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

and that

Now we

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.