Functional Differential Geometry

Differential geometry is deceptively simple. It is surprisingly easy to get the right answer with unclear and informal symbol ma-nipulation. To address this problem we use computer programs to communicate a precise understanding of the computations in differential geometry. Expressing the methods of differential ge-ometry in a computer language forces them to be unambiguous and computationally effective. The task of formulating a method as a computer-executable program and debugging that program is a powerful exercise in the learning process. Also, once formal-ized procedurally, a mathematical idea becomes a tool that can be used directly to compute results. 2A manifold is a generalization of our idea of a smooth sur-face embedded in Euclidean space. The critical feature of an n-dimensional manifold is that locally (near any point) it looks like n-dimensional Euclidean space: around every point there is