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yungman
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Homework Statement
Suppose [itex]u[/itex] is harmonic ([itex]\nabla^2 u = 0 [/itex]) and [itex]v=0 \;\hbox{ on } \;\Gamma [/itex] where [itex]\Gamma[/itex] is the boundary of a simple or multiply connected region and [itex] \Omega[/itex] is the region bounded by [itex]\Gamma[/itex].
Using Green's identities, show:
[tex] \int \int_{\Omega} \nabla u \cdot \nabla v \; dx dy = 0 [/tex]
Homework Equations
Green's identities:
[tex] \int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds [/tex]
[tex] \int \int_{\Omega} \;(u\nabla^2 v \;-\; v\nabla^2 u )\; dx dy \;=\; \int_{\Gamma} \;(u\frac{\partial v}{\partial n} - v\frac{\partial u}{\partial n}) \;ds [/tex]
[tex] \frac{\partial u}{\partial n} = \nabla u \; \cdot \; \widehat{n}[/tex]
The Attempt at a Solution
I use
[tex] \int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds [/tex]
If [itex]v=0 \hbox { on } \Gamma \;\;\Rightarrow\;\; v \hbox { is a constant = 0 } \;\;\Rightarrow\;\; v= 0 \;\hbox{ on } \;\Omega[/itex].
[tex] \int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \int \int_{\Omega}\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds = 0[/tex]
Is this the right way?
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