  We review a simple but instructive application of the formalism of covariant
bitensors, to use a deviation vector field along a fiducial geodesic to
describe a neighboring worldline, in an exact and manifestly covariant manner,
via the exponential map. Requiring the neighboring worldline to be a geodesic
leads to the usual linear geodesic deviation equation for the deviation vector,
plus corrections at higher order in the deviation and relative velocity. We
show how these corrections can be efficiently computed to arbitrary orders via
covariant bitensor expansions, deriving a form of the geodesic deviation
equation valid to all orders, and producing its explicit expanded form through
fourth order. We also discuss the generalized Jacobi equation, action
principles for the higher-order geodesic deviation equations, results useful
for describing accelerated neighboring worldlines, and the formal general
solution to the geodesic deviation equation through second order.
