Grassmann Algebra

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TheComplement.nb 9 propagate through the algebra, and in consequence maintain a certain simplicity of structure in the resulting formulae, or we can reformulate our approach to allow 1 �� s � ��1� �1, where the parity of s is even if �gij� is positive, and odd if it is negative. This latter approach would add a factor ��1� s to many of the axioms and formulae developed from this chapter on. Because of the introductory nature of the book however, we have chosen to adopt the former approach, allowing readers less versed in the algebra to be undistracted by the extra complexities of pseudo-Riemannian metrics. From this point on then, we take the value of � to be the reciprocal of the positive square root of the determinant of the coefficients of the complement mapping gij . � � 1 �� gij� This relation clearly also implies that the array �gij� is required to be non-singular. 5.4 The Euclidean Complement Tabulating Euclidean complements of basis elements The Euclidean complement of a basis m-element may be defined as its cobasis element. Conceptually this is the simplest correspondence we can define between basis m-elements and basis (nÐ1)-elements. In this case the matrix of the metric tensor is the identity matrix, with the result that the 'volume' � of the basis n-element is unity. We tabulate the basis-complement pairs for spaces of two, three and four dimensions. Basis elements and their Euclidean complements in 2-space 2001 4 5 � BASIS COMPLEMENT � 0 1 e1 � e2 � l e1 e2 � l e2 �e1 � 2 e1 � e2 1 5.14
TheComplement.nb 10 Basis elements and their Euclidean complements in 3-space � BASIS COMPLEMENT � 0 � l � l � l � 2 � 2 1 e1 � e2 � e3 e1 e2 e3 e1 � e2 e1 � e3 e2 � e3 ��e1 � e3� e1 � e2 e3 �e2 � 2 e2 � e3 e1 � 3 e1 � e2 � e3 1 Basis elements and their Euclidean complements in 4-space 2001 4 5 � BASIS COMPLEMENT � 0 � l � l � l � l � 2 � 2 � 2 � 2 � 2 � 2 � 3 � 3 � 3 � 3 1 e1 � e2 � e3 � e4 e1 e2 e3 e4 e1 � e2 e1 � e3 e1 � e4 e2 � e3 e2 � e4 e3 � e4 e1 � e2 � e3 e1 � e2 � e4 e1 � e3 � e4 e2 � e3 � e4 e2 � e3 � e4 ��e1 � e3 � e4� e1 � e2 � e4 ��e1 � e2 � e3� e3 � e4 ��e2 � e4� e2 � e3 e1 � e4 ��e1 � e3� e1 � e2 e4 �e3 e2 �e1 � 4 e1 � e2 � e3 � e4 1
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Grassmann Algebra Exploring applica
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TableOfContents.nb 2 1.7 Exploring
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TableOfContents.nb 4 3.5 The Common
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TableOfContents.nb 6 5.2 Axioms for
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TableOfContents.nb 8 6.7 The Induce
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TableOfContents.nb 10 9 Exploring G
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Preface The origins of this book Th
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Preface.nb 3 Chapters 7 to 13: Expl
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1. Introduction 1.1 Background The
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Introduction.nb 3 the scientific wo
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Introduction.nb 5 A plane can be re
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Introduction.nb 7 We see here also
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Introduction.nb 9 �a�Α m �
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Introduction.nb 11 ¥ Scalars facto
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Introduction.nb 13 Here Det[x,y,v]
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Introduction.nb 15 Lines and planes
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Introduction.nb 17 Graphic showing
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Introduction.nb 19 x � ae1 � be
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Introduction.nb 21 Calculating inte
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Introduction.nb 23 1.11 Exploring H
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TheExteriorProduct.nb 2 � Calcula
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TheExteriorProduct.nb 4 This may be
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TheExteriorProduct.nb 6 At any stag
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TheExteriorProduct.nb 8 Note that w
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TheExteriorProduct.nb 10 � �1:
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TheExteriorProduct.nb 12 Grassmann
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TheExteriorProduct.nb 14 when gener
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TheExteriorProduct.nb 16 � Genera
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TheExteriorProduct.nb 18 BasisTable
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TheExteriorProduct.nb 20 That is: e
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TheExteriorProduct.nb 22 CobasisPal
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TheExteriorProduct.nb 24 � 1 The
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TheExteriorProduct.nb 26 ��
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TheExteriorProduct.nb 28 Let ei m b
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TheExteriorProduct.nb 30 Α1 � Α
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TheExteriorProduct.nb 32 2.9 Soluti
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TheExteriorProduct.nb 34 Evaluating
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TheExteriorProduct.nb 36 ��Α 2
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TheExteriorProduct.nb 38 Division b
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TheRegressiveProduct.nb 2 3.9 Produ
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TheRegressiveProduct.nb 4 The grade
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TheRegressiveProduct.nb 6 � �8:
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TheRegressiveProduct.nb 8 The inver
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TheRegressiveProduct.nb 10 1. Repla
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TheRegressiveProduct.nb 12 Example
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TheRegressiveProduct.nb 14 Here we
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TheRegressiveProduct.nb 16 Extendin
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TheRegressiveProduct.nb 18 Finally,
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TheRegressiveProduct.nb 20 Example:
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TheRegressiveProduct.nb 22 � Auto
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TheRegressiveProduct.nb 24 Y � Cr
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TheRegressiveProduct.nb 26 e1 �
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TheRegressiveProduct.nb 28 Similarl
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TheRegressiveProduct.nb 30 A 3 �
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TheRegressiveProduct.nb 32 Provided
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TheRegressiveProduct.nb 34 n � i
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Page 117 and 118: TheRegressiveProduct.nb 40 �Α m
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Page 157 and 158: TheComplement.nb 2 � Converting t
Page 159 and 160: TheComplement.nb 4 5.2 Axioms for t
Page 161 and 162: TheComplement.nb 6 essential underp
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Page 167 and 168: TheComplement.nb 12 ei m ��
Page 169 and 170: TheComplement.nb 14 Relating exteri
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Page 173 and 174: TheComplement.nb 18 The default met
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Page 181 and 182: TheComplement.nb 26 � Simplifying
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Page 197 and 198: TheComplement.nb 42 Α m �Α1 �
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Page 201 and 202: TheInteriorProduct.nb 4 An alternat
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TheInteriorProduct.nb 18 Det�Deve
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TheInteriorProduct.nb 20 If Α is e
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TheInteriorProduct.nb 22 Measure�
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TheInteriorProduct.nb 24 The measur
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TheInteriorProduct.nb 26 6.7 The In
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TheInteriorProduct.nb 28 ��
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TheInteriorProduct.nb 30 Interior p
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TheInteriorProduct.nb 32 �Α m
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TheInteriorProduct.nb 34 Implicatio
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TheInteriorProduct.nb 36 Α m �
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TheInteriorProduct.nb 38 Or, more s
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TheInteriorProduct.nb 40 � ��
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TheInteriorProduct.nb 42 � The an
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TheInteriorProduct.nb 44 6.15 Histo
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ExploringScrewAlgebra.nb 2 The obje
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ExploringScrewAlgebra.nb 4 so that
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ExploringScrewAlgebra.nb 6 The comp
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ExploringScrewAlgebra.nb 8 S ��
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8 Exploring Mechanics 8.1 Introduct
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ExploringMechanics.nb 3 every 'line
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ExploringMechanics.nb 5 � � P
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ExploringMechanics.nb 7 8.3 Momentu
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ExploringMechanics.nb 9 The bound v
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ExploringMechanics.nb 11 8.4 Newton
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ExploringMechanics.nb 13 8.7 The Ve
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ExploringGrassmannAlgebra.nb 2 �
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ExploringGrassmannAlgebra.nb 4 Decl
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ExploringGrassmannAlgebra.nb 6 �3
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ExploringGrassmannAlgebra.nb 12 Ξ0
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ExploringGrassmannAlgebra.nb 16 How
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ExploringGrassmannAlgebra.nb 22 Inv
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ExploringGrassmannAlgebra.nb 24 Ver
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ExploringGrassmannAlgebra.nb 26 The
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ExploringGrassmannAlgebra.nb 30 S
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ExploringGrassmannAlgebra.nb 32 Div
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ExploringGrassmannAlgebra.nb 36 9.9
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ExploringGrassmannAlgebra.nb 40 Log
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ExploringGrassmannAlgebra.nb 42 Gra
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10 Exploring the Generalized Grassm
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ExpTheGeneralizedProduct.nb 3 For b
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ExpTheGeneralizedProduct.nb 5 Α m
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ExpTheGeneralizedProduct.nb 7 It is
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ExpTheGeneralizedProduct.nb 9 To ex
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ExpTheGeneralizedProduct.nb 11 Exam
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ExpTheGeneralizedProduct.nb 13 B1
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ExpTheGeneralizedProduct.nb 17 1�
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ExpTheGeneralizedProduct.nb 19 ToIn
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ExpTheGeneralizedProduct.nb 23 10.9
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ExpTheGeneralizedProduct.nb 25 �
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ExpTheGeneralizedProduct.nb 27 B
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ExpTheGeneralizedProduct.nb 33 The
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ExpTheGeneralizedProduct.nb 35 Flat
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ExpTheGeneralizedProduct.nb 37 Prod
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11 Exploring Hypercomplex Algebra 1
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ExploringHypercomplexAlgebra.nb 7 p
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12 Exploring Clifford Algebra 12.1
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ExploringCliffordAlgebra.nb 3 12.17
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ExploringCliffordAlgebra.nb 5 � T
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ExploringCliffordAlgebra.nb 33 Or,
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ExploringCliffordAlgebra.nb 39 C2
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ExploringCliffordAlgebra.nb 41 scal
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ExploringCliffordAlgebra.nb 47 Hist
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ExploringCliffordAlgebra.nb 53 2001
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ExploringCliffordAlgebra.nb 55 numb
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ExplorGrassmannMatrixAlgebra.nb 2 1
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ExplorGrassmannMatrixAlgebra.nb 4 X
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ExplorGrassmannMatrixAlgebra.nb 8 A
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A Brief Biography of Grassmann Herm
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ABriefBiographyOfGrassmann.nb 3 I h
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Declarations �n �n declare a li
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Glossary To be completed. Ausdehnun
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Glossary.nb 3 Grade The grade of an
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Glossary.nb 5 Physical entities Poi
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Bibliography To be completed Armstr
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Bibliography2.nb 3 Clifford W K 187
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Bibliography2.nb 5 quaternions and
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Grassmann Algebra

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