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Benny May10-06 03:54 AM

Integrals of normal derivatives
 
Hi can someone please help me out with the following questions?

Supposed that [itex]\phi [/itex] is a C^2 function which satisfies Laplace's equation [itex]\nabla ^2 \phi = \nabla \bullet \nabla \phi = 0[/itex] everywhere in a region V bounded by a closed surface S.

a) If [tex]\frac{{\partial \phi }}{{\partial n}} = \mathop n\limits^ \to \bullet \nabla \phi [/tex] denotes the direction derivative of [itex]\phi[/itex] in the direction of the outward unit normal n, establish the following.

[tex]
\int\limits_{}^{} {\int\limits_S^{} {\frac{{\partial \phi }}{{\partial n}}} \partial S = 0}
[/tex]

[tex]
\int\limits_{}^{} {\int\limits_S^{} {\phi \frac{{\partial \phi }}{{\partial n}}dS = \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_V^{} {\left\| {\nabla \phi } \right\|^2 } } } } } dV
[/tex]

b) If [tex]\frac{{\partial \phi }}{{\partial n}}[/tex] vanishes everywhere on S, deduce from the second result that [itex]\phi[/itex] must be constant in V.

c) If [itex]\phi[/itex] vanishes everywhere on S, deduce from the second result that [itex]\phi[/itex] must vanish everywhere in V.

Part a: Phi is a function of three variables, otherwise the question wouldn't really make sense. I tried to visualise what the statement meant so I tried some specific cases. If S is a hemisphere then V is a half unit ball. The gradient of phi is pendicular to level sets of phi so in this case the gradient of phi is a normal to V (since phi is a function of three variables). So in this case, the unit normal n, to S, is parallel to grad(phi). So how can the integral be zero? The integrand is non-zero. I'm not sure what to do here.

For the second result I'm fairly sure that I need to apply the divergence theorem. But I don't really know how to 'div' the integrand of the surface integral because I don't what the normal, n, is. I think I might need to use a general property somewhere here but again I really don't know what to do.

Part b: If the partial derivative with respect to n is zero on S then from the second result, the triple integral on the RHS must be equal to zero everywhere in V. This can only be the case if the integrand (grad(phi))^2 is identically zero. So phi is constant in V? Not sure.

Part c: I can't think of a way to do this.

ANy help would be good thanks.


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