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I am trying to find some sort of motivation as to why we integrate manifold over differential form and why especially does it in some form corresponds to integrating the surface of the area. I have already passed my university courses that have covered these topics, including Stokes's theorem with proof (My uni has this approach of definition - theorem - proof structure in all courses with very little motivation). The issue is that sadly I've never intuitively seen that it could have anything to do with the real area if the manifold would be an earth-like object or any other intuitive geometric structure.

Is there any free material/youtube class that illustrates that it truly corresponds well? I don't need to see proof as I quite understand the technical side of things. My issue also is that I cannot see differential form in any other way than only technical mathematical extremely abstract definition.

Please keep in mind that my knowledge is limited to European standards of a bachelor's degree...

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  • $\begingroup$ The brief answer is: The change of variables theorem for multiple integrals. You might check out my YouTube lectures on forms, etc., linked in my profile. $\endgroup$
    – Ted Shifrin
    Commented Jun 2, 2023 at 0:12
  • $\begingroup$ @TedShifrin and directed at OP, too, since this is just for my own education. Coming from a physics background and then to the formalization of differential forms, I didn't have as much of a problem with visualization as my colleagues who only developed the math intuition. My question is, what is it about the definitions that make it hard to look at an object like $dx$ and consider it like a length or a toothpick, and ascribe further "cute" visualizations for $k$ forms, etc? $\endgroup$
    – Ninad Munshi
    Commented Jun 2, 2023 at 0:18
  • $\begingroup$ I pick on the length thing because that's the thing that stuck with me that my colleagues had significant trouble seeing at the time. But the struggles continued to visualizing things like areas, volumes, etc especially when it differed from the measure-theoretic sense of size. $\endgroup$
    – Ninad Munshi
    Commented Jun 2, 2023 at 0:22
  • $\begingroup$ @Ninad It is oriented length, not length, of the projection of the object onto the $z$-axis. Your “differed from measure-theoretic sense of size” is beyond vague. $\endgroup$
    – Ted Shifrin
    Commented Jun 2, 2023 at 0:35
  • $\begingroup$ You don't integrate manifolds over differential forms, you integrate differential forms over manifolds... $\endgroup$
    – Hans Lundmark
    Commented Jun 2, 2023 at 8:11

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motivation as to why we integrate manifold over differential form differential form over manifold

An n-dimensional manifold could have a top, n-form, and if it's defined smoothly it's called differential n-form. It's also called a volume form, however in 2d it's called area form. It takes n vectors (or equivalently, an n-vector) as arguments and calculates the oriented (signed) volume they enclose. If you don't want to deal with orientations and signed volumes you could also integrate a https://en.wikipedia.org/wiki/Density_on_a_manifold

Integration intuitively is a procedure that 1) facets the manifold into infinitesimal n-vectors, 2) feeds those into the integrand (the n-form) and 3) sums up the results.

In 1d that's the familiar limiting Riemannian sums for $\int_a^b dx$, for example. By the way, you could even apply Stokes theorem to it noting that the $[a, b]$ is a 1-manifold with boundary consisting of two oriented points $\{-a,b\}$ and the 1-form $dx$ becomes 0-form $x$ eating oriented points. Integral supplies those and sums up the results: $$\int_a^b dx=\int_{\{-a,b\}} x = -a + b$$

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