I'm trying to understand differential forms. My instructor explained them in terms of operators on vector fields that spit out continuous functions. I was trying to understand the geometric meaning of them but he says in his notes "take a differential form $\omega$ supported on a compact set $K$", could someone explain what this means, for example, if I took the differential form, $$\omega = (xy)dx + (z-y)dy + z^2dz$$ could someone the whole "supported on a compact set" and the geometric idea here?
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You can think of support as follows: If two vector fields agree on the set K then the diffferential form acting on the vector fields gives you same answer. In other words if the vector field vanishes on K then the differential form acting on it gives you the zero function. I think the cleanest way to understand differential forms is perhaps as sections of vector bundles. – DBS Mar 14 '15 at 17:29
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Your form does not have compact support. To say that $\omega=\sum f_i(x)dx_i$ has compact support is to say that there is some compact set $K$ outside of which all the coefficient functions $f_i$ are zero.
I'm not sure why you're worrying about compact support at the outset. Ordinarily you worry about that only for integrating over a non-compact manifold.
Ted Shifrin
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