Understanding the scope of the printed work Volume 1 This page is an extract from “Grassmann Algebra Volume 1: Foundations - Exploring extended vector algebra with Mathematica” by John Browne. First Edition 2012.

1  Introduction 

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1.1  Background

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The mathematical representation of physical entities 5 The central concept of the Ausdehnungslehre  5 Comparison with the vector and tensor algebras 6 Algebraicizing the notion of linear dependence 6 Grassmann algebra as a geometric calculus 7

1.2  The Exterior Product

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The anti-symmetry of the exterior product 8 Exterior products of vectors in a three-dimensional space 9 Terminology: elements and entities 10 The grade of an element 11 Interchanging the order of the factors in an exterior product 12 A brief summary of the properties of the exterior product 12

1.3  The Regressive Product

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The regressive product as a dual product to the exterior product 13 Unions and intersections of spaces 14 A brief summary of the properties of the regressive product 14 The Common Factor Axiom 15 The intersection of two bivectors in a three-dimensional space 17

1.4  Geometric Interpretations

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Points and vectors 17 Sums and differences of points 18 Determining a mass-centre 20 Lines and planes 21 The intersection of two lines 22

1.5  The Complement

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The complement as a correspondence between spaces 23 The Euclidean complement 24 The complement of a complement 26 The Complement Axiom 27

1.6  The Interior Product

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The definition of the interior product 28 Inner products and scalar products 29 Sequential interior products 29 Orthogonality 30 Measure and magnitude 30 Calculating interior products from their definition 31 Expanding interior products 32 The interior product of a bivector and a vector   32 The cross product 33

1.7  Summary

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Summary of operations 34 Summary of objects 35

2  The Exterior Product

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2.1  Introduction

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2.2  The Exterior Product

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Basic properties of the exterior product 38 Declaring scalar and vector symbols in GrassmannAlgebra  40 Entering exterior products 40

2.3  Exterior Linear Spaces

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Composing m-elements 40 Composing elements automatically 41 Spaces and congruence 42 The associativity of the exterior product 42 Transforming exterior products 43

2.4  Axioms for Exterior Linear Spaces

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Summary of axioms 44 Grassmann algebras 46 On the nature of scalar multiplication 46 Factoring scalars 47 Grassmann expressions 47 Calculating the grade of a Grassmann expression 48

2.5  Bases

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Bases for exterior linear spaces 49 Declaring a basis in GrassmannAlgebra  49 Composing bases of exterior linear spaces 50 Composing palettes of basis elements 50 Standard ordering 51 Indexing basis elements of exterior linear spaces 52

2.6  Cobases

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Definition of a cobasis 52 The cobasis of unity 53 Composing palettes of cobasis elements 54 The cobasis of a cobasis 54

2.7  Determinants

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Determinants from exterior products 55 Properties of determinants 56 The Laplace expansion technique 56 Calculating determinants 57

2.8  Cofactors

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Cofactors from exterior products 58 The Laplace expansion in cofactor form 59 Transformations of cobases 60

2.9  Solution of Linear Equations

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Grassmann's approach to solving linear equations 61 Example solution: 3 equations in 4 unknowns 62 Example solution: 4 equations in 4 unknowns 62

2.10  Simplicity

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The concept of simplicity 63 All (n-1)-elements are simple 63 Conditions for simplicity of a 2-element in a 4-space 64 Conditions for simplicity of a 2-element in a 5-space 64 Factorizing simple elements from first principles 65

2.11  Exterior Division

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The definition of an exterior quotient 67 Division by a 1-element 67 Division by a k-element 68 Automating the division process 69

2.12  Multilinear Forms

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The span of a simple element 69 Composing spans 70 Example: Refactorizations 72 Multilinear forms 73 Defining m:k-forms 74 Composing m:k-forms 75 Expanding and simplifying m:k-forms 76 Developing invariant forms 76 The invariance of m:k-forms 77 The complete span of a simple element 78 The Zero Form Theorem 81 Zero Form formulae 82

2.13  Unions and Intersections

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Union and intersection as a multilinear form 85 Where the intersection is evident 86 Where the intersections is not evident 88 Intersection with a non-simple element 89 Factorizing simple elements 90

2.14  Summary

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3  The Regressive Product  

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3.1  Introduction

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3.2  Duality

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The notion of duality 93 Examples:  Obtaining the dual of an axiom 94 Summary:  The duality transformation algorithm  96

3.3  Properties of the Regressive Product

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Axioms for the regressive product 96 The unit n-element 97 The inverse of an n-element 99 Grassmann's notation for the regressive product       100

3.4  The Grassmann Duality Principle

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The dual of a dual               101 The Grassmann Duality Principle       101 Using the GrassmannAlgebra function Dual               102

3.5  The Common Factor Axiom

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Motivation       104 The Common Factor Axiom       105 Extension of the Common Factor Axiom to general elements       106 Special cases of the Common Factor Axiom       107 Dual versions of the Common Factor Axiom       107 Application of the Common Factor Axiom       108 When the common factor is not simple       110

3.6  The Common Factor Theorem

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Development of the Common Factor Theorem             110 Proof of the Common Factor Theorem       113 The A and B forms of the Common Factor Theorem       115 Example: The decomposition of a 1-element       116 Example: Applying the Common Factor Theorem       117 Automating the application of the Common Factor Theorem               118 A special form of the Common Factor Theorem        120

3.7  The Regressive Product of Simple Elements

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The regressive product of simple elements       122 The regressive product of (n-1)-elements       122 Regressive products leading to scalar results       122 The cobasis form of the Common Factor Axiom       123 The regressive product of cobasis elements       124

3.8  Expressing an Element in another Basis

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Expressing an element in terms of another basis       125 Using the computable form of the Common Factor Theorem               126 Automating the process       127 The symmetric expansion of a 1-element in terms of another basis       128

3.9  Factorization of Simple Elements

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Factorization using the regressive product       129 Factorizing elements expressed in terms of basis elements       131 The factorization algorithm       133 Factorization of (n-1)-elements       135 Factorizing simple m-elements       136 Factorizing contingently simple m-elements       138 Determining if an element is simple       140

3.10  Product Formulae for Regressive Products

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The Product Formula       141 Deriving Product Formulae       143 Deriving Product Formulae automatically       143 Computing the General Product Formula       145 Comparing the two forms of the Product Formula       149 The invariance of the General Product Formula       150 Alternative forms for the General Product Formula       150 The Decomposition Formula       152 Exploration: Dual forms of the General Product Formulae       153 The double sum form of the General Product Formula       154

3.11  Summary

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