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[SOLVED] Greens Functions
show:
\int\int\int_{D}\vec{F}\cdot\vec{G}dV = 0
where:
\vec{F}=\nabla\phi
\vec{G}=\nabla\psi
\nabla\cdot\vec{F}=0
\psi|_{\partial D}=0
This looks like a problem for Greens first theorem:
\int\int\int_{D}\phi\nabla^{2}\psi dV = \int\int_{\partial D}\phi\nabla\psi dS - \int\int\int_{D}\nabla\psi\cdot\nabla\phi dV
The very right term is clearly the integral that I'm looking for. So, i will set it to look like the requested answer. Also, I know that
\psi|_{\partial D}=0
meaning that I can also throw out the second term because that term wants me to integrate the gradient of psi over the surface, while I know that psi is 0 over the surface. So, I am left with this:
\int\int\int_{D}\phi\nabla^{2}\psi dV = - \int\int\int_{D}\vec{F}\cdot\vec{G} dV
So, this means that the term on the left mus equal zero. Does anyone know how I can show this? Psi is not zero through the domain, and the problem doesn't specify that it is a harmonic potential (although I suppose it could be). Could someone please help me with this step? Any help at all is greatly appreciated.
Homework Statement
show:
\int\int\int_{D}\vec{F}\cdot\vec{G}dV = 0
where:
\vec{F}=\nabla\phi
\vec{G}=\nabla\psi
\nabla\cdot\vec{F}=0
\psi|_{\partial D}=0
The Attempt at a Solution
This looks like a problem for Greens first theorem:
\int\int\int_{D}\phi\nabla^{2}\psi dV = \int\int_{\partial D}\phi\nabla\psi dS - \int\int\int_{D}\nabla\psi\cdot\nabla\phi dV
The very right term is clearly the integral that I'm looking for. So, i will set it to look like the requested answer. Also, I know that
\psi|_{\partial D}=0
meaning that I can also throw out the second term because that term wants me to integrate the gradient of psi over the surface, while I know that psi is 0 over the surface. So, I am left with this:
\int\int\int_{D}\phi\nabla^{2}\psi dV = - \int\int\int_{D}\vec{F}\cdot\vec{G} dV
So, this means that the term on the left mus equal zero. Does anyone know how I can show this? Psi is not zero through the domain, and the problem doesn't specify that it is a harmonic potential (although I suppose it could be). Could someone please help me with this step? Any help at all is greatly appreciated.