Grassmann Algebra

TheInteriorProduct.nb 19
Y � ToScalarProducts�X�
1 � ax� c �e1 � e3� �e2 � e2� � c �e1 � e2� �e2 � e3� �
b �w � z� x � y � b �w � y� x � z � b �w � x� y � z
ToMetricForm�Y�
1 � ax� c �1,3 �2,2 � c �1,2 �2,3 �
b �w � z� x � y � b �w � y� x � z � b �w � x� y � z
Inner products of basis elements
The inner product of two basis m-elements has been expressed in various forms previously.
Here we collect together the results for reference. Here, gij is the metric tensor, g is its
determinant, and Α and Β refer to any basis elements.
m m
ei
m
ei
m
e i
m
Α m ���� Β m
� Β m
���� ej m �∆ j
i
���� ej
m
���� ej
m
� ei
m
� ei
m
���� Α �
����� ���� ����
Α ���� Βm � Βm ����
�����
Αm
m m
e i
m
���� ej
m
�����
����
����� i
ej � g�e ���� ej
m ����� m ����� m
�∆ j i
� gm
ij
����� �����
j 1
���� e � ���� ei ���� ej � g
m g ����� m
����� m
mij
6.6 The Measure of an m-element
The definition of measure
The measure of an element Α m is denoted �Α m � and defined as the positive square root of the
inner product of Α m with itself.
2001 4 5
�Α m � � ������������
Α m ���� Α m
6.37
6.38
6.39
6.40
6.41