Mmmm
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Homework Statement
Show that
\frac{\partial}{\partial t} \int_{B(x,t)} \nabla^2 p(x') dx'=\int_{S(x,t)} \nabla^2 p(x') d\sigma_t
[Hint: Introduce spherical coordinates.]
Homework Equations
The Attempt at a Solution
I thought the divergence thm would be necessary to get from the ball to the surface of the ball and so I will need to construct a unit normal to the surface
x'=x+t\alpha
where x is the vector to the centre of the ball, t is the radius and \alpha is the unit vector in the direction of the radius, so x' is the vector to the surface of the ball
that makes \alpha the unit normal, so using the divergence thm,
\frac{\partial}{\partial t}\int_{B(x,t)}\nabla^2p(x')dx' =\frac{\partial}{\partial t}\int_{S(x,t)} \nabla p(x+t \alpha ). \alpha d\sigma_t
this is where I get (more?) lost...
converting to spherical polars...
d \sigma_t = r^2 sin\phi d\theta d\phi
so
= \frac{\partial}{\partial t}\int_{S(x,t)} \nabla p(x+t \alpha ). \alpha r^2 sin\phi d\theta d\phi
and really... I'm stuck...
what is r in terms of t ? I must be going the wrong way here.
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