- #1
Chingon
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Homework Statement
∫Bdot[∇×A]dV=∫Adot[∇×B]dV
Prove this by integration by parts. A(r) and B(r) vanish at infinity.
Homework Equations
I'm getting stuck while trying to integrate by parts - I end up with partial derivatives and dV, which is dxdydz?
The Attempt at a Solution
I can break things down to Cartesian components, but integrating by parts is where I get stuck.
Essentially, I'm simplifying by stating the identity that
BdotCurlA - AdotCurlB = Div(A×B) = 0 in this case (Subtract right hand side from left and combine under one ∫dV)
The components look like this:
∂x(AyBz−AzBy)=(∂xAy)Bz+Ay(∂xBz)−(∂xAz)By−Az(∂xBy)
plus the y,z terms as well.
How would one integrate ∫[(∂xAy)Bz]dV by parts? I have Ay and Bz which are potentially functions of x,y,z and a partial wrt x, and dxdydz...
Thanks!