are , (with endpoints fixed), in the fixed endpoint case: ; or in the closed case: .
, if is closed in which is homotopic to a point, then This is the case if is the boundary of a subregion such that is to a closed disk.
We first introduce some embedding and approximation theorems.
Let be a and let be a function. Given any positive function , there exists a function that is close to .
Let be , and let be a map. Then is homotopic to a map .
If are homotopic smooth maps, then they are If is homotopic to relative to some closed subset , then they are smoothly homotopic relative to .
By virtue of these properties of homotopy, one can apply Stoke’s Theorem which demands both , the oriented dimensional manifold with boundary, and , the compactly supported form on , are smooth.
: such that . : .
: a closed covector field on a smooth manifold . Then every has a neighborhood on which is exact.