Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and PhysicsGeometric Calculus is a language for expressing and analyzing the full range of geometric concepts in mathematics. Clifford Algebra provides the grammar. Complex number, quaternions, matrix algebra, vector, tensor and spinor calculus and differential forms are integrated into a single 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 details 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 thinks 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-dimension spaces and explicit integral formula for the inverse of a transformation; a new approach to Lie groups and Lie algebras. --From cover. |
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Contents
Geometric Algebra | 1 |
11 Axioms Definitions and Identities | 3 |
12 Vector Spaces Pseudoscalars and Projections | 16 |
13 Frames and Matrices | 27 |
14 Alternating Forms and Determinants | 33 |
15 Geometric Algebras of PseudoEuclidean Spaces | 41 |
Differentiation | 44 |
22 Multivector Derivatives Differentials and Adjoints | 53 |
47 Complex Numbers and Conform Transformations | 180 |
Differential Geometry of Vector Manifolds | 188 |
51 Curl and Curvature | 189 |
52 Hypersurfaces in Euclidean Space | 196 |
53 Related Geometries | 201 |
54 Parallelism and Projectively Related Geometries | 203 |
55 Conformally Related Geometries | 210 |
56 Induced Geometries | 220 |
23 Factorization and Simplicial Derivatives | 59 |
Linear and Multilinear Functions | 63 |
31 Linear Transformations and Outermorphisms | 66 |
32 Characteristic Multivecton and the CaylcyHamilton Theorem | 71 |
33 Eigenblades and Invariant Spaces | 75 |
34 Symmetric and Skewsymmetric Transformations | 78 |
35 Normal and Orthogonal Transformations | 86 |
36 Canonical Forms for General Linear Transformations | 94 |
37 Metric Tensors and Isometries | 96 |
38 Isometrics and Spinors of PseudoEuclidean Spaces | 102 |
39 Linear Multivector Functions | 111 |
310 Tensors | 130 |
Calculus on Vector Manifolds | 137 |
41 Vector Manifolds | 139 |
42 Projection Shape and Curl | 147 |
43 Intrinsic Derivatives and Lie Brackets | 155 |
44 Curl and Pseudoscalar | 162 |
45 Transformations of Vector Manifolds | 165 |
46 Computation of Induced Transformations | 173 |
The Method of Mobiles | 225 |
62 Mobiles and Curvature | 230 |
63 Curves and Comoving Frames | 237 |
64 The Calculus of Differential Forms | 240 |
Directed Integration Theory | 249 |
72 Derivatives from Integrals | 252 |
73 The Fundamental Theorem of Calculus | 256 |
74 Antidcrivatives Analytic Functions and Complex Variables | 259 |
75 Changing Integration Variables | 266 |
76 Inverse and Implicit Functions | 269 |
77 Winding Numbers | 272 |
78 The GaussBonnet Theorem | 276 |
Lie Groups and Lie Algebras | 283 |
82 Computation | 291 |
83 Classification | 296 |
| 305 | |
| 309 | |
Other editions - View all
Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics ... David Hestenes,Garret Sobczyk No preview available - 1984 |
Common terms and phrases
analytic function applied axioms biform bivector blades canonical form chain rule Chapter coderivative coefficients complex numbers computations conventional coordinate-free coordinates curl tensor curvature tensor curve defined definition determined developed differential and adjoint differential forms differential geometry dimension dimensional directed integral dual eigenblade eigenvalue eigenvectors equation equivalent Euclidean expressed extensor exterior differential factor follows formula frame Geometric Algebra Geometric Calculus geometric product grade Hence identity inner product integrability condition inverse isometry Lie algebra Lie bracket Lie group linear function linear transformation mathematical matrix metric tensor multiform multilinear multivector nonsingular notation obtained operator orthogonal transformation outer products outermorphism plane projection proof properties prove pseudoscalar r-blade r-form r-vector right side rotation rotor satisfies scalar Section 2-1 shows simple simplicial derivative spinor spinor group subspace symmetric tangent algebra tangent vector theory tion unique unit pseudoscalar vanishes variable vector field vector manifold vector space versor
Popular passages
Page 306 - MR Hestenes, An analogue of Green's theorem in the calculus of variations, Duke Math.
Bibliographic information
| Title | Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics Volume 5 of Fundamental Theories of Physics, ISSN 0168-1222 Volume 5 of Fundamental Theories of Physics, A New International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application |
| Authors | David Hestenes, Garret Sobczyk |
| Edition | illustrated, reprint |
| Publisher | Springer Science & Business Media, 1987 |
| ISBN | 9027725616, 9789027725615 |
| Length | 314 pages |
| Subjects | › › Mathematics / Algebra / Abstract Mathematics / Algebra / General Mathematics / Geometry / General Mathematics / Group Theory Science / Physics / General Science / Physics / Mathematical & Computational |
| Export Citation | BiBTeX EndNote RefMan |