Harmonic coordinates

In Riemannian geometry, a branch of mathematics, harmonic coordinates are a coordinate system (x,...,x) on a Riemannian manifold each of whose coordinate functions x is harmonic, meaning that it satisfies Laplace's equation

Here Δ is the Laplace–Beltrami operator. Equivalently, regarding a coordinate system as a local diffeomorphism φ : M → R, the coordinate system is harmonic if and only if φ is a harmonic map of Riemannian manifolds, roughly meaning that it minimizes the elastic energy of "stretching" M into R. The elastic energy is expressed via the Dirichlet energy functional