  In a 1991 paper by Buttig and Eichhorn, the existence and uniqueness of a
differential forms heat kernel on open manifolds of bounded geometry was
proven. In that paper, it was shown that the heat kernel obeyed certain
properties, one of which was a relationship between the derivative of heat
kernel of different degrees. We will give a proof of this condition for
complete manifolds with Ricci curvature bounded below, and then use it to give
an integral representation of the heat kernel of degree $k$.
