  We present a Melnikov method to analyze two-dimensional stable and unstable
manifolds associated with a saddle point in a three-dimensional non-volume
preserving autonomous systems. The time-varying perturbed locations of such
manifolds is obtained under very general, non-volume preserving and with
arbitrary time-dependence, perturbations. In unperturbed situations with a
two-dimensional heteroclinic manifold, we adapt our theory to quantify the
splitting into a stable and unstable manifold, and thereby obtain a Melnikov
function characterizing the time-varying locations of transverse intersections
of these manifolds, as well as a formula for the volume of each lobe generated
via such intersections. We demonstrate our theory using both the classical and
the swirling versions of Hill's spherical vortex.
