Lots of integrals in life look like Gaussians.

\[ \int_{\mathbb{R}^n} \frac{e^{-\frac{|y|^2}{2}}}{(2\pi)^{n/2}} \mathrm{d}y = 1. \] In the heat kernel, we often have some rescaling $y \to \sqrt{\lambda} y$, scaling the integral to \[ \int_{\mathbb{R}^n} \frac{e^{-\frac{|y|^2}{2\lambda}}}{(2\pi \lambda)^{n/2}} \mathrm{d}y = 1, \] where perhaps we might take $\lambda = \frac{1}{2\pi}$ to get the familiar \[ \int_{\mathbb{R}^n} e^{-\pi |y|^2} \mathrm{d}y = 1. \] More generally, we might change coordinates by an invertible linear transformation $x = Sy$, whence $|y|^2 = |Sx|^2$ and \[ \int_{\mathbb{R}^n} \frac{e^{-\frac{1}{2} |Sx|^2}}{(2\pi)^{n/2}\det(S)} \mathrm{d}y = 1; \] that is, \[ \int_{\mathbb{R}^n} \frac{e^{-\frac{1}{2} |Sx|^2}}{\det(\sqrt{2\pi} S)} \mathrm{d}y = 1; \]