Grassmann Algebra

TheInteriorProduct.nb 31
�Α � x� ���� Β � ��1�
m k
�m�k���n�m��n�k�1 �Β
k
Α ���� �Β ���� x� � ��1�
m k
�m�k���n�m��m�1 �� Β
k
�����
���� �
�����
Αm ���� x�
����� �����
� x� ���� Αm
Hence equation 6.69 may be expressed in a variety of ways, for example:
�Α ���� Β��x � �Α � x� ���� Β � ��1�
m k
m k
�m�k���n�m� �� Β
k
Interior Product Formula 4
�����
� x� ����
�����
Αm
Taking the interior product of equation 6.69 with a second 1-element x2 gives:
��Α ���� Β��x1� ���� x2 �
m k
�Α � x1� ���� �Β � x2� � ��1�
m k
m�k �� Α ���� x2� ���� �Β ���� x1�
m k
Interior product formulae for p-elements from regressive product
formulae
Interior Product Formula 5
Consider equation 3.44.
�Α m � Β k
x p � x1
r
Now replace Β k
p
��x � � ���1�
p r�0
r��n�m� Ν
���
i�1
�
�
� x1 � x2 � x2 � � � xΝ
p�r r p�r
r
����� �����
with Β , Β � xi
n�k n�k
�Α m ���� Β k
x p � x1
r
Interior Product Formula 6
r
���Α � xi
m p�r
����
��Β
� k
� xi�
r
� xΝ , Ν��
p�r
p
r �
with ��1�r��n�r� � �
�����
xi
n�k r
p
��x � � ���1�
p r�0
r��m�r� Ν
���
i�1
�
�
� x1 � x2 � x2 � � � xΝ
p�r r p�r
r
By a similar process equation 3.45 becomes:
2001 4 5
������������� , and nÐk with k to give:
���Α � xi
m p�r
����
���� �Β
� k
���� xi�
r
� xΝ , Ν��
p�r
p
r �
6.71
6.72
6.73