Grassmann Algebra

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Introduction.nb 22 �Α � Β� �� x � �Α �� x��Β � ��1� m k m k m�Α ��Β�� x� m k The interior product of a bivector and a vector Suppose x is a vector and x1 � x2 is a simple bivector. The interior product of the bivector x1 � x2 with the vector x is the vector �x1 � x2� �� x. This can be expanded by the Interior Common Factor Theorem, or the formula above derived from it, to give: �x1 � x2� �� x � �x �� x1� x2 � �x �� x2� x1 Since the result is expressed as a linear combination of x1 and x2 it is clearly contained in the bivector x1 � x2 . The result is also orthogonal to x. We can show this by taking its scalar product with x: 1.24 ��x �� x1� x2 � �x �� x2� x1� �� x � �x �� x1�� x2 �� x� � �x �� x2�� x1 �� x� � 0 Graphic showing the interior product of a vector with a bivector: Shows orthogonal projection and vector in planar direction at right angle to orthogonal projection. These concepts may easily be extended to geometric entities of arbitrary dimension. 1.7 Exploring Screw Algebra To be completed 1.8 Exploring Mechanics To be completed 1.9 Exploring Grassmann Algebras To be completed 1.10 Exploring The Generalised Product To be completed 2001 4 5
Introduction.nb 23 1.11 Exploring Hypercomplex Algebras To be completed 1.12 ExploringClifford Algebras To be completed 1.13 ExploringGrassmann Matrix Algebras To be completed 2001 4 5
Page 1 and 2: Grassmann Algebra Exploring applica
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Page 5 and 6: TableOfContents.nb 4 3.5 The Common
Page 7 and 8: TableOfContents.nb 6 5.2 Axioms for
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Page 13 and 14: Preface The origins of this book Th
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Page 19 and 20: Introduction.nb 3 the scientific wo
Page 21 and 22: Introduction.nb 5 A plane can be re
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Page 27 and 28: Introduction.nb 11 ¥ Scalars facto
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Page 37: Introduction.nb 21 Calculating inte
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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 20 Example:
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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 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 10 Basis elements
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TheComplement.nb 18 The default met
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TheComplement.nb 28 ComplementTable
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TheComplement.nb 38 ei m The comple
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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 22 Measure�
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TheInteriorProduct.nb 34 Implicatio
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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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8 Exploring Mechanics 8.1 Introduct
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ExploringMechanics.nb 7 8.3 Momentu
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ExploringGrassmannAlgebra.nb 2 �
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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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11 Exploring Hypercomplex Algebra 1
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12 Exploring Clifford Algebra 12.1
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ExploringCliffordAlgebra.nb 53 2001
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A Brief Biography of Grassmann Herm
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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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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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