For a planar curve, this result reduces to Green's theorem.
This statement provides the duality between the homology of chains and the de Rahm cohomology. This statement can be used to show that the codifferential is the adjoint of the exterior derivative when considering the Hodge dual on manifolds. This statement is used to generate a contradiction when smoothly retracting a ball onto its boundary in the proof of the Brouwer fixed point theorem. The most general form of this statement relates the integral of the exterior derivative of a differential form over some manifold to its integral over the boundary of the same manifold. For a planar curve, this result reduces to Green's theorem. For 10 points, name this theorem from vector calculus, which states that the surface integral of the curl of a function dotted with the normal is equal to the line integral of the function around a boundary of the surface.
This statement provides the duality between the homology of chains and the de Rahm cohomology.
Stokes's theorem