  We derive general expressions for the sum rules of the eigenvalues of drums
of arbitrary shape and arbitrary density, obeying different boundary
conditions. The formulas that we present are a generalization of the analogous
formulas for one dimensional inhomogeneous systems that we have obtained in a
previous paper. We also discuss the extension of these formulas to higher
dimensions. We show that in the special case of a density depending only on one
variable the sum rules of any integer order can be expressed in terms of a
single series. As an application of our result we derive exact sum rules for
the a homogeneous circular annulus with different boundary conditions, for a
homogeneous circular sector and for a radially inhomogeneous circular annulus
with Dirichlet boundary conditions.
