(Laplace's equation) Show that for a solution w(x, y) of Laplace's equation \,2U- = 0 in
Chapter 10, Problem 10.1.78(choose chapter or problem)
Show that for a solution w(x, y) of Laplace's equation \(\nabla^{2} w=0\) in a region R with boundary curve C and outer unit normal vector \(\mathbf{n}\),
(10) \(\iint_{R}\left[\left(\frac{\partial w}{\partial x}\right)^{2}+\left(\frac{\partial w}{\partial y}\right)^{2}\right] d x d y\)
\(=\oint_{C} w \frac{\partial w}{\partial n} d s\)
Text Transcription:
nabla^2 w = 0
n
iint_R[(partial w / partial x)^2 + (partial w / partial y)^2] dx dy
= oint_C w partial w / partial n ds
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