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The Dirichlet energy of a differentiable scalar function is one half the integral of its squared gradient. On a Euclidean domain it is $E(u)=\frac{1}{2}\int |\nabla u{|}^{2}dx$; the Dirichlet energy on a Riemannian manifold uses its metric and volume measure. Its stationary functions satisfy the Laplace equation with fixed boundary values.

The metric and orientation determine the Riemannian volume form: in a positively oriented coordinate chart $(x^1,\dots ,x^n)$,Writing $|g|=\det (g_{ij})$ and $(g^{ij})=(g_{ij}{)}^{-1}$, the Dirichlet energy on a Riemannian manifold with source $f$ is