integration of a form

Differential forms were built to be integrated, and the matching rule is beautifully tight: a k-form is exactly what you integrate over a k-dimensional region. A 0-form (a function) is 'integrated' over a 0-dimensional region (a point) just by evaluating it there. A 1-form is integrated over a curve, a 2-form over a surface, a 3-form over a solid. The degree of the form must equal the dimension of the domain, or the integral does not even make sense.

Mechanically, you integrate a k-form by parametrizing the region and pulling the form back to the parameter space, where it becomes an ordinary multiple integral. To integrate omega over a surface S given by a map from a u-v rectangle, you compute the pullback F-star omega, which lands as (some function) du^dv, and then you integrate that function as a plain double integral over the rectangle. Because pullback automatically carries the Jacobian, you never separately remember a change-of-variables factor — the dr d(theta) and the surface-element scaling appear on their own. The orientation of the region sets the overall sign: reverse the orientation and the integral flips sign, mirroring the antisymmetry of the form itself.

This single framework swallows the line integral, the surface flux integral, and the ordinary volume integral as special cases — they are all 'integrate a form of the matching degree over an oriented region'. The payoff is that the generalized Stokes' theorem can be stated once for all of them: the integral of d omega over a region equals the integral of omega over the boundary. And because the construction is built on pullback, the answer is genuinely geometric — independent of the parametrization you happened to choose.