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

Let . We define the spherical mean of as .

Demonstrate that if is a harmonic function (i.e. ) then u satisfies the equation for all .

Hint: Use .

According to my class notes, , where .

So I guess I should start with .

Now I'm not really sure how to proceed. I guess there's a trick with . 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.