Grassmann Algebra

ExpTheGeneralizedProduct.nb 38
ToScalarProducts��ae1 � e2������ 1 ��b e1 � e2��
0
Products of the second order
Products of the second order of two 1-elements, or of a 1-element and a 2-element are
necessarily zero since the order of the product is greater than the minimum of the grades of the
factors.
ToScalarProducts��ae1 � be2������ 2 ��c e1 � de2��
0
Products of the second order of two 2-elements reduce to their inner product:
�a e1 � e2������ 2 ��b e1 � e2� � �a e1 � e2� ���� �b e1 � e2�
ToInnerProducts��a e1 � e2������ 2 ��b e1 � e2��
ae1 � e2 � be1 � e2
In sum: The only non-zero generalized products of elements (other than where one is a scalar)
in a space of two dimensions reduce to exterior, interior, inner or scalar products.
10.16 Generalized Products in 3-Space
To be completed
2001 4 26