Grassmann Algebra

TheInteriorProduct.nb 39
Note that the second factor in the interior product of the right-hand side is of grade 1. We apply
the Interior Common Factor Theorem to give:
� � m � �
�Α1 � � � Αm � x� ���� ��Α � x� ���� Αm � � ��1� ����Αm � x� ���� Αm � ���� x��Α1 � � � Αm �
m
����1� i i�1 � �
����Α � x� ���� Αm � ���� Αi��Α1 � � � ���� i � � � Αm � x
m
But the terms in this last sum are zero since:
� � � �
��Α � x� ���� Αm � ���� Αi � �Α � x� ���� �Αm � Αi� � 0
m
m
This leaves only the first term in the expansion. Thus:
� � �
�Α � x� ���� ��Αm � x� ���� Αm � �
m
��1� m � � � m � � �
����Α � x� ���� Αm � ���� x��Αm � ��1� ��Αm � x� ���� �Αm � x��Αm
m
� �
Since Α ���� Αm � 1, we have finally that:
m
� � � � � �
��Α � x� ���� Αm � ���� ��Αm � x� ���� Αm � � �Αm � x� ���� �Αm � x�
m
� � �
The measure of �Α � x� ���� Αm is therefore the same as the measure of Αm � x.
m
� � �
��Α � x� ���� Αm � � �Αm � x�
m
Equivalent forms for the triangle components
6.107
There is an interesting relationship between the two terms �Α � x� ���� Α and Α ���� �Α ���� x� which
m m m m
enables formula 6.104 to be proved in an identical manner to formula 6.103. Using the identity
[6.13] we find that each form may be expressed (apart from a possible sign) in the other form
provided Α is replaced by its complement
�����
Α .
m m
�Α � x� ���� Α � ��1�
m m n�m�1�Α �����
����
��������� n�m�1
�Αm � x�
� ��1�
�����
�Αm ���� �
�����
Αm ���� x�
m
In a similar manner it may be shown that:
2001 4 5
�Α � x� ���� Α � ��1�
m m n�m�1 �Α
�����
���� �
�����
Αm ���� x�
m
Α ���� �Α ���� x� � ��1�
m m m�1 ����� �����
�� Α � x� ���� Αm
m
6.108
6.109