= Dirichlet energy of a map
{c}
{title2=$E(u)=\frac12\int|du|^2$}
For a smooth map between <Riemannian manifolds>, its Dirichlet energy is half the integral of the squared <norm> of its differential, using the source and target metrics. When the source is two-dimensional, a conformal change of its metric leaves this energy unchanged. For a <J-holomorphic curve> into a <symplectic manifold> with a <compatible almost complex structure>, the <Energy identity for a J-holomorphic curve> identifies it with the pulled-back <symplectic area>. The norm of a differential here is the full tensor norm, summing its squared values on an orthonormal source frame.
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