Simplicial Calculus with Geometric Algebra c Garret Sobczyk (Posted with permission) ABSTRACT. We construct geometric calculus on an oriented k-surface embedded in Eu- clidean space by utilizing the notion of an oriented k-surface as the limit set of a sequence of k-chains. This method provides insight into the relationship between the vector derivative, and the Fundamental Theorem of Calculus and Residue Theorem It should be of practical value in numerical nite dierence calculations with integral and dierential equations in Cliord algebra. 0. Introduction In 1968, D. Hestenes showed how Geometric Algebra can be used to advantage in refor- mulating ideas from multivariable calculus [1], [2]. For example, he showed that the proof of Stokes’ theorem becomes a one-line identity in geometric algebra, if the integral denition of the vector derivative is adopted. In this paper, we systematically build up calculus on a k-surface in order to more closely examine the content of these theorems. Section 1 gives a brief introduction to the geometric algebra of Euclidean n-space, includ- ing basic denitions and identities which are be used in later sections. The inner, outer, and geometric products of vectors are discussed, as is the notion of a reciprocal frame of vectors. The material in this section is taken from [3], [4], and, primarily [5]. In section 2 the concept of an oriented simplex is introduced and related notions from homology theory are reviewed; more details can be found in [6], [7]. Peculiar to the present approach are the concepts of the directed content of a simplex, made possible by the intro- duction of geometric algebra, and the simplicial variable of a k-surface. These concepts are the basic building blocks for our theory of simplicial calculus developed in later sections. In section 3 a k-surface is dened to be the limit set of an appropriate sequence of chains of simplices. A k-surface ϕ is said to be smooth if there exists a smooth k-vector eld, called the pseudoscalar eld, at each point of ϕ. Our approach is most closely related to [8]. In section 4 the directed integral on a k-surface is dened in terms of the directed content of the limit of the sequence of chains which denes it. The theory of directed integrals is rst developed over a k-simplex, and then generalized. The most remarkable theorem of this section, which has no counterpart in the related theory of scalar-valued dierential forms, expresses the directed moment of the boundary of a k-simplex as the inner product of the directed content of the simplex with the direction of the axis. In section 5 the vector derivative is dened in terms of the limit of the directed integral over the boundary of the simplicial variable as this variable approaches zero. The vec- 1 tor derivative is shown to be equivalent to the ordinary gradient when applied to scalar functions. Section 6 presents a simple proof of the Fundamental Theorem of Calculus using the simplicial calculus developed in earlier sections. This theorem relates the directed integral of the vector derivative of a function over a k-surface to the directed integral of the function over the boundary of the k-surface. In section 7 the Dirac delta function is introduced, [9], [10], to quickly obtain a powerful Residue Theorem. Cauchy’s integral formula is shown to be a special case of this theorem. 1. The Geometric Algebra of Euclidean Space Let En denote Euclidean n-space represented as an n-dimensional vector space with a positive denite inner product, which we denote by x y for x, y ∈E. (1.1) The elements of En will alternatively be referred to as points or vectors, depending upon their usage [3,p.15]. The geometric algebra Gn of En, is the associative algebra generated by geometric multi- plication of vectors in En. In the literature it is often referred to as the Cliord algebra of the 2 quadratic form x , [4]. The geometric product of the vectors x, y ∈En can be decomposed into xy = x y + x ∧ y, (1.2) where the inner product 1 x y = 2 (xy + yx) , (1.3) is the symmetric part of the geometric product, and the outer product ∧ 1 x y = 2 (xy yx) , (1.4) is the antisymmetric part of the geometric product. The quantity x ∧ y is called a 2-vector, or bivector; it can be interpreted geometrically as a directed area. Let x1,...,xk be k vectors in En. The outer product X 1 x ∧∧x = (1)x x x , (1.5) 1 k k! 1 2 k is dened as the totally antisymmetric part of the geometric product of these vectors. The sum in (1.5) is taken over all permutations of the indices i;ifis an even or odd permutation we set = 0 or 1 respectively. The geometric number x1 ∧∧xk is called a k-vector, it can be interpreted geometrically as a directed k-volume. The magnitude or k-volume of x1 ∧∧xk will he denoted by | x1 ∧∧xk |. A systematic construction of the geometric algebra G can be found in [5]. The notation and algebraic identities used here are largely taken from this reference. 2 We gather here a number of identities and relationships which are indispensable in this work. For vectors b and ai, Xk i+1 ∨ b (a1 ∧ a2 ∧∧ak)= (1) (b ai)(a1 ∧ ∧ai ∧∧ak) i=1 (1.6) k1 =(1) (a1 ∧ a2 ∧∧ak)b ∨ The symbol over the ai in (1.6) means that ai is to be deleted from the product. Identity (1.6) explicitly shows that when a k-vector is “dotted” with a vector, the result is a (k 1)- vector. Dotting a k-vector with a vector is closely related to contracting an r-form by a vector. For a complete discussion of the relationship between r-forms and r-vectors see [5,p.33]. The following cancellation property will be used in a later section. If Ar and Br are r-vectors where r<n, and c ∧ Ar = c ∧ Br (1.7) for all vectors c ∈En, then Ar = Br. Given a set of linear independent vectors, {ei : i =1,...,k}, spanning a k-dimensional j subspace of En we can construct a reciprocal frame {e : j =1,...,k} spanning the same subspace, and satisfying Xk j j i ei e = i and e ∧ ei =0, (1.8) i=1 j where i = 1 or 0 according to whether i = j or i =6 j, respectively. The explicit construc- tion of the reciprocal frame is given in [5,p.28]. We close this section with a useful lemma regarding the volume of a regular k-simplex, and its moment with respect to any of its coordinate axes. The proof, by induction, is omitted. Z Z Z s s tk s tk t2 sk LEMMA. dt1dt2 dtk = (1.9) 0 0 0 k! Z Z Z s s tk s tk t2 sk+1 ti dt1dt2 dtk = (1.10) 0 0 0 (k + 1)! 2. Simplices Let {a0,a1,...,ak} be an ordered set of points in En. The oriented k-simplex (a)(k) of these points is dened by Xk (a)(k) (a0,a1,...,ak)= a|a= ta , (2.1) =0 P k where =0 t = 1 and 0 t 1, are the barycentric coordinates of the point a.We say that the simplex (a)(k) is located at the point a = a0. Alternatively, we will use the 3 symbolism ((k)a0) to denote a k-simplex at the point a0. By the boundary of (a)(k),we mean the (k 1) chain Xk i+1 ∨ ∂(a)(k) = (1) (aki)(k1) (2.2) i=0 ∨ where (aki)(k1) is the (k 1)-simplex dened by ∨ ∨ (aki)(k1) (a0,a1,...,aki,...ak). (2.3) ∨ As before, aj indicates that this point is omitted. It is not dicult to establish the basic result of homology theory: 2 ∂ (a)(k) =0 (2.4) For a discussion of simplices, chains, and their boundaries, see [6,p.57], [7,p.206]. By the directed content of the simplex (a)(k), we mean the k-vector (for k 1) 1 a D[(a) ]= (a a )∧(a a )∧∧(a a ). (2.5a) (k) (k) k! 1 0 2 0 k 0 With the abbreviated notation ai = ai a0, (2.5a) takes the form 1 a = a ∧∧a . (k) k! 1 k The simplex (a)(k) is said to be non-degenerate if its directed content a(k) =6 0. In the special case that k = 0, we dene a(0) D[(a)(0)]=1. (2.5b) More generally, it is possible to dene D[(a0)] = (a0), where (a0) is the mass density at the point a0, but we will not do this here. We have the following useful 1 LEMMA. a = (a a ) ∧ (a a ) ∧∧(a a ) (2.6) (k) k! 1 0 2 1 k k 1 1 Proof.a = (aa)∧(aa)∧∧(a a ) (k) k! 1 0 2 0 k 0 1 = (a a )∧∧(a a )∧(a a +a a ) k! 1 0 k 1 0 k k 1 k 1 0 1 = (a a )∧∧(a a )∧(a a ) k! 1 0 k 1 0 k k 1 1 = (a a )∧(a a )∧∧(a a ). k! 1 0 2 1 k k 1 Lemma (2.6) is needed in establishing that the directed content of the boundary of a simplex vanishes, as proven in the following 4 THEOREM. D[∂(a)(k)] = 0 (2.7) Proof.