The Fundamental Theorem of Geometric Calculus via a Generalized Riemann Integral

Here V is the tangent to M and A is the tangent to ∂M . (By the tangent, we mean, e.g., that V (X) is the unit positively oriented pseudoscalar in the tangent algebra to M at X.) Recall the important relationship V A = N , where N is the unit outward normal to M [4, p. 319]. The relationships dV = |dV |V and dA = |dA|A define the integrals componentwise as Lebesgue integrals on M and ∂M [4, p. 317]. Eq. 1 is a generalization of the fundamental theorem of calculus and the integral theorems of vector calculus [4, p. 323], Cauchy’s theorem [5], and an arbitrary dimension multivector version of Cauchy’s theorem [3, 5].