  The van Vleck determinant is an ubiquitous object, arising in many physically
interesting situations such as: (1) WKB approximations to quantum time
evolution operators and Green functions. (2) Adiabatic approximations to heat
kernels. (3) One loop approximations to functional integrals. (4) The theory of
caustics in geometrical optics and ultrasonics. (5) The focussing and
defocussing of geodesic flows in Riemannian manifolds. While all of these
topics are interrelated, the present paper is particularly concerned with the
last case and presents extensive theoretical developments that aid in the
computation of the van Vleck determinant associated with geodesic flows in
Lorentzian spacetimes. {\sl A fortiori} these developments have important
implications for the entire array of topics indicated. PACS: 04.20.-q,
04.20.Cv, 04.60.+n. To appear in Physical Review D47 (1993) 15 March.
