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Mmmm
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Homework Statement
Show that
[tex]\frac{\partial}{\partial t} \int_{B(x,t)} \nabla^2 p(x') dx'=\int_{S(x,t)} \nabla^2 p(x') d\sigma_t[/tex]
[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
[tex]x'=x+t\alpha[/tex]
where x is the vector to the centre of the ball, t is the radius and [itex]\alpha[/itex] is the unit vector in the direction of the radius, so x' is the vector to the surface of the ball
that makes [itex]\alpha[/itex] the unit normal, so using the divergence thm,
[tex]\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[/tex]
this is where I get (more?) lost...
converting to spherical polars...
[tex]d \sigma_t = r^2 sin\phi d\theta d\phi[/tex]
so
[tex] = \frac{\partial}{\partial t}\int_{S(x,t)} \nabla p(x+t \alpha ). \alpha r^2 sin\phi d\theta d\phi[/tex]
and really... I'm stuck...
what is r in terms of t ? I must be going the wrong way here.
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