information:classification, structural:expansion
Generalizations

Many of the critical theorems of vector calculus generalize elegantly to statements about the integration of differential forms on manifolds. In the language of differential forms and exterior derivatives, the gradient theorem states that

∫ ∂ γ ϕ = ∫ γ d ϕ {\displaystyle \int _{\partial \gamma }\phi =\int _{\gamma }\mathrm {d} \phi }

for any 0-form, ϕ, defined on some differentiable curve γ ⊂ R (here the integral of ϕ over the boundary of the γ is understood to be the evaluation of ϕ at the endpoints of γ).

Notice the striking similarity between this statement and the generalized Stokes’ theorem, which says that the integral of any compactly supported differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative d ω over the whole of Ω, i.e.,

∫ ∂ Ω ω = ∫ Ω d ω {\displaystyle \int _{\partial \Omega }\omega =\int _{\Omega }\mathrm {d} \omega }

This powerful statement is a generalization of the gradient theorem from 1-forms defined on one-dimensional manifolds to differential forms defined on manifolds of arbitrary dimension.

The converse statement of the gradient theorem also has a powerful generalization in terms of differential forms on manifolds. In particular, suppose ω is a form defined on a contractible domain, and the integral of ω over any closed manifold is zero. Then there exists a form ψ such that ω = d ψ. Thus, on a contractible domain, every closed form is exact. This result is summarized by the Poincaré lemma.