understanding of what it does, I cannot show what exactly it does. Also, I have very little understanding of the equivalents of the fundamental theorem of calculus in multivariable calculus.
So, considering this... Should I embark upon trying to relearn calculus through differential forms?
I am considering A Geometric Approach to Differential Forms by David Bachman.
I'm not familiar with that particular book, but any good treatment of the subject should be very helpful to you. It will definitely answer your questions about the Jacobian. And it should also give you a good grounding in the general version of Stoke's theorem of which all the other "equivalents of the fundamental theorem of calculus" are special cases.