Grassmann Algebra

TheComplement.nb 38
ei
m
The complement of a basis element
� g m
ij �ej
m
The complement of a basis 1-element of the contravariant basis has already been defined by:
g � �gij�
����� 1
ei � ���������� �gij�ej
����� �����
g
Here g � �gij� is the determinant of the metric tensor. Taking the complement of equation
5.22 and substituting for ei
����� in equation 5.24 gives:
The reciprocal relation to this is:
e i
����� 1
�
����������
�����
g
ei
�����
����� ����� i
ei � ge �����
These formulae can be extended to basis elements of � m .
Particular cases of these formulae are:
2001 4 5
����� �����
ei �
m
e i
����� 1
�
m
����������
�����
g
g �e i
m
�����
�ei
m
�����
1
���������� �e1 �����
� e2 � � � en � 1
g
����� � �����
ge1�e2 � � � en ���������������������������� �����
e1 � e2 � � � en � g
5.49
5.50
5.51
5.52
5.53
5.54
5.55
5.56