Section II. Geometric Calculus

Section II is concerned with the extension of CA to GC, especially vector derivatives and directed integrals. The basic ideas were originally set forth in the papers Multivector Calculus and Multivector Functions -- subsequently elaborated in CA to GC. These ideas provide the foundation for "Clifford Analysis," a new branch of mathematics emerging during the last two decades. The first paper in Section II explains that the crucial synthesis of Clifford algebra with differential forms that opens this branch was made independently by several investigators. However, the crux of the synthesis is seldom understood, so more commentary is needed. It is no small irony that Clifford algebra and differential forms emerged from the work of Grassmann, but are combined in a kind of hybrid in most accounts of Clifford analysis. In line with Grassmann's vision, Geometric Calculus reconstructs the theory of differential forms in terms of GA from the ground up. The consequence, of course, is greater simplicity, clarity and efficiency. The paper on Hamiltonian mechanics opens up a new approach to the subject. An important by-product is that the characterization of symplectic relations given there provided a key to the whole development of Lie groups as spin groups.

Unified Language for Mathematics and Physics

Abstract: To cope with the explosion of information in mathematics and physics, we need a unified mathematical language to integrate ideas and results from diverse fields. Clifford algebra provides the key to a unified Geometric Calculus for expressing, developing, integrating and applying the large body of geometrical ideas running through mathematics and physics.

D. Hestenes, in: Clifford Algebras and their Applications in Mathematical Physics, 1-23, 1986 (J.S.R. Chisholm and A.K. Common, Eds.), Kluwer Academic Publishers, Dordrecht.
© Kluwer

Differential Forms in Geometric Calculus

Abstract: Geometric calculus and the calculus of differential forms have common origins in Grassmann algebra but different lines of historical development, so mathematicians have been slow to recognize that they belong together in a single mathematical system. This paper reviews the rationale for embedding differential forms in the more comprehensive system of Geometric Calculus. The most significant application of the system is to relativistic physics where it is referred to as Spacetime Calculus. The fundamental integral theorems are discussed along with applications to physics, especially electrodynamics.

D. Hestenes, in: Clifford Algebras and their Applications in Mathematical Physics, 269-285, 1993 (F. Brackx et al, Eds.), Kluwer Academic Publishers, Dordrecht.
© Kluwer

Simplicial Calculus with Geometric Algebra

Abstract: We construct geometric calculus on an oriented k-surface embedded in Euclidean 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 finite difference calculations with integral and differential equations in Clifford algebra.

© G. Sobczyk.

Multivector Calculus

Abstract: The object of this paper is to show how differential and integral calculus in many dimensions can be greatly simplified by using Clifford algebra. Here the necessary notations, definitions, and fundamental theorems are developed to make the calculus ready to be used. Those features of Clifford algebra which are needed for this task are described without proof.

The discussion of differentiation and integration omits without comment many important problems in analysis, because they are in no way affected by the special features of the approach advanced here. The object throughout is to show how Clifford algebra can be used to advantage.

D. Hestenes, Journal of Mathematical Analysis and Applications,24, No. 2, November 1968, 313-325.
© Academic Press.

Multivector Functions

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 E_n 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.

D. Hestenes, Journal of Mathematical Analysis and Applications,24, No. 3, December 1968, 467-473.
© Academic Press.

The Shape of Differential Geometry in Geometric Calculus

Abstract: We review the foundations for coordinate-free differential geometry in Geometric Calculus. In particular, we see how both extrinsic and intrinsic geometry of a manifold can be characterized a single bivector-valued oneform called the Shape Operator. The challenge is to adapt this formalism to Conformal Geometric Algebra for wide application in computer science and engineering.

D. Hestenes, Chapter 19 in Guide to Geometric Algebra in Practice, L.Dorst and J.Lasenby, eds., 2011, 393-410.
© Springer.


I. Saccadic and Compensatory Eye Movements

Abstract: A new invariant formulation of 3D eye-head kinematics improves on the computational advantages of quaternions. This includes a new formulation of Listing's Law parameterized by gaze direction leading to an additive rather than a multiplicative saccadic error correction with a gaze vector difference control variable. A completely general formulation of compensatory kinematics characterizes arbitrary rotational and translational motions, vergence computation, and smooth pursuit. The result is an invariant, quantitative formulation of the computational tasks that must be performed by the oculomotor system for accurate 3D gaze control. Some implications for neural network modeling are discussed.

D. Hestenes, Neural Networks, 7, No. 1, 1994, 65-77.
© Elsevier Science Ltd., all rights reserved.

II. Reaching and Neurogeometry

Abstract: Invariant methods for formulating and analying the mechanics of the skeleto-muscular system with geometric algebra are further developed and applied to reaching kinematics. This work is set in the context of a neurogeometry research program to develop a coherent mathematical theory of neural sensory-motor control systems.

D. Hestenes, Neural Networks, 7, No. 1, 1994, 79-88.
© Elsevier Science Ltd., all rights reserved.

Hamiltonian Mechanics with Geometric Calculus

Abstract: Hamiltonian mechanics is given an invariant formalism in terms of Geometric Calculus, a general differential and integral calculus with the structure of Clifford algebra. Advantages over formulations in terms of differential forms are explained.

D. Hestenes, in: Spinors, Twistors, Clifford Algebras and Quantum Deformations, 203-214, 1993 (Z. Oziewicz et al, Eds.), Kluwer Academic Publishers, Dordrecht.
© Kluwer
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