Thus

Abstract

In a previous paper [1], the fundamentals of differential and integral calculus on Euclidean n-space were expressed in terms of multivector algebra. The theory is used here to derive some powerful theorems which generalize well-known theorems of potential theory and the theory of functions of a complex variable. Analytic multivector functions on En are defined and shown to be appropriate generalizations of analytic functions of a complex variable. Some of their basic properties are pointed out. These results have important applications to physics which will be discussed in detail elsewhere. 1. INTEGRAL OF THE GRADIENT OF A FUNCTION A multivector function f defined on a region R in En is said to be differentiable on R if its gradient ∇f(x) exists in some sense at each point x in R. If f and g are differentiable on R, then ∫ R g dv ∇f + (−1)n+1 R (g∇) dvf = ∂R g da f. (1.1) This formula still holds if either f or g is a generalized function [2] (distribution), a fact which often simplifies integration. Here it is used to integrate ∇f. Let ∇ represent the gradient operating at the point x and ∇ ′ the gradient operating at the point x′. If r = x − x′, then ∇|r | = r|r | = −∇ ′|r | (1.2) ∇r = n = −∇′r. (1.3) It can be readily verified that the equation g(r) ∇ = δ(r) (1.4) admits the particular solution g(r)