Riemannian geometry is customarily developed by tensor methods, which is not necessarily the most computationally efficient approach.  Using the language of differential forms, Elie Cartan's formulation of the Riemannian geometry can be elegantly summarized in two structural equations.  Essentially, the local curvature of the manifold is a measure of how the connection varies from point to point.  This Maple worksheet uses the DifferentialGeometry package to solves three problems in Harley Flanders' book on differential forms to demonstrate the implementation of Cartan's method.