Stokes’ Beautiful Theorem: Differential Forms, Boundaries, Exterior Derivatives, and Manifolds

Stokes' Theorem in its general form is a remarkable theorem with many applications in calculus, starting with the Fundamental Theorem of Calculus. The pattern in each case is that the integral of a function over a region is equal to the integral of a related function over the boundary of the region. We can use information about the boundary of a region to get information about the entire region, which is both useful and mathematically elegant.