[Math] Are “differential forms” an algebraic approach to multivariable calculus

Question

I am recently learning some basic differential geometry. As I understand, differential forms provide a neat way to deal with the topics in calculus such as Stoke's theorem. In order to define the differential forms, one needs to learn the basic concepts in multi-linear algebra.

Here are my questions:

• Are "differential forms" basically an algebraic approach to (multivariable) calculus?
• If the answer is YES, what would be a analysis counterpart and what are the advantages and disadvantages of these two approaches?
• If the answer is NO, how should I understand it properly? (EDITED: Are differential forms the only approach to multivariable analysis calculus?)

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MOTIVATION:

When I look for a rigorous theoretical approach to multivariable calculus, such as I asked in this question , differential forms are almost always included in the books recommended. I am thus wondering if this is the only approach to multivariable calculus.

I may not being asking a good question. Any suggestion for improving the questions above will be really appreciated.

Answer 1

Any approach to multivariable calculus necessitates some modicum of multilinear algebra ( for instance because the change of variables formula for integrals uses determinants), and the cleaner the interface between calculus and algebra, the better. The language of differential forms is a clean interface between the two, and this language generalizes from $\mathbb{R}^n$ to arbitrary manifolds. That is, differential forms are how calculus is done on manifolds. Or, to be more accurate, they are a convenient algebraic formalism for doing calculus on manifolds. To abandon differential forms is not to go to analysis, but to lose oneself in an ocean of notation.