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Homework Statement
Show that for a solution w of Laplace's equation in a region R with boundary curve C and outer unit normal vector N,
\int_{R}\left\| \nabla w\right\| dxdy = \oint_{C}w\frac{\partial w}{\partial N}ds
Homework Equations
The book goes through the steps to show that the following is true (which is very similar to the problem I am doing so I thought it would be relevant):
eq(1): \int_{R}\left( \nabla^{2}w \right) dxdy = \oint_{C}\frac{\partial w}{\partial N}ds
Also, Green's theorem, which is just Stokes theorem in the plain:
eq(2): \int_{R}\left( \frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{\partial y}\right) dx dy = \oint_{C}\left( F_{1}dx + F_{2}dy\right)
Also:
eq(3): (\nabla w)\cdot \hat{N} = \frac{\partial w}{\partial N}
The Attempt at a Solution
I have been trying to follow the steps in the book that they use to get eq(1) but it just doesn't seem the same.
What they do is to define F_{2} = \frac{\partial w}{\partial x} and F_{1} = -\frac{\partial w}{\partial y} so that when those F's are plugged into eq(2) you end up with the laplacian of w. That is how they get the left side, and using that this laplacian can now be written in the form of eq(2) they get the relation on the right side.
But in my case the integrand on the left side is now:
\left\| \nabla w \right\| = \left( \frac{\partial w}{\partial x} \right)^{2} + \left( \frac{\partial w}{\partial y} \right)^{2}
I don't see how I can write this in the form of eq(2), so I don't know if the book's steps will help me.
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