= Conformal change of half-plane capacity
Suppose a capacity-parameterized <Loewner chain> lies in a domain on which $\psi$ is a <conformal map>. For its image chain set $\psi_t=\widetilde g_t\circ\psi\circ g_t^{-1}$. Then the image <Loewner driving function> is $\widetilde U_t=\psi_t(U_t)$ and
$$
d\operatorname{hcap}(\widetilde A_t)=2\psi_t'(U_t)^2\,dt.
$$
The derivative is real by the <Schwarz reflection principle>. Cancellation of the pole in the differentiated conjugacy identity proves the squared-derivative rule.
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