Differential Forms: High Level Approach to Real Analysis?

Question

I am currently skimming through the differential forms book by Edwards. I was wondering whether real analysis is basically just a special case of differential forms? I am learning about flows, 1-forms, 2-forms, Fundamental Theorem of Calculus, etc...

These seem to be analogous to the topics in a typical real analysis course. So is the differential forms approach just a high level approach to real analysis? Would I appreciate real analysis more if I first go through differential forms? It seems to illuminate the machinery behind multivariable calculus.

Answer 1

You might want to consider Shlomo Sternberg's Advanced Calculus. It's freely available online:

Harvard: Books of Shlomo Sternberg

Though the primary approach in that book isn't based on differential forms, if you're looking for a "high level" approach to real analysis, the book can prove very useful.

Answer 2

I think what Person means by real analysis is multivariate calculus. Differential forms are infinitesimal volume elements. One thinks of a manifold as piecing together local pieces of euclidean space and one integrates differential forms over a manifold. However, the definition of integration of differential form is usually given in terms of local coordinates. That is to say, in order to understand calculus on manifolds, one must first understand calculus on euclidean space.

Thus, multivariate calculus and differential geometry touches on different things. The thrust of multivariate calculus is Stokes' theorem, which relates neighboring dimensions and the operations of integration and differentiation. The touchstone of differential geometry, however, is the study of differentiable maps. There, one seeks to find that which is invariant under maps that preserve the derivative. This is the differential geometric approach to understanding differentiation.

PS. The comments basically which to point out that real analysis includes topics like measure theory etc. Originally, real analysis used to be about real numbers. However, due to abstraction, real analysis now includes many topics beyond calculus.