Stokes' theorem

Le $\mathrm{\Omega }$ be a compact oriented manifold with boundary of dimension $k$ and $\omega$ a $(k-1)$-form. It holds

It can be understood intuitively from the interpretation of the exterior derivative.
Related to infinitesimal Stokes' theorem.

From here can be proven that:

• A 1-form $E$ is closed if and only if and only if

for all contractible loops $\gamma$.

• A 1-form $E$ is exact if and only if and only if

for all loops $\gamma$.

This results are closely related to Poincare lemma.