Grassmann Algebra

TheInteriorProduct.nb 13
the interior product, the Interior Common Factor Theorem may then be applied to each of the
simple terms.
Α m ���� Β k
1 2 3 � �Α �Αm �Αm � �� ���� Βk
m
1 2 3 � Α ���� Βk �Α ���� Βk �Α ���� Βk � �
m m m
Thus, the Interior Common Factor Theorem may be applied to the interior product of any two
elements, simple or non-simple. Indeed, the application to components in the preceding
expansion may be applied to the components of Α m even if it is simple.
To make the Interior Common Factor Theorem more explicit, suppose Α �Α1� Α2 � � � Αm ,
m
then:
�
i1 �� ik
k
where �k � �Γ�1 �Α1 � Α2 � � � Αm� ���� Β k
��Αi1 � � � Αik � ���� Β����1� k
�k �Α1 � � � ���� i1 � � � ���� ik � � � Αm
�iΓ � 1
����
2 �k��k � 1�, and ���� j means Αj is missing from the product.
� Examples of the Interior Common Factor Theorem
In this section we take the formula just developed and give examples for specific values of m
and k. We do this using the GrassmannAlgebra function
InteriorCommonFactorTheorem which generates the specific case of the theorem when
given an argument of the form Α ���� Β. The grades m and k must be given as positive integers.
m k
Β is a scalar
Β is a 1-element
2001 4 5
�Α1 � Α2 � � � Αm � ���� Β 0
�Α1 � Α2 � � � Αm � ���� Β��
i�1
InteriorCommonFactorTheorem�Α 2 ���� Β�
m
�
�Β��Α1 � Α2 � � � Αm�
0
�Αi ���� Β ����1� i�1 �Α1 � � � ���� i � � � Αm
Α 2 � Β��Α1 � Α2 � Β����Β � Α2� Α1 � �Β � Α1� Α2
6.29
6.30
6.31