Grassmann Algebra

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ExploringClifford<strong>Algebra</strong>.nb 52 Step 1: Replace symbols for entities and operations C7 � �C6 �� ReplaceNegativeUnit� �. �1 ��,e1 �Γ0 , e2 �Γ1 ,e3 �Γ2, e4 �Γ3 ��. Wedge � Dot; PaletteForm�C7 � � Γ 0 Γ 1 Γ 2 Γ 3 Γ 0 � Γ 0.Γ 1 Γ 0 .Γ 2 Γ 0.Γ 3 Γ1 �Γ0.Γ1 �� Γ1 .Γ2 Γ1.Γ3 Γ2 �Γ0.Γ2 �Γ1 .Γ2 �� Γ2.Γ3 Γ3 �Γ0.Γ3 �Γ1 .Γ3 �Γ2.Γ3 �� Γ0.Γ1 �Γ1 �Γ0 Γ0 .Γ1.Γ2 Γ0 .Γ1 .Γ3 Γ0.Γ2 �Γ2 �Γ0.Γ1 .Γ2 �Γ0 Γ0 .Γ2 .Γ3 Γ 0.Γ 3 �Γ 3 �Γ 0.Γ 1 .Γ 3 �Γ 0 .Γ 2 .Γ 3 �Γ 0 Γ 1.Γ 2 Γ 0 .Γ 1 .Γ 2 Γ 2 �Γ 1 Γ 1 .Γ 2 .Γ 3 Γ1.Γ3 Γ0 .Γ1 .Γ3 Γ3 �Γ1 .Γ2 .Γ3 �Γ1 Γ2.Γ3 Γ0 .Γ2 .Γ3 Γ1 .Γ2 .Γ3 Γ3 �Γ2 Γ0 .Γ1 .Γ2 Γ1.Γ2 Γ0.Γ2 �Γ0.Γ1 Γ0.Γ1.Γ2 .Γ3 Γ0 .Γ1 .Γ3 Γ1.Γ3 Γ0.Γ3 �Γ0 .Γ1.Γ2.Γ3 �Γ0 .Γ1 Γ0 .Γ2 .Γ3 Γ2.Γ3 Γ0.Γ1.Γ2.Γ3 Γ0 .Γ3 �Γ0 .Γ2 Γ 1 .Γ 2 .Γ 3 �Γ 0 .Γ 1.Γ 2.Γ 3 �Γ 2 .Γ 3 Γ 1 .Γ 3 �Γ 1 .Γ 2 Γ0 .Γ1.Γ2.Γ3 �Γ1 .Γ2 .Γ3 �Γ0.Γ2 .Γ3 Γ0 .Γ1.Γ3 �Γ0 .Γ1 .Γ2 Step 2: Substitute matrices and calculate 2001 4 26 � 1 0 0 0 �� 1 0 0 0 �� 0 0 0 �1 �� 0 1 0 0 0 1 0 0 0 0 �1 0 C8 � C7 �. �� , Γ0 � , Γ1 � 0 0 1 0 0 0 �1 0 0 1 0 0 � 0 0 0 1 � � 0 0 0 �1 � � 1 0 0 0 Γ2 � � 0 0 0 � �� 0 0 �1 0 �� 0 0 �� 0 0 0 0 1 , Γ3 � �; MatrixForm�C8 � 0 �� 0 0 1 0 0 0 � � 0 0 0 � � 0 �1 0 0 � � �� , �
ExploringClifford<strong>Algebra</strong>.nb 53 2001 4 26 �� 1 0 0 0 �� 1 0 0 0 �� 0 0 0 �1 �� 0 0 0 � �� 0 0 �1 0 �� 0 1 0 0 0 1 0 0 0 0 �1 0 0 0 �� 0 0 0 0 1 0 0 1 0 0 0 �1 0 0 1 0 0 0 �� 0 0 1 0 0 0 � 0 0 0 1 � � 0 0 0 �1 � � 1 0 0 0 � � � 0 0 0 � � 0 �1 0 0 � � 1 0 0 0 �� 1 0 0 0 �� 0 0 0 �1 �� 0 0 0 � �� 0 0 �1 0 �� 0 1 0 0 0 1 0 0 0 0 �1 0 0 0 �� 0 0 0 0 1 0 0 �1 0 0 0 1 0 0 �1 0 0 0 � 0 0 �1 0 0 0 � 0 0 0 �1 � � 0 0 0 1 � � �1 0 0 0 � � �� 0 0 0 � � 0 1 0 0 � � 0 0 0 �1 �� 0 0 0 1 �� 1 0 0 0 �� 0 0 0 �� 0 1 0 0 �� 0 0 �1 0 0 0 1 0 0 �1 0 0 0 � 0 0 �1 0 0 0 0 1 0 0 0 1 0 0 0 0 �1 0 0 0 �� 0 0 0 0 1 � 1 0 0 0 � � 1 0 0 0 � � 0 0 0 �1 � � 0 0 0 � � � 0 0 �1 0 � � 0 0 0 � �� 0 0 0 �� 0 0 0 �� 1 0 0 0 �� 0 �� 0 0 �� 0 0 �� 0 0 0 � 0 0 �� 0 0 0 �1 0 0 �� 0 0 0 0 �� 0 0 0 �� 0 0 0 0 � 0 0 0 �1 0 0 0 0 �� 0 0 0 � � � 0 0 0 � � 0 0 0 �� 0 0 0 �1 � � 0 0 �� 0 � 0 0 �1 0 �� 0 0 1 0 �� 0 �1 0 0 �� 0 � 0 0 �� 1 0 0 0 �� 0 0 0 1 0 0 0 �1 1 0 0 0 � 0 0 0 0 �1 0 0 1 0 0 0 1 0 0 0 0 0 0 �1 0 0 0 � 0 0 �1 0 � 0 �1 0 0 � � 0 �1 0 0 � � 0 0 1 0 � � 0 0 � 0 � � 0 0 0 �1 � 0 0 0 �1 �� 0 0 0 1 �� 1 0 0 0 �� 0 0 0 �� 0 1 0 0 �� 0 0 �1 0 0 0 1 0 0 �1 0 0 0 � 0 0 �1 0 0 0 0 �1 0 0 0 �1 0 0 0 0 1 0 0 0 � 0 0 0 0 �1 � �1 0 0 0 � � �1 0 0 0 � � 0 0 0 1 � � 0 0 0 �� 0 0 1 0 � � 0 0 0 � �� 0 0 0 �� 0 0 0 �� 1 0 0 0 �� 0 �� 0 0 �� 0 0 �� 0 0 0 � 0 0 �� 0 0 0 �1 0 0 �� 0 0 0 0 � 0 0 0 � 0 0 0 0 �� 0 0 0 1 0 0 0 0 � � �� 0 0 0 � � �� 0 0 0 � � 0 0 0 � � � 0 0 0 1 � � 0 0 � 0 � � 0 0 �1 0 �� 0 0 1 0 �� 0 �1 0 0 �� 0 � 0 0 �� 1 0 0 0 �� 0 0 0 1 0 0 0 �1 1 0 0 0 � 0 0 0 0 �1 0 0 �1 0 0 0 �1 0 0 0 0 0 0 1 0 0 0 �� 0 0 1 0 � 0 1 0 0 � � 0 1 0 0 � � 0 0 �1 0 � � 0 0 �� 0 � � 0 0 0 1 � � �� 0 0 0 �� 0 0 0 �� 0 0 0 � �� 0 0 0 1 �� 0 0 � 0 �� 0 � 0 0 0 � 0 0 0 0 �� 0 0 0 1 0 0 0 0 � 0 0 �� 0 0 0 � 0 0 �� 0 0 0 �1 0 0 �� 0 0 0 � 0 0 0 � � � 0 0 0 �� 0 0 0 � � �1 0 0 0 � � 0 �� 0 0 � � 0 1 0 0 �� 0 1 0 0 �� 0 0 �1 0 �� 0 0 �� 0 �� 0 0 0 1 �� 1 0 0 0 �1 0 0 0 0 0 0 1 0 0 0 �� 0 0 1 0 0 0 0 1 0 0 0 �1 1 0 0 0 � 0 0 0 0 �1 0 0 � 0 0 �1 0 � � 0 0 1 0 � � 0 �1 0 0 � � 0 � 0 0 � � �1 0 0 0 � � 0 �� 0 0 �� 0 �� 0 0 �� 0 0 � 0 �� 0 0 �1 0 �� 0 0 0 �� 0 0 0 �� 0 0 0 0 0 0 � 0 0 0 1 0 0 � 0 0 0 0 �� 0 0 0 � �� 0 0 0 1 0 0 0 0 � 0 0 � 0 0 �� 0 � � 0 0 � 0 � � 0 �� 0 0 � � 0 �1 0 0 � � �� 0 0 0 � � �� 0 0 0 �� 0 0 0 �� 0 0 0 � �� 0 0 0 1 �� 0 0 � 0 �� 0 � 0 0 0 � 0 0 0 0 �� 0 0 0 1 0 0 0 0 � 0 0 � 0 0 0 �� 0 0 � 0 0 0 1 0 0 � 0 0 0 � 0 0 0 �� 0 0 0 � � � �� 0 0 0 � � 1 0 0 0 � � 0 � 0 0 � � 0 1 0 0 �� 0 1 0 0 �� 0 0 �1 0 �� 0 0 �� 0 �� 0 0 0 1 �� 1 0 0 0 �1 0 0 0 0 0 0 1 0 0 0 �� 0 0 1 0 0 0 0 �1 0 0 0 1 �1 0 0 0 �� 0 0 0 0 1 0 0 � 0 0 1 0 � � 0 0 �1 0 � � 0 1 0 0 � � 0 �� 0 0 � � 1 0 0 0 � � 0 �� 0 0 �� 0 �� 0 0 �� 0 0 � 0 �� 0 0 �1 0 �� 0 0 0 �� 0 0 0 �� 0 0 0 0 0 0 � 0 0 0 1 0 0 � 0 0 0 0 � 0 0 0 �� 0 0 0 �1 0 0 0 0 �� 0 0 � 0 0 � 0 � � 0 0 �� 0 � � 0 � 0 0 � � 0 1 0 0 � � � 0 0 0 � � 0 0 � 0 �� 0 0 �� 0 �� 0 � 0 0 �� 0 1 0 0 �� 0 0 0 �� 0 0 0 � 0 0 0 �� 0 0 0 �1 0 0 0 0 �� 0 0 �� 0 0 0 �� 0 0 0 0 0 0 � 0 0 0 1 0 0 � 0 � 0 �� 0 0 � � 0 �� 0 0 � � 0 0 � 0 � � 0 0 �1 0 � � 0 0 0 �� 0 0 � 0 �� 0 0 �� 0 �� 0 � 0 0 �� 0 1 0 0 �� 0 0 0 �� 0 0 0 � 0 0 0 �� 0 0 0 �1 0 0 0 0 �� 0 0 � 0 0 0 � 0 0 0 0 0 0 �� 0 0 0 �1 0 0 �� 0 � � 0 � 0 0 � � 0 � 0 0 � � 0 0 �� 0 � � 0 0 1 0 � � 0 0 0 � �
Page 1 and 2:
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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TheRegressiveProduct.nb 36 � A 2-
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TheRegressiveProduct.nb 38 SimpleQ
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TheRegressiveProduct.nb 40 �Α m
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TheRegressiveProduct.nb 42 x p �
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TheRegressiveProduct.nb 44 Here the
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4 Geometric Interpretations 4.1 Int
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GeometricInterpretations.nb 3 and t
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GeometricInterpretations.nb 5 Sir W
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GeometricInterpretations.nb 7 A not
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GeometricInterpretations.nb 9 The g
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GeometricInterpretations.nb 11 simp
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GeometricInterpretations.nb 13 Deco
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GeometricInterpretations.nb 15 The
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GeometricInterpretations.nb 17 The
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GeometricInterpretations.nb 19 L
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GeometricInterpretations.nb 21 �P
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GeometricInterpretations.nb 23 �
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GeometricInterpretations.nb 27 Alte
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GeometricInterpretations.nb 29 �
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GeometricInterpretations.nb 33 P1
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TheComplement.nb 2 � Converting t
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TheComplement.nb 4 5.2 Axioms for t
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TheComplement.nb 6 essential underp
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TheComplement.nb 8 ��
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TheComplement.nb 10 Basis elements
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TheComplement.nb 12 ei m ��
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TheComplement.nb 14 Relating exteri
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TheComplement.nb 18 The default met
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TheComplement.nb 20 �3; DeclareMe
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TheComplement.nb 22 We can transfor
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TheComplement.nb 24 2 2 2 ��1,3
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TheComplement.nb 26 � Simplifying
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TheComplement.nb 28 ComplementTable
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TheComplement.nb 32 hence can be fa
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TheComplement.nb 38 ei m The comple
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TheComplement.nb 40 e i ��
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TheComplement.nb 42 Α m �Α1 �
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TheInteriorProduct.nb 2 6.9 The Cro
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TheInteriorProduct.nb 4 An alternat
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TheInteriorProduct.nb 6 ��
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TheInteriorProduct.nb 8 ToScalarPro
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TheInteriorProduct.nb 14 InteriorCo
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TheInteriorProduct.nb 16 Thus Α 3
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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 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 28 ? G
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ExploringGrassmannAlgebra.nb 30 S
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ExploringGrassmannAlgebra.nb 32 Div
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ExploringGrassmannAlgebra.nb 34 ? G
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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 15 B1
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ExpTheGeneralizedProduct.nb 17 1�
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ExpTheGeneralizedProduct.nb 19 ToIn
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ExpTheGeneralizedProduct.nb 21 We c
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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 3 W
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ExploringHypercomplexAlgebra.nb 5
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ExploringHypercomplexAlgebra.nb 7 p
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ExploringHypercomplexAlgebra.nb 9 M
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Page 365 and 366: 12 Exploring Clifford Algebra 12.1
Page 367 and 368: ExploringCliffordAlgebra.nb 3 12.17
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Page 373 and 374: ExploringCliffordAlgebra.nb 9 ToSca
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Page 387 and 388: ExploringCliffordAlgebra.nb 23 But
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Page 405 and 406: ExploringCliffordAlgebra.nb 41 scal
Page 411 and 412: ExploringCliffordAlgebra.nb 47 Hist
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Page 419 and 420: ExploringCliffordAlgebra.nb 55 numb
Page 421 and 422: ExplorGrassmannMatrixAlgebra.nb 2 1
Page 423 and 424: ExplorGrassmannMatrixAlgebra.nb 4 X
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Page 427 and 428: ExplorGrassmannMatrixAlgebra.nb 8 A
Page 455 and 456: A Brief Biography of Grassmann Herm
Page 457 and 458: ABriefBiographyOfGrassmann.nb 3 I h
Page 459 and 460: Declarations �n �n declare a li
Page 461 and 462: Glossary To be completed. Ausdehnun
Page 463 and 464: Glossary.nb 3 Grade The grade of an
Page 465 and 466: 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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