  We give a short proof of a strong version of the short time asymptotic
expansion of heat kernels associated to Laplace type operators acting on
sections of vector bundles over compact Riemannian manifolds, including
exponential decay of the difference of the approximate heat kernel and the true
heat kernel. We use this to show that repeated convolution of the approximate
heat kernels can be used to approximate the heat kernel on all of $M$, which is
related to expressing the heat kernel as a path integral. This scheme is then
applied to obtain a short-time asymptotic expansion of the heat kernel at the
cut locus.
