The Jet Formula and Heat Kernel Coefficients The differential information of a section of a vector bundle over a manifold contained in its partial derivatives up to some order $d$ at a given point $x$ with respect to a trivialisation over a local coordinate chart can be encoded in its symmetrized iterated covariant derivatives up to order $d$ at $x$. In consequence it is possible to express every linear differential operator in terms of symmetrized covariant derivatives alone. In particular there exists an explicit formula expressing the iterated covariant derivatives themselves in terms of their symmetrizations. This jet formula tames myriads of seemingly unrelated curvature terms and can be used to give an explicit formula for the heat kernel coefficients of all generalized Laplacians with potentials given by Weitzenb\"ock-like curvature terms. After formulating the jet formula I will discuss its application to heat kernel coefficients as well as other applications if time permits.