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## Uniqueness Theorem

There is a uniqueness theorem for Laplace's equation such that if a solution is found, by whatever means, it is the solution. The proof follows a proof by contradiction.

Suppose that, in a given finite volume bounded by the closed surface , we have

and that

Now we assume that there are two different functions and satisfying these conditions. Then,

satisfies

and, since both and both satisfy the same boundary conditions,

Now we calculate the integral of over the volume, , of the plasma. This integral must be positive and we will show that it must, in fact, be zero. Hence,

We now use a vector identity to rewrite the right hand side in a form suitable for the divergence theorem. Thus, the right hand side becomes

The second term is automatically zero since . Using the divergence theorem we obtain

where is a unit vector normal to the surface, . This integral is equal to zero since on the surface .

Finally, since the original integrand , is positive, the only way that the integral can be zero is if is a constant. However, the condition on the surface tells us that the constant must in fact be zero. Hence,

Hence, there is a unique solution to Laplace's equation.

Next: Basics Potential Solutions Up: Physical Applications Previous: Boundary Conditions
Andrew Wright
2002-09-16