This is something I don't understand from my partial differential equations course. Let us be in $\mathbb{R}^3$. The average of a function $u(x,t)$ on the sphere $\|x\|=r$ of center $0$ and radius $r$ is denoted $$\overline{u}(r,t) = \frac{1}{4\pi r^2} \int\int_{\|x\|=r} u(x,t) dS = \frac{1}{4\pi} \int_{0}^{2\pi}\int_{0}^{\pi}u(x,t) \sin (\theta) d \theta d \phi.$$
My textbook uses that $\Delta \overline{u} = \overline{\Delta u}$. I don't know why this is true. Would anyone have a nice explanation for this? (It mentions that a possible hint is to use spherical polar coordinate form of the Laplacian.)