  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.
