Question

Use Green's first identity to show that if \(f\)is harmonic on \(D\), and if \(f(x,y) = 0\)on the boundary curve \(C\), then . (Assume the same hypotheses as in Exercise 31.)

Short answer

If \(g\)is harmonic on \(D\), and if \(f(x,y) = 0\)on the boundary curve \(C\), then

\[\iint_{D}{|}\nabla f{{|}^{2}}dA=0\]

Step-by-step solution

To Find theGreen’s first identity

Start with Green's first identity ,

\[\iint_{D}{f}{{\nabla }^{2}}gdA=\oint_{C}{f}(\nabla g)\cdot nds-\iint_{D}{\nabla }f\cdot \nabla gdA\]

and rewrite the identity with\(g = f\):

\[\iint_{D}{f}{{\nabla }^{2}}fdA=\oint_{C}{f}(\nabla f)\cdot nds-\iint_{D}{\nabla }f\cdot \nabla fdA\]

Now, if\(f\)is harmonic, then\({\nabla ^2}f = 0\)on\(D\), and . Also, if\(f(x,y) = 0\)on the curve\(C\), then\(\oint_C f (\nabla f) \cdot {\bf{n}}ds = 0\)Thus, equation (1) is reduced to

\[\iint_{D}{\nabla }f\cdot \nabla fdA=0\]

or, using the property \(a \times a = |a{|^2}\), we have

\[\iint_{D}{|}\nabla f{{|}^{2}}dA=0\]

Final Answer

If \(g\)is harmonic on \(D\), and if \(f(x,y) = 0\)on the boundary curve \(C\), then

\[\iint_{D}{|}\nabla f{{|}^{2}}dA=0\]