Grassmann Algebra

TheComplement.nb 40
e i
����� 1
�
����� m
����������
�����
g
���1� m��n�m� �ei
m
The complement of a complement of a basis element
To verify the complement of a complement axiom, we begin with the equation for the
complement of a covariant basis element, take the complement of the equation, and then
substitute for ei �����
from equation 5.37.
����� m
����� ����� ����� �����
i
ei � g �e
m ����� m
� � �����
g �� ������
1 ������
����������
�����
� g �
���1�m��n�m� �ei
m
� ��1� m��n�m� �ei
m
A similar result is obtained for the complement of the complement of a contravariant basis
element.
The exterior product of basis elements
5.63
The exterior product of a basis element with the complement of the corresponding basis element
in the reciprocal basis is equal to the unit n-element. The exterior product with the complement
of any other basis element in the reciprocal basis is zero.
ei
m
e i
m
� e j
�����
� ei �
m m
������
1 ������
����������
�����
� g �
�ej
j �����
�∆i� 1
����� m
����� i � ej � e
m
m ������� g ��ej ����� m
�∆j i � 1
�����
ei
m
� ej �����
j �����
�∆i� 1
m
e i
m
�
����� i �����
ej �∆j� 1
m
The exterior product of a basis element with the complement of any other basis element in the
same basis is equal to the corresponding component of the metric tensor times the unit nelement.
2001 4 5
ei
m
�
�����
ej � ei ��g
m m
m
jk�ek �����
� � ei � g
m m
m
jk�� �����
�
1 ������
����������
�����
g �
�ek � g
����� m
m
ij� 1
�����
5.64