= Logarithmic polar coordinate change
{title2=$x=e^{-s}\sin t,\quad y=e^{-s}\cos t$}
This <change of variables> replaces radius by its negative <logarithm>: $r=\sqrt{x^2+y^2}=e^{-s}$. The angular convention in the title measures $t$ from the positive $y$ axis toward the positive $x$ axis. Its <Jacobian determinant> is $e^{-2s}>0$, so it supplies local coordinates on the punctured <plane>. For a single-valued <function> $u$, its expression $v(s,t)=u(x,y)$ is periodic in $t$ with period $2\pi$.
The <multivariable chain rule> gives
$$
v_s=-xu_x-yu_y,\qquad v_t=yu_x-xu_y,
$$
and, for a twice continuously differentiable <function>,
$$
v_{tt}-v_s=y^2u_{xx}-2xyu_{xy}+x^2u_{yy}.
$$
The first-order correction comes from differentiating the coefficients $x,y$ in the angular operator. Consequently the <partial differential equation> with the right-hand expression equal to zero becomes the <heat equation> $v_s=v_{tt}$, with the logarithmic radial coordinate acting as the evolution variable.
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