  The Green function has a complex dependence upon its underlying domain and
differential operator. We briefly review Hadamard's formula for the first
variation of the Green function due to a perturbation of the domain. We then
take a different avenue and approximate the change in the Green function when
the Laplacian is perturbed into a number of different operators: Helmholtz,
Schr\"odinger and Laplace--Beltrami.
