Clifford Algebra to Geometric Calculus

David Hestenes and Garret Sobczyk

© Kluwer. First published in 1984; reprinted with corrections in 1992.

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Geometric Calculus is a language for expressing and analyzing the full range of geometric concepts in mathematics. Clifford algebra provides the grammar. Complex numbers, quaternions, matrix algebra, vector, tensor and spinor calculus and differential forms are integrated into a singe comprehensive system. The geometric calculus developed in this book has the following features: a systematic development of definitions, concepts and theorems needed to apply the calculus easily and effectively to almost any branch of mathematics or physics; a formulation of linear algebra capable of detailed computations without matrices or coordinates; new proofs and treatments of canonical forms including an extensive discussion of spinor representations of rotations in Euclidean n-space; a new concept of differentiation which makes it possible to formulate calculus on manifolds and carry out complete calculations of such things as the Jacobian of a transformation without resorting to coordinates; a coordinate-free approach to differential geometry featuring a new quantity, the shape tensor, from which the curvature tensor can be computed without a connection; a formulation of integration theory based on a concept of directed measure, with new results, including a generalization of Cauchy's integral formula to n-dimensional spaces and an explicit integral formula for the inverse of a transformation; a new approach to Lie groups and Lie algebras.

Table of Contents

Symbols and Notation

Chapter 1 / Geometric Algebra
      1-1. Axioms, Definitions and Identities
      1-2. Vector Spaces, Pseudoscalars and Projections
      1-3. Frames and Matrices
      1-4. Alternating Forms and Determinants
      1-5. Geometric Algebras of PseudoEuclidean Spaces

Chapter 2 / Differentiation
      2-1. Differentiation by Vectors
      2-2. Multivector Derivative, Differential and Adjoints
      2-3. Factorization and Simplicial Derivatives

Chapter 3 / Linear and Multilinear Functions
      3-1. Linear Transformations and Outermorphisms
      3-2. Characteristic Multivectors and the Cayley-Hamilton Theorem
      3-3. Eigenblades and Invariant Spaces
      3-4. Symmetric and Skew-symmetric Transformations
      3-5. Normal and Orthogonal Transformations
      3-6. Canonical Forms and General Linear Transformations
      3-7. Metric Tensors and Isometries
      3-8. Isometries and Spinors of PseudoEuclidean Spaces
      3-9. Linear Multivector Functions
      3-10. Tensors

Chapter 4 / Calculus on Vector Manifolds
      4-1. Vector Manifolds
      4-2. Projection, Shape and Curl
      4-3. Intrinsic Derivatives and Lie Brackets
      4-4. Curl and Pseudoscalar
      4-5. Transformations of Vector Manifolds
      4-6. Computation of Induced Transformations
      4-7. Complex Numbers and Conformal Transformations

Chapter 5 / Differential Geometry of Vector Manifolds
      5-1. Curl and Curvature
      5-2. Hyperspaces in Euclidean Spaces
      5-3. Related Geometries
      5-4. Parallelism and Projectively Related Geometries
      5-5. Comformally Related Geometries
      5-6. Induced Geometries

Chapter 6 / The Method of Mobiles
      6-1. Frames and Coordinates
      6-2. Mobiles and Curvature
      6-3. Curves and Comoving Frames
      6-4. The Calculus of Differential Forms

Chapter 7 / Directed Integration Theory
      7-1. Directed Integrals
      7-2. Derivatives from Integrals
      7-3. The Fundamental Theorem of Calculus
      7-4. Antiderivatives, Analytic Functions and Complex Variables
      7-5. Changing Integration Variables
      7-6. Inverse and Implicit Functions
      7-7. Winding Numbers
      7-8. The Gauss-Bonnet Theorem

Chapter 8 / Lie Groups and Lie Algebras
      8-1. General Theory
      8-2. Computation
      8-3. Classification


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