Introduction to Discrete Calculus

Abstract

In this chapter we review conventional vector calculus from the standpoint of a generalized exposition in terms of exterior calculus and the theory of forms. This generalization allows us to distill the important elements necessary to operate the basic machinery of conventional vector calculus. This basic machinery is then redefined in a discrete setting to produce appropriate definitions of the domain, boundary, functions, integrals, metric and derivative. These definitions are then employed to demonstrate how the structure of the discrete calculus behaves analogously to the conventional vector calculus in many different ways.