Riemannian volume form ($d{\mathrm{vol}}_g$)

On an oriented Riemannian $n$-manifold, the volume form is the unique positive $n$-form taking value one on every positively oriented orthonormal frame. In positive local coordinates,

A differential form is parallel when its covariant derivative vanishes. Parallel transport then preserves it, and every parallel form is harmonic.

For a function $u$ on a compact Riemannian manifold, its Dirichlet energy is $\frac{1}{2}{\int }_M|\nabla u{|}_g^{2}d{\mathrm{vol}}_g$. Adding a linear source term $-{\int }_{M}fud{\mathrm{vol}}_g$ gives Euler-Lagrange equation $-{\Delta }_{g}u=f$.