So I understand that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω.(adsbygoogle = window.adsbygoogle || []).push({});

And I understand that one can pull back the integral of a 1-form over a line to the line integral between the two endpoints.

I also understand (from my poor calculus training) how to do an integral of some function * dx between two endpoints. I KNOW what that means.

But I do NOT know what it means to integrate a one form. I mean, I get the integral, I get the presence of the integrand, I get the idea of "performing" it over a line. But if there is no "d" symbol on the integral, I cannot figure out what it means.

Note: I am teaching myself differential geometry, forms, calculus on manifolds, all at once. The clouds are clearing and rote operations are becoming more clear. I get that forms make doing these operatoins easier. I can see how the integral of force along a line bewtween two endpoints is work. I get that. But I do not understand what it means to integrate the 1-form force over the line. I cannot attach physical meaning to it.

So... What does it mean to integrate a one form? Where does it come from? (I get how I can pull it back, but I cannot figure out where the integral comes from or what it means when there is no "d")

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# A The meaning of an integral of a one-form

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