Note that is the directional derivative of u in the direction of the unit vector , thus it equals . So the integral evaluates
So is constant for t. And obviously when t is 0 the equation holds.
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.
Thank you very much for the reply. I think I could understand everything if someone tells me what is the notation here.
By the way I'm also stuck to understand which unit vector is . Is it the unit vector of the cylindrical or spherical coordinate system? Or is it associated with the solution of the wave equation?
Any calculus book will cover directional derivatives, probably with figures?
According to the chain rule, directional derivative along a vector v=(a, b, c) at point p will be
Yes that's Gauss's theorem or Green's or Stokes, anyway it turns a surface integral to a volume one.
see http://en.wikipedia.org/wiki/Divergence_theorem