Editing Harmonic function (section)

===Harmonic maps between manifolds===
{{main|Harmonic map}}
If {{mvar|M}} and {{mvar|N}} are two Riemannian manifolds, then a harmonic map <math>u: M \to N</math> is defined to be a critical point of the Dirichlet energy
<math display="block">D[u] = \frac{1}{2} \int_M \left\|du\right\|^2 \, d\operatorname{Vol}</math>
in which <math>du: TM \to TN </math> is the differential of {{mvar|u}}, and the norm is that induced by the metric on {{mvar|M}} and that on {{mvar|N}} on the tensor product bundle <math>T^\ast M \otimes u^{-1} TN.</math>

Important special cases of harmonic maps between manifolds include [[minimal surface]]s, which are precisely the harmonic immersions of a surface into three-dimensional Euclidean space.  More generally, minimal submanifolds are harmonic immersions of one manifold in another.  [[Harmonic coordinates]] are a harmonic [[diffeomorphism]] from a manifold to an open subset of a Euclidean space of the same dimension.