Book review: Discrete Calculus: Applied Analysis on Graphs for Computational Science, Leo Grady, Jonathan Polimeni. Springer (2010). $129.00, ISBN: 978-1-84996-289-6

摘要

The discrete exterior calculus describes the analogs of differentiation and integration on general discrete spaces such as trees, graphs, and other discrete structures in a way that preserves the inherent topological duality of these operators. Whereas the discretization of a continuous differential operator has always been the standard way to numerically compute gradients, it does not preserve the underlying topological properties of this operator (the exterior derivative) and does not yield relations that satisfy the fundamental theorem of calculus, nor Stokes' Theorem, its grand generalization to any orientable compact manifold. The discrete calculus gradient, defined as a coboundary operator on an inherently discrete structure does enjoy this property, which has some remarkable consequences.