nLab closed differential form

Idea

A differential form $\omega \in \Omega^p(X)$ is called closed if the de Rham differential $d \colon \Omega^{p}(X) \to \Omega^{p+1}(X)$ sends it to zero: $d \omega = 0$, hence if it is in the kernel of the de Rham.

A differential form $\omega \in \Omega^{p+1}(X)$ is called exact if it is in the image of the de Rham differential: $\omega = d \alpha$, for some $\alpha \in \Omega^{p}(X)$.

The quotient of the vector space of closed differential forms by the exact differential forms of degree $p$ is the de Rham cohomology of $X$ in degree $p$.

Formalization of closed and co-exact differential forms in cohesive homotopy theory is discussed at differential cohomology hexagon.

Properties

• Stokes theorem

• kernel of integration is the exact differential forms