Grassmann Algebra

ExpTheGeneralizedProduct.nb 11
Example 3
A245 � ToInteriorProductsB�Α����� �Β� 2 4 5
Α1 � Α2 � Β1 � Β2 � Β3 � Β4 � Β5 �
Α1 � Α2 � Β2 � Β1 � Β3 � Β4 � Β5 �Α1 � Α2 � Β3 � Β1 � Β2 � Β4 � Β5 �
Α1 � Α2 � Β4 � Β1 � Β2 � Β3 � Β5 �Α1 � Α2 � Β5 � Β1 � Β2 � Β3 � Β4
ToScalarProducts�A245�
0
10.5 The Generalized Product Theorem
The A and B forms of a generalized product
There is a surprising alternative form for the expansion of the generalized product in terms of
exterior and interior products. We distinguish the two forms by calling the first (definitional)
form the A form and the new second form the B form.
A: Α m ����� Λ �Β k
B: Α m ����� Λ �Β k
Β k
� Β 1
Λ
� Β 1
k�Λ
� k
Λ �
� �
j�1
� k
Λ �
� �
j�1
� Β 2
Λ
�Α ���� Β
m j��Β
Λ
j
k�Λ
�Α � Β
m j � ���� Β
k�Λ
j
Λ
� Β 2
� �
k�Λ
The identity between the A form and the B form is the source of many useful relations in the
Grassmann and Clifford algebras. We call it the Generalized Product Theorem.
10.10
In the previous sections, the identity between the expansions in terms of either of the two factors
was shown to be at the inner product level. The identity between the A form and the B form is
at a further level of complexity - that of the scalar product. That is, in order to show the identity
between the two forms, a generalized product may need to be reduced to an expression
involving exterior and scalar products.
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