Grassmann Algebra

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ExplorGrassmannMatrixAlgebra.nb 7 M �� MatrixForm � a1,1 � e1 b1,1 � e2 c1,1 � d1,1 e1 � e2 a1,2 � e1 b1,2 � e2 c1,2 � d1,2 e1 � e2 � a2,1 � e1 b2,1 � e2 c2,1 � d2,1 e1 � e2 a2,2 � e1 b2,2 � e2 c2,2 � d2,2 e1 � e2 The body of M is given by: and its soul by: Body�M� ��MatrixForm � a1,1 a1,2 � a2,1 a2,2 Soul�M� ��MatrixForm � e1 b1,1 � e2 c1,1 � d1,1 e1 � e2 e1 b1,2 � e2 c1,2 � d1,2 e1 � e2 � e1 b2,1 � e2 c2,1 � d2,1 e1 � e2 e1 b2,2 � e2 c2,2 � d2,2 e1 � e2 The even components are extracted by using EvenGrade. EvenGrade�M� ��MatrixForm � a1,1 � d1,1 e1 � e2 a1,2 � d1,2 e1 � e2 � a2,1 � d2,1 e1 � e2 a2,2 � d2,2 e1 � e2 The odd components are extracted by using OddGrade. OddGrade�M� ��MatrixForm � e1 b1,1 � e2 c1,1 e1 b1,2 � e2 c1,2 � e1 b2,1 � e2 c2,1 e1 b2,2 � e2 c2,2 Finally, we can extract the elements of any specific grade using ExtractGrade. ExtractGrade�2��M� ��MatrixForm � d1,1 e1 � e2 d1,2 e1 � e2 � d2,1 e1 � e2 d2,2 e1 � e2 � Determining the type of elements We take the matrix A created in Section 13.2. 2001 4 26
ExplorGrassmannMatrixAlgebra.nb 8 A �� MatrixForm 2 � 3e1 � 5 � 9e1 � e2 e1 � 6e2 � e2 � e1 � e2 EvenGradeQ, OddGradeQ, EvenSoulQ and GradeQ all interrogate the individual elements of a matrix. EvenGradeQ�A� ��MatrixForm False False � True False � OddGradeQ�A� ��MatrixForm False True � False False � EvenSoulQ�A� ��MatrixForm False False � False False � GradeQ�1��A� ��MatrixForm False True � False False � To check the type of a complete matrix we can ask whether it is free of either True or False values. 2001 4 26 FreeQ�GradeQ�1��A�, False� False FreeQ�EvenSoulQ�A�, True� True
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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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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 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 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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12 Exploring Clifford Algebra 12.1
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ExploringCliffordAlgebra.nb 3 12.17
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ExploringCliffordAlgebra.nb 9 ToSca
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Page 455 and 456: A Brief Biography of Grassmann Herm
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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
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Page 467 and 468: Bibliography To be completed Armstr
Page 469 and 470: Bibliography2.nb 3 Clifford W K 187
Page 471 and 472: Bibliography2.nb 5 quaternions and
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Grassmann Algebra

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