# Harmonic Function

by billiards
Tags: function, harmonic
 P: 748 1. The problem statement, all variables and given/known data Derive that: $$\left[r\frac{\partial\overline{f}}{\partial r}\right]}^{r=R}_{r=0}=0$$ 2. Relevant equations I have taken the Laplacian $$\nabla^{2}f=0$$ for a disk in cylindrical co-ordinates and have found that: $$\int^{R}_{0}\int^{2\pi}_{0} \left[\frac{\partial}{\partial r}(r\frac{\partial f}{\partial r})\right] d\varphi dr=0$$ And the definition of the average of the function around the circle of radius r is provided: $$\overline{f}(r)\equiv\frac{1}{2\pi}\int^{2\pi}_{0}f(r,\varphi)d\varphi$$ 3. The attempt at a solution This ones seems to have me stumped. I've tried setting $$\int^{2\pi}_{0} \left[\frac{\partial}{\partial r}(r\frac{\partial f}{\partial r})\right] d\varphi = \int^{2\pi}_{0}f(r,\varphi)d\varphi = 2\pi\overline{f}(r)$$ But that didn't seem to be fruitful. I've tried expanding $$\left[\frac{\partial}{\partial r}(r\frac{\partial f}{\partial r})\right]d\varphi = \frac{\partial f}{\partial r}d\varphi + r\frac{\partial^{2} f}{\partial r^{2}}d\varphi$$ That looks a little bit like a Taylor series but I don't know what to do with it. I've been playing around with the algebra but can't seem to find my break through.
 HW Helper P: 3,307 i think you've pretty much got it, just need to work backwards so you have $$\overline{f}(r)=\frac{1}{2\pi}\int^{2\pi}_{0} f(r,\phi)d\phi$$ differentiate that whole expression w.r.t. r, multiply by r then intergate over r from 0 to R and see what you end up with
 Quote by lanedance i think you've pretty much got it, just need to work backwards so you have $$\overline{f}(r)=\frac{1}{2\pi}\int^{2\pi}_{0} f(r,\phi)d\phi$$ differentiate that whole expression w.r.t. r, multiply by r then intergate over r from 0 to R and see what you end up with
 P: 748 Thanks lanedance, I didn't know about differentiating under the integral sign, good stuff! I think I have the answer, would appreciate feedback as this is not 100% comfortable stuff for me. So far i get: 1) Use the definition of $$\overline{f}(r)\equiv\frac{1}{2\pi}\int^{2\pi}_{0}f(r,\varphi)d\varphi$$ to find $$(r\frac{\partial \overline{f}}{\partial r}) = \frac{1}{2\pi}\int^{2\pi}_{0} r\frac{\partial f(r,\varphi)}{\partial r} d\varphi$$ in terms of $$f$$ 2) Note that $$\left[r\frac{\partial\overline{f}}{\partial r}\right]}^{r=R}_{r=0}=\int^{R}_{0} \frac{\partial}{\partial r} (r\frac{\partial \overline{f}}{\partial r})dr = \frac{1}{2\pi}\int^{R}_{0}\int^{2\pi}_{0} \frac{\partial}{\partial r} (r\frac{\partial f(r,\varphi)}{\partial r}) d\varphi dr$$ 3) Sub in the Laplacian expression $$\int^{R}_{0}\int^{2\pi}_{0} \left[\frac{\partial}{\partial r}(r\frac{\partial f}{\partial r})\right] d\varphi dr=0$$ To find that $$\left[r\frac{\partial\overline{f}}{\partial r}\right]}^{r=R}_{r=0}=0$$ QED?