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Tutorial on Geometric Calculus
David Hestenes Arizona State University The book Clifford Algebra to Geometric Calculus is the first and still the most complete exposition of Geometric Calculus (GC). But it is more of a reference book than a textbook, so can it be a difficult read for beginners. This tutorial is a guide for serious students who want to dig deeply into the subject. It presents helpful background and aims to clarify objectives, important results and methods in the book. It is supplemented by a hot-linked annotated bibliography of papers elaborating on various aspects of Geometric Algebra and Calculus. Objectives of this Tutorial Only a handful of people have mastered Clifford Algebra to Geometric Calculus [2] to the point of using it in their research. Among the first are Chris Doran, Steve Gull and Anthony Lasenby. I am proud to say that it helped them produce some truly innovative theoretical physics, most notably, an improvement of General Relativity called Gauge Theory Gravity [4, 5]. The book [2] has many ideas and results that remain to be exploited, but first it is necessary to master the core concepts and mathematical tools. Readers of this tutorial are presumed to be familiar with the basics of Geometric Algebra (GA) , so we can concentrate on more ambitious objectives. The tutorial emphasizes the following fundamental concepts of Geometric Calculus, explaining their unique features and advantages:
- Universal Geometric Algebra – arbitrary dimension and signature
- Linear and multilinear algebra (tensors, determinants)
- Vector manifolds – for representing any manifold
- Directed integrals and differential forms
- Vector derivative and the fundamental theorem of calculus
- Differentials and codifferentials for mappings and fields
- Coordinate-free differential geometry
- Lie groups as Spin groups Origins of the Geometric Calculus book Though Geometric Calculus (GC) had important precursors [6], its systematic development as a unified language for mathematics and physics began in 1966 with publication of the book SpaceTime Algebra (STA) [1]. That book shows how STA provides compact, coordinate-free formulations for basic equations of physics that provide new insights into their geometric structure. Specifically,
- Maxwell’s four electromagnetic equations are reduced to a single equation: (^)! F = J
- Dirac’s electron equation is given a new manifestly geometrical form.
- Einstein’s General Relativity is formulated with a spinor form of the Principle of Local Relativity.
However, to solve and apply these equations by standard mathematical methods at the time, it was necessary to translate them into standard formulations, which detracted from their compact and elegant structure. To take full advantage of the new formulations, new computational tools and methods or, at least, coordinate-free reformulations and adaptations of old methods were needed. Creation of new and more powerful mathematical tools began immediately with extraction of the concepts of vector derivative and directed integral from [1] and further development in mathematical papers in {3, 4}. [Note: Papers available for direct online access are numbered in curly brackets {…} and commented on below before the references, while book references are numbered in square brackets […]. ] This led immediately to reduction of the integral theorems of Gauss, Stokes, Green and Cauchy into a single formula, and, more remarkably, to generalization of Cauchy’s integral theorem to arbitrary dimension. In this way, it unified real and complex variable theory. Moreover, it raised the questions about how GC relates to the Calculus of Differential forms, in particular, with respect to transformations (change of variables) in integrals. I was fortunate to have a capable student, Garret Sobczyk [7] to help me answer this. When Sobczyk completed his thesis in 1971, I combined it with ideas of my own into a series of three papers submitted for publication in mathematics journals. He went off on his own, ending up as a postdoc in Poland. The papers were rejected by three different journals in three successive years, but each with the recommendation that they be published as a book. As I had a lot more to say on the subject, I was not averse to writing a book, though I thought it was premature. Clifford Truesdell directed me to Mario Bunge, then editor of an advanced book series published by D. Reidel Company, who accepted my book plan immediately. That was the easy part. Little did I know that it would take a decade to get the book in print. Writing was the easy part. In those days before desktop publishing technical manuscripts were written by hand and then typed by a secretary. Arizona State University had only one technical typist for the departments of physics, mathematics and chemistry, so it took three years to get my manuscript typed. Anyway, the manuscript was finished by 1976 and shipped off to Reidel for publication, only to be rejected by the publisher some six months later. I had made the mistake of submitting directly to the publisher instead of going through the editor Bunge. Back to square one! I wasn’t worried though, because I was confident of the book’s quality. In 1978 I was pleased to get a letter from the distinguished mathematician Gian-Carlo Rota requesting a copy of my book SpaceTime Algebra [1]. I looked him up shortly thereafter when I attended a Maximum Entropy Conference at MIT and asked him why he was interested. He gave me copies of several papers of his on Invariant Theory. I was astounded by how close it was to my treatment of GA identities in the GC book, so in just a few days I was able to work it into my treatment of determinants in Chap.1. That was the last change made to the manuscript. It opened up rich opportunities for integrating GA with Invariant Theory that are yet to be fully exploited. Rota agreed to consider my manuscript for publication by Addison-Wesley, for which he was editor of the Encyclopedia of Mathematics series. He requested six copies of the manuscript to be sent to reviewers. What happened thereafter is too involved to recount in detail. The net result was many delays. After several years, Rota surprised me one Saturday morning with the most gratifying phone call of my life: First, he explained that I had been unable to reach him because he had been in the hospital for the better part of a year, but he had already strongly
Readers need not go through all the calculations in Chap. 1; it is advisable just to sample the calculations and proofs to see how they work. The rest can be left for reference as needed. Some years after writing this chapter, when Grassmann’s original work was translated into English, I learned that he had derived the main identities a hundred years before. An important feature of Universal Geometric Algebra defined in the book is that it is generated by an infinite dimensional vector space. All finite dimensional vector spaces and their geometric algebras are then defined by choice of a pseudoscalar, as indicated in the next to last line of Fig. 1. The linear structure of such algebras is schematized in Fig. 2. Fig. 2 The book starts out by assuming a Euclidean signature for the algebra, because that was done in the three original mathematical papers from which Chap. 1 is mostly composed. I wasn’t sure that all the proofs would carry over to non-Euclidean signatures, and I didn’t want to take the trouble to check, so I introduced non-Euclidean signatures only at the end of Chap. 1. This brings up a common misconception about GA, namely, that use of an inner product limits its applicability to metric spaces. On the contrary, as shown in Fig. 3, the inner product can serve as a contraction without any notion of metric. Thus it can be seen that the infinite dimensional GA is equivalent to the algebra of fermion operators in quantum field theory. The formulation of finite dimensional GA with all possible signatures is schematized in Fig. 4.
surprise a few years later, the Cambridge group used it for elegant derivations of conservation laws in Lagrangian field theory. More recently, it has been applied to robotics by Lasenby and Doran as well as Valkenburg and Alwesh in [9]. Chap. 2 also introduces a concept of “simplcial derivative,” but my current opinion is that its applications are better done by other means. Fig. 6 Chapter 3: Linear and Multilinear Functions Geometric Algebra introduces the powerful new concept of “outermorphism” to the field of linear algebra. The outermorphism of a linear function defined on a vector space is a unique linear extension to the entire geometric algebra of the vector space (as defined in Fig. 2). In Chap. 3 the simplicial derivative is used to define it, but I now prefer the alternative approach used in {6}, which also extends the treatment of linear algebra. This material deserves to be extended to a complete book on linear algebra. The typical reader is advised to peruse the chapter to get a sense of the approach, and then to refer back to it for details as needed. Chapter 4: Calculus on Vector Manifolds The preceding definition of vector derivative presumes the vector variable is defined on a vector space. This chapter removes that restriction by defining the concept of vector manifold , and that enables us in subsequent chapters to create a completely coordinate-free approach to differential geometry. The concept of vector manifold is schematized in Fig. 7, and its relation to the use of coordinates is outlined in Fig. 8.
Fig. 7 Fig. 8
–– a set on which differential and integral calculus is well-defined!
M
m
What is a manifold? of dimension m
R
m
- Calculus done indirectly by local mapping to = R! R!! R
- Proofs required to establish results independent of coordinates.
Geometric Calculus defines a manifold as any set
isomorphic to a vector manifold
M
m
Vector manifold =^ { x }is a set of vectors in GA that generates
at each point x a tangent space with pseudoscalar Im ( x )
Advantages:
- Manifestly coordinate-free vs. nominally coordinate-free!
- Calculus done directly with algebraic operations on points
- Geometry completely determined by derivatives of Im ( x ).
M
m
Remark: It is unnecessary to assume that is embedded in a
vector space, though embedding theorems can be proved.
As illustrated in Fig. 9, a diffeomorphism (map) on a vector manifold induces a transformation of the tangent space at each point called the differential (also called the “push-forward”). The adjoint of this transformation (also called the “pull-back”) goes in the opposite direction. Note that the differential and adjoint are defined, respectively, by the directional derivative and the gradient of the map. As indicated in Fig. 10, the differential and adjoint transformations of tangent vectors are extended to the entire tangent algebra by outermorphism. The outermorphism of the pseudoscalar gives the Jacobian of the map, notably, without introducing a local coordinate system. This coordinate-free treatment of diffeormorphism has been applied to great effect in defining the concept of position gauge transformation , a keystone in the Gauge Theory of Gravity of Lasenby, Doran and Gull [4,5]. That resolves a longstanding problem of providing a precise definition of Einstein’s General Relativity Principle [5]. Note that the whole apparatus of differential outermorphism applies equally well to linear algebra. Indeed, a map is linear if it is equal to its differential. Chapters 5 & 6: Differential Geometry Now we have the necessary tools for addressing coordinate-free differential geometry. There are two distinct methods, each with distinct advantages. The method of mobiles (comoving frames in Chap. 6 is originated in the 19 th century, but GA greatly enhances it with a spinor treatment of rotations and their derivatives. The method of sliding pseudoscalar in Chap. 5 is reviewed in {9} and summarized in Fig. 11. Fig. 11
Chapter 7: Directed Integration Theory The directed integral is defined in Fig. 12 in terms of the more familiar multiple integral. Its power is shown by the formulation of the Fundamental Theorem of Geometric Calculus in Fig. 13. Fig. 12 Fig. 13
The most general form of the Fundamental Theorem is given in Fig. 15. It reduces precisely to the standard theory of differential forms when the integrands are scalar-valued. But non-scalar integrands lead to Cauchy’s Theorem and Cauchy’s Integral Theorem in complex variable theory and their generalizations to higher dimensions, as first demonstrated in {4}. This is illustrated in Fig. 16, which gives the integral form of Maxwell’s equation! E = " for a static electric field with charge density! = !( x ). This, of course, is a solution of Maxwell’s equation if the charge density inside the region and the electric field on the boundary are given. Note that in the 2D case the formula reduces to a generalization of Cauchy’s integral formula that includes an area integral. That generalization was first given by Pompieu in 1910, but is seldom mentioned in books on complex variable theory, which are hung up on the notion that complex line integrals are something special. Fig 16 A more detailed discussion of differential forms in Geometric Calculus is given in {7}, and a valuable alternative approach is given in {8}. Chapter 8: Lie Groups and Lie Algebras This chapter is a prospectus for incorporating the theory of Lie Groups and Lie Algebras. It was originally proposed as a doctoral thesis for a student of mine, but he got engaged in other things, so I wrote it up myself. Incomplete as it is, it has served well as a stimulus for further developments. For example, I worked out an explicit representation for the conformal group, which has played a key role in developing Conformal Geometric Algebra [6,9]. In another direction, it stimulated me to represent generators of the symplectic group as bivectors in phase space, and Chris Doran extended that to a complete treatment of the classical groups {10}. The general idea is summarized in Fig. 17. The crystallographic point groups and space groups are fully worked out in {11,12}.
Fig. 17
Selected online papers
Here is a brief description of selected papers available online that elaborate the fundamental concepts of GA or treat them at a more elementary level. The first two papers {1, 2} develop the fundamentals for undergraduate physics majors. The first published papers on Geometric Calculus {3, 4} referred to the subject more modestly as Multivector Calculus, because development was not yet sufficient to claim it as a universal mathematical language. That claim could be confidently made in {5}, because adequate foundations had been laid in the books [1, 2, 3]. Of the remaining papers listed below, the most recent {9} reviews the powerful approach to differential geometry using the Shape Operator and suggests directions for further research. Comments on the other papers are given in the chapter discussions above. (Note: titles are linked to web pages on which the papers can be found.) {1} Synopsis of Geometric Algebra Summarizes and extends some of the basic ideas and results of GA. To make the summary self-contained, all essential definitions and notations are explained, and geometric interpretations of algebraic expressions are reviewed. {2} Geometric Calculus (Fundamentals) A thorough treatment of differentiation and integration with respect to vector variables
{11} Point and Space Groups in Geometric Algebra A detailed treatment of the 3D point groups and crystal classes with GA. Minimal background with GA required. {12} The Crystallographic Space Groups in Geometric Algebra Complete treatment of the 17 different 2D space groups and 230 different 3D space groups demonstrating the considerable advantages of formulation with Conformal GA.
Books about Geometric Algebra
[1] D. Hestenes, Space-Time Algebra , Gordon & Breach, New York, (1966). [2] D. Hestenes and G. Sobczyk, CLIFFORD ALGEBRA to GEOMETRIC CALCULUS, A Unified Language for Mathematics and Physics, Kluwer: Dordrecht/Boston (1984), paperback (1985). Fourth printing 1999. {Reviewed by James S. Marsh (1984), American Journal of Physics 53 (5): 510-11.) [3] D. Hestenes, New Foundations for Classical Mechanics , Kluwer: Dordrecht/Boston (1986), paperback (1987). Second Edition (1999). [4] C. Doran and A. Lasenby. Geometric Algebra for Physicists. Cambridge: The Univer- sity Press, 2003. [5] D. Hestenes, Gauge Theory Gravity with Geometric Calculus, Foundations of Physics 36 , 903 - 970 (2005). [6] D. Hestenes, Grassmann’s Legacy. In H-J. Petsche, A. Lewis, J. Liesen, S. Russ (Eds.) From Past to Future: Grassmann’s Work in Context (Birkhäuser: Berlin, 2011 ). [7] G. Sobszyk, Mappings of Surfaces in Euclidean Space using Geometric Algebra, Thesis, Arizona State University (1971). [8] H. Li, Invariant Algebras and Geometric Reasoning. (Beijing: World Scientific, 2008) [9] L. Dorst and J. Lasenby (Eds.) Guide to Geometric Algebra in Practice (Springer: London, 2011 ). [10] L. Dorst, D. Fontijne, and S. Mann. Geometric Algebra for Computer Science
