Most notably, part of Maxwell's equations states that the Faraday 2-form is closed:
$$dF=0$$
From this we can infer from Poincare's lemma that there exists a 1-form $A$ such that $dA=F$. In some elementary treatments $F$ is considered to be an exact form. But when considering magnetic monopoles is it important to treat it as a closed form because of the "locally" clause in the Poincare lemma.

A really trivial example is the following: let $g$ be an orthonormal metric. Then it is a closed 0-form
$$dg=0$$
This is merely the equation for the antisymmetry of the spin connection on a Riemannian manifold with orthonormal metric.

Cohomology is used quite extensively in a little sector of physics called String Theory. I'm sure you know how important closed forms are for that. A really important closed form is the Kahler form:
$$dJ=0$$

EDIT: Those weren't 1-forms. The curl operator is $\star d$. Thus a closed one-form is isomorphic to a vector that has zero curl! Some examples I can think of off the top of my head:

Take Faraday's law $\nabla\times\mathbf{E}+\dot{\mathbf{B}}=0$. Suppose the fields are static. Then $\dot{\mathbf{B}}=0$ and $\nabla\times\mathbf{E}=0$. If $\mathcal{E}=\mathbf{E}^\flat$
$$d\mathcal{E}=0$$

The same works for the Maxwell-Ampere law in a vacuum. Then the magnetic 1-form $\mathcal{B}=\mathbf{B}^\flat$ is closed
$$d\mathcal{B}=0$$

Suppose the integral of some force $\mathbf{F}$ is path-independent. Work is defined by
$$W_P=\int_P\mathbf{F}\cdot d\mathbf{x}$$
If $\mathcal{F}=\mathbf{F}^\flat$ then
$$W_P=\int_P\mathcal{F}$$
The difference of work along two different paths vanishes ($P'-P$ is a closed curve which is the boundary of a surface $S$)
$$W_{P'}-W_P=\int_{P'-P}\mathcal{F}=\int_S d\mathcal{F}=0$$
by Stokes' theorem. This implies for any conservative force
$$d\mathcal{F}=0$$

This post imported from StackExchange Physics at 2015-01-06 10:51 (UTC), posted by SE-user 0celo7