The importance of manifolds and Riemannian geometry in mathematics is
spreading to applied fields in which the need to model non-linear structure has
spurred wide-spread interest in geometry. The transfer of interest has created
demand for methods for computing classical constructs of geometry on manifolds
occurring in practical applications. This paper develops initial value problems
for the computation of the differential of the exponential map and Jacobi
fields on parametrically and implicitly represented manifolds.