The metric and orientation determine the <Riemannian volume form>: in a positively oriented coordinate chart $(x^1,\ldots,x^n)$,
$$
d\operatorname{vol}_g=\sqrt{\det(g_{ij})}\,dx^1\wedge\cdots\wedge dx^n.
$$
Writing $|g|=\det(g_{ij})$ and $(g^{ij})=(g_{ij})^{-1}$, the <Dirichlet energy on a Riemannian manifold> with source $f$ is
$$
E(u)=\int_M\left(\frac12g^{ij}(x)\,\partial_i u\,\partial_j u-fu\right)\sqrt{|g|}\,dx^1\cdots dx^n.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 115 / 2 / a / Solution

{title2=$E(u)$}
{wiki=Dirichlet_energy}

For a function $u$ on a compact Riemannian manifold, its Dirichlet energy is $\frac12\int_M|\nabla u|_g^2d\operatorname{vol}_g$. Adding a linear source term $-\int_Mfu\,d\operatorname{vol}_g$ gives Euler-Lagrange equation $-\Delta_gu=f$.


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