Solving Laplace Operator Integral with Partial Integration

[jet10]

I have read in an electrodynamics book that
<br /> \int d^3r \frac{\Delta\Phi(r)}{r}=\int d^3r \Phi(r)}\Delta \frac{1}{r}<br />
is possible through partial integration. But how?? Can some one help me on this by showing me the steps? Thanks

 

[Physics Monkey]

Start with the following two identities:
\int d^3 r \nabla \cdot \left( \frac{1}{r} \nabla \Phi \right) = \int d^3 r \left( \nabla \frac{1}{r} \cdot \nabla \Phi + \frac{1}{r} \Delta \Phi \right)

\int d^3 r \nabla \cdot \left( \Phi \nabla \frac{1}{r} \right) = \int d^3 r \left( \nabla \frac{1}{r} \cdot \nabla \Phi + \Phi \Delta \frac{1}{r}\right)

Now the left hand side of each equation can written as a surface integral which gives zero. This gives
<br /> \int d^3 r \left( \nabla \frac{1}{r} \cdot \nabla \Phi + \frac{1}{r} \Delta \Phi \right) = 0 <br />
<br /> \int d^3 r \left( \nabla \frac{1}{r} \cdot \nabla \Phi + \Phi \Delta \frac{1}{r} \right) = 0 <br />
but since these two expressions have the same first term you can easily see that
<br /> \int d^3 r \frac{1}{r} \Delta \Phi = \int d^3 r \Phi \Delta \frac{1}{r} <br />

 

[jet10]

Thanks for your reply. Just one thing I am not able to see yet. How are these equal to zero?
<br /> \int d^3 r \nabla \cdot \left( \frac{1}{r} \nabla \Phi \right)=\oint dA \frac{1}{r} \nabla \Phi=0?<br />
<br /> \int d^3 r \nabla \cdot \left( \Phi \nabla \frac{1}{r} \right)=\oint dA \Phi \nabla \frac{1}{r}=0?<br />

 

[Physics Monkey]

You integrate on a surface at infinity, and with very weak assumptions about the fall off of \Phi with distance, you can show those area integrals go to zero.

 

[jet10]

That's clever. Thanks!