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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 origininal integrand is positive, the ony 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
Prof. Alan Hood
2000-11-06