## 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 | |

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Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics ... David Hestenes,Garret Sobczyk No preview available - 1984 |

### Common terms and phrases

According adjoint analysis applied approach argument assume basis bivector blades bracket Calculus called canonical form Chapter classical complex computations condition consider conventional coordinates corresponding course curl curvature curve defined definition derivative describe determined developed differential dimension directed discussion easily element equation equivalent established Euclidean example expressed exterior exterior derivative fact factor follows formula frame function fundamental Geometric Algebra Geometric Calculus given gives grade Hence identity implies important integral introduce linear transformation manifold mathematical matrix method metric multiform multivector notation obtained operator orthogonal outer problem projection proof properties prove pseudoscalar reduces reference regarded rotation rule satisfies scalar Section shows simple space specific spinor structure symmetric tangent tangent vector tensor theorem theory unique unit values vanishes variable vector field vector space write written

### Popular passages

Page 306 - MR Hestenes, An analogue of Green's theorem in the calculus of variations, Duke Math.