Logarithmic polar coordinate change ($x=e^{-s}\sin t,y=e^{-s}\cos t$)

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 givesand, for a twice continuously differentiable function,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.