I'm stuck on this problem I've been assigned in an electromagnetism class (it's a mathematical problem though).

Let $\displaystyle u: \mathbb{R}^3 \to \mathbb{R}$. We define the spherical mean of $\displaystyle f$ as $\displaystyle M_t(u)(x)=\frac{1}{4 \pi} \int _{|\psi |=1} u(x+t \psi ) dS_{\psi}$.

Demonstrate that if $\displaystyle u$ is a harmonic function (i.e. $\displaystyle \triangle u =0$) then u satisfies the equation $\displaystyle u(x)=M_t (u)(x)$ for all $\displaystyle t$.

Hint: Use $\displaystyle \partial _t M_t (u)$.

According to my class notes, $\displaystyle \partial _t M_t ( g (\vec x))=\frac{1}{4\pi} \int _{S^2} \partial _t \tilde \phi (\vec x +t \hat n) d \Omega$, where $\displaystyle \tilde \phi (\vec x) =\partial _t \phi (t,\vec x)|_{t=0}$.

So I guess I should start with $\displaystyle \partial _t M _t (u)(x)= \frac{1}{4 \pi} \int _{|\psi|=1} \partial _t u (x+t \psi) dS _ \psi$.

Now I'm not really sure how to proceed. I guess there's a trick with $\displaystyle \partial _t u(x+t \psi)$. I've been stuck here for days. I've seen the gradient appearing on the sheets of some people in my class, but I've no idea how I could introduce it.

Thanks for any help.