(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Evaluate

[tex]\int_{\Gamma} x\frac{\partial}{\partial n} G(x,y,\frac{1}{2}, \frac{1}{3}) ds [/tex]

On a unit disk region [itex]\Omega[/tex] with positive oriented boundary [itex]\Gamma[/itex]

2. Relevant equations

[tex]u(x_0, y_0) = \frac{1}{2\pi}\int_{\Gamma} ( u\frac{\partial v}{\partial n} - v\frac{\partial u}{\partial n}) ds[/tex]

[tex]u(x_0, y_0) = \frac{1}{2\pi}\int_{\Gamma} [ u\frac{\partial }{\partial n} G(x,y,x_0,y_0)] ds[/tex]

In this case [itex] u= x \Rightarrow \nabla^2 u = 0 [/itex] which means u is harmonic in [itex]\Omega[/tex]

[tex]G=v + h \hbox { where }\; v=\frac{1}{2}ln[(x-x_0)^2 + (y-y_0)^2] = ln|r| [/tex]

[tex] h = -v \hbox { on }\; \Gamma \hbox { and h is harmonic in } \Omega [/tex]

Since v is not harmonic in [itex] \Omega [/tex] because [itex] v\rightarrow -\infty \hbox { as } (x,y) \rightarrow (x_0,y_0) [/itex]. This mean G is not harmonic.

3. The attempt at a solution

I have no idea how to approach this and no idea how to find G. Please help.

Thanks

Alan

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# Homework Help: Please help evaluate this Green's problem.

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