For questions about Stokes' theorem. Stokes' theorem relates the integral of a differential form over the boundary of some orientable manifold M is equal to the integral of its exterior derivative over the whole of M.
Stokes' theorem (also called the generalized Stokes theorem or the Stokes–Cartan theorem) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
Theorem:
For a differential $(n-1)$-form $\omega$ with compact support on an oriented $n$-dimensional manifold with boundary $M$, $$\int_{M} d\omega = \int_{\partial M} \omega \text{,}$$ where $d\omega$ is the exterior derivative of the differential form $\omega$.
The classical Kelvin–Stokes theorem, along with the divergence theorem, the fundamental theorem of calculus, and Green's theorem are simply special cases of the general formulation stated above.
