Derivation of Jacobian matrices for the error propagation of charged particles traversing magnetic fields and materials

In high-energy physics experiments, the trajectories of charged particles are reconstructed using track reconstruction algorithms. Such algorithms need to both identify the set of measurements from a single charged particle and to fit the parameters by propagating tracks along the measurements. The propagation of the track parameter uncertainties is an important component in the track fitting to get the optimal precision in the fitted parameters. The error propagation is performed at the surface intersections by calculating a Jacobian matrix corresponding to the surface-to-surface transport. This paper derives the Jacobian matrix in a general manner to harmonize with semi-analytical numerical integration methods developed for inhomogeneous magnetic fields and materials. The Jacobian and transported covariance matrices are validated by simulating the charged particles between two surfaces and comparing with the results of numerical methods.