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Building on an insight due to Avramidi, we provide a system of transportequations for determining key fundamental bi-tensors, including derivatives ofthe world-function, \sigma(x,x'), the square root of the Van Vleck determinant,\Delta^{1/2}(x,x'), and the tail-term, V(x,x'), appearing in the Hadamard formof the Green function. These bi-tensors are central to a broad range ofproblems from radiation reaction to quantum field theory in curved spacetimeand quantum gravity. Their transport equations may be used either in asemi-recursive approach to determining their covariant Taylor seriesexpansions, or as the basis of numerical calculations. To illustrate the powerof the semi-recursive approach, we present an implementation in\textsl{Mathematica} which computes very high order covariant series expansionsof these objects. Using this code, a moderate laptop can, for example,calculate the coincidence limit a_7(x,x) and V(x,x') to order (\sigma^a)^{20}in a matter of minutes. Results may be output in either a compact notation orin xTensor form. In a second application of the approach, we present a schemefor numerically integrating the transport equations as a system of coupledordinary differential equations. As an example application of the scheme, weintegrate along null geodesics to solve for V(x,x') in Nariai and Schwarzschildspacetimes.

Transport Equation Approach to Calculations of Hadamard Green functions and non-coincident DeWitt coefficients