This book presents a unified treatment of discrete calculus, beginning with the classical calculus of finite differences and culminating in the discrete Hodge decomposition theorem on simplicial complexes. The Hodge decomposition asserts that every discrete differential form on a finite simplicial complex decomposes uniquely and orthogonally into an exact, coexact, and harmonic component, with the harmonic forms recovering the cohomology. This single result ties together the algebraic, topological, and analytic dimensions of discrete calculus.
The exposition is organized around three strands. The first develops the algebraic calculus of differences: difference operators, falling factorials, summation, the umbral calculus, and the Euler–Maclaurin formula. The second treats linear difference equations and discrete dynamics, including the Z-transform, stability theory, and an introduction to chaos through the logistic map. The third—and the heart of the book—constructs calculus on graphs and simplicial complexes: gradient, divergence, the graph Laplacian, simplicial homology and cohomology, discrete differential forms, and the Hodge Laplacian. A central thesis of the book is that these three strands are not merely analogous but convergent, sharing a common algebraic structure that finds its deepest expression in the Hodge decomposition.
The book is intended for advanced undergraduates in the mathematical sciences seeking a coherent framework that unifies the calculus of finite differences, spectral graph theory, and discrete exterior calculus. No prior knowledge of graph theory, algebraic topology, or differential geometry is assumed; all necessary concepts are developed from first principles.