Turning the Black-Scholes PDE into the heat equation

Show how a change of variables converts the Black-Scholes equation into the heat equation.

Set $\tau=T-t$, $x=\ln S+(r-\tfrac12\sigma^2)\tau$ and $V=e^{-r\tau}u(x,\tau)$; then $u_\tau=\tfrac12\sigma^2u_{xx}$. The log variable removes the $S$ and $S^2$ coefficients, shifting $x$ by the drift absorbs the first-derivative term, and the discount factor cancels the $-rV$ term. Rescaling time by $\tfrac12\sigma^2$ gives the standard $u_\tau=u_{xx}$.