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 Exercise [10.13] 
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Joined: 12 Mar 2008, 10:57
Posts: 69
Location: India
Post Exercise [10.13]
Differentiate as follows:
\frac{\partial\alpha}{\partial{x}}=\frac{\partial\beta}{\partial{y}}\Rightarrow\frac{\partial}{\partial{x}}(\frac{\partial\alpha}{\partial{x}})=\frac{\partial}{\partial{x}}(\frac{\partial\beta}{\partial{y}})
\frac{\partial\alpha}{\partial{y}}=-\frac{\partial\beta}{\partial{x}}\Rightarrow\frac{\partial}{\partial{y}}(\frac{\partial\alpha}{\partial{y}})=\frac{\partial}{\partial{y}}(-\frac{\partial\beta}{\partial{x}})
Adding both these equations we have:
\nabla^2\alpha=0(because \frac{\partial^{2}\beta}{\partial{x}\partial{y}}=\frac{\partial^{2}\beta}{\partial{y}\partial{x}})
Similarly we can prove:
\nabla^2\beta=0by differentiating the first equation wrt y and the second wrt to x.


02 Apr 2008, 08:59
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