For real <smooth functions> of <compact support> in a planar <domain> $D$, use the <Dirichlet inner product>
$$
\boxed{(u,v)_{\nabla,D}=\frac1{2\pi}\int_D\nabla u\cdot\nabla v\,dA.}
$$
Changing the positive normalization factor does not change <orthogonality>. For complex functions, insert <complex conjugation> in the second factor to obtain the corresponding <Hermitian form>.

Let $f:D\to\widetilde D$ be a <conformal bijection>, and let $\widetilde u,\widetilde v$ be <smooth functions> of <compact support> on $\widetilde D$. Its real <Jacobian matrix> is $Df=|f'|R$, where $R$ is a rotation. By the <chain rule>,
$$
\nabla(\widetilde u\circ f)\cdot\nabla(\widetilde v\circ f)=|f'|^2(\nabla\widetilde u\cdot\nabla\widetilde v)\circ f.
$$
The <change of variables formula> has <Jacobian determinant> $|f'|^2$, so this factor cancels:
$$
\boxed{(\widetilde u\circ f,\widetilde v\circ f)_{\nabla,D}=(\widetilde u,\widetilde v)_{\nabla,\widetilde D}.}
$$
This proves <conformal invariance of the planar Dirichlet inner product>. By completion it is also an <isometry> between the corresponding <Dirichlet energy spaces>. It asserts invariance of the energy form, not of the inhomogeneous <Sobolev norm>, whose $L^2$ term has a different transformation rule.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 203 / 2 / c / Solution

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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 Incoming links: Conformal invariance of the planar Dirichlet inner product