= Conformal invariance of the planar Dirichlet inner product
For a <conformal bijection> $f:D\to\widetilde D$ and <smooth> $u,v$ of <compact support> on $\widetilde D$, $\boxed{(u\circ f,v\circ f)_{\nabla,D}=(u,v)_{\nabla,\widetilde D}}$. The <Jacobian matrix> of $f$ is a rotation times $|f'|$. Consequently the <chain rule> introduces $|f'|^2$ into the <gradient> pairing, while the <change of variables formula> introduces precisely the same factor into area. They cancel. The identity extends by completion to the <Dirichlet energy spaces>; it does not preserve the $L^2$ term of an inhomogeneous <Sobolev norm>.
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