Grassmann Algebra
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Grassmann Algebra

Grassmann Algebra

Grassmann Algebra

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TheInteriorProduct.nb 27<br />

M2 � DevelopScalarProductMatrix�M1�; MatrixForm�M2�<br />

�����������������������<br />

�<br />

� e1 � e1<br />

e2 � e1<br />

e1 � e2<br />

�<br />

e2 � e2<br />

� e1 � e1<br />

e2 � e1<br />

e1 � e3<br />

�<br />

e2 � e3<br />

� e1 � e2<br />

e2 � e2<br />

e1 � e3<br />

�<br />

e2 � e3<br />

� e1 � e1<br />

e3 � e1<br />

e1 � e2<br />

�<br />

e3 � e2<br />

� e1 � e1<br />

e3 � e1<br />

e1 � e3<br />

�<br />

e3 � e3<br />

� e1 � e2<br />

e3 � e2<br />

e1 � e3<br />

�<br />

e3 � e3<br />

� e2 � e1<br />

e3 � e1<br />

e2 � e2<br />

�<br />

e3 � e2<br />

� e2 � e1<br />

e3 � e1<br />

e2 � e3<br />

�<br />

e3 � e3<br />

� e2 � e2<br />

e3 � e2<br />

e2 � e3<br />

�<br />

e3 � e3<br />

Fourth, we substitute the metric element �i,j for ei ���� ej in the matrix.<br />

DeclareMetric���; M3� ToMetricForm�M2�; MatrixForm�M3�<br />

�������������������������<br />

���<br />

�<br />

�<br />

�1,1<br />

�1,2<br />

�1,2 ���<br />

�2,2 �<br />

���<br />

�<br />

�1,1<br />

�1,2<br />

�1,3 ���<br />

�2,3 �<br />

���<br />

�<br />

�1,2<br />

���<br />

�<br />

�2,2<br />

�1,3 ���<br />

�2,3 �<br />

�1,1<br />

�1,3<br />

�1,2 ���<br />

�2,3 �<br />

���<br />

�<br />

�1,1<br />

�1,3<br />

�1,3 ���<br />

�3,3 �<br />

���<br />

�<br />

�1,2<br />

���<br />

�<br />

�2,3<br />

�1,3 ���<br />

�3,3 �<br />

�1,2<br />

�1,3<br />

�2,2 ���<br />

�2,3 �<br />

���<br />

�<br />

�1,2<br />

�1,3<br />

�2,3 ���<br />

�3,3 �<br />

���<br />

�<br />

�2,2<br />

�2,3<br />

�������������������������<br />

�2,3 ���<br />

�3,3 � �<br />

Fifth, we compute the determinants of each of the elemental matrices.<br />

M4 � M3 �. G_?�MetricElementMatrixQ�2�� � Det�G�;<br />

MatrixForm�M4�<br />

��2 1,2 � �1,1 �2,2 ��1,2 �1,3 � �1,1 �2,3 ��1,3 �2,2 � �1,2 �2,3<br />

�����������<br />

�����������<br />

��1,2 �1,3 � �1,1 �2,3 ��2 1,3 � �1,1 �3,3 ��1,3 �2,3 � �1,2 �3,3<br />

� ��1,3 �2,2 � �1,2 �2,3 ��1,3 �2,3 � �1,2 �3,3 ��2 2,3 � �2,2 �3,3 �<br />

We can have Mathematica check that this is the same matrix as obtained from the Metric�<br />

function.<br />

M4 � Metric��2�<br />

True<br />

� Displaying induced metric tensors as a matrix of matrices<br />

<strong>Grassmann</strong><strong>Algebra</strong> has an inbuilt function MetricMatrix for displaying an induced metric<br />

in matrix form. For example, we can display the induced metric on � 3 in a 4-space.<br />

2001 4 5<br />

�4; DeclareMetric���<br />

���1,1, �1,2, �1,3, �1,4�, ��1,2, �2,2, �2,3, �2,4�,<br />

��1,3, �2,3, �3,3, �3,4�, ��1,4, �2,4, �3,4, �4,4��<br />

�����������������������<br />

�

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