= Conformal flattening of a surface of revolution
{title2=$\rho^{-2}g=d\sigma^2+d\phi^2$}
For a <surface of revolution> whose <induced metric> is $g=E(\rho)d\rho^2+\rho^2d\phi^2$, with $\rho>0$ and $E>0$ smooth on a coordinate interval, define $\sigma(\rho)=\int_{\rho_*}^{\rho}\sqrt{E(r)}\,dr/r$. Then $\sigma'>0$ and the <conformal factor> $\rho^{-1}$ gives $\rho^{-2}g=d\sigma^2+d\phi^2$. This proves the rescaled metric is locally the <Euclidean metric> without solving a curvature equation. Poles and places where $\rho$ is not a coordinate must be treated in other charts; the angular coordinate is also understood locally.
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