According to Tongs notes on Classical Mechanics; a force is called conservative when $F=-\nabla V$ And iff $\nabla \times F = 0$. This is in $R^3$.
Also the potential $V=\int_{x_o}^{x^1} F(x)$ $dx$ where the integration is along a path $C$ whose end-points are $x_0, x_1$.
How can this be written in the language of forms? (if it is done so it generalises to curved manifolds of any finite dimension).
Since classically $V$ is a number; do we say it is a now 0-form; then $F=-dV$ is a 1-form.
We have $V=\int_C F=\int_C$ $dV$ which is well-defined as it is a 1-form $dV=-F$ integrated on a curve.
Going from a different angle, given a force $F$; then
We say $F$ is conservative if it is exact; that is written as $F=-dV$.
(But this means in general a force may not be a 1-form).
Further if $F$ is closed; that is if $dF=0$ then by the Poincare lemma we have that $F$ is locally exact ie can be written as $F=-dV$.
How correct is this?