# Method of spherical means/average

These are a few steps 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 $$\overline{u}(r,t) = \frac{1}{4\pi r^2} \int\int_{||x||=r} u(x,t) dS$$.

My first question is, why is $$\overline{u}(r,t) = \frac{1}{4\pi} \int_{0}^{2\pi}\int_{0}^{\pi}u(x,t) \sin (\theta) d \theta d \phi$$? I know that we are changing to spherical coordinates, but I'm not sure about the intermediate details.

## 1 Answer

In the integral $$\overline{u}(r,t) = \frac{1}{4\pi r^2} \int\int_{||x||=r} u(x,t) dS$$ the area in the spharical polar coordinates is element $$dS = r^2\sin(\theta)d\theta d\phi$$, where $$0\le \theta \le \pi$$ and $$0 \le \phi \le 2\pi$$.

• Thanks for your answer. But how would your second comment imply $\Delta \overline{u} = \overline{\Delta u}$? Commented Oct 7, 2020 at 0:07
• I should have written "it might help to use it", but I don't know yet! I think you removed that part from the question Commented Oct 7, 2020 at 9:07
• I moved it to a different question : ) Commented Oct 8, 2020 at 2:30