Grassmann Algebra

ExploringCliffordAlgebra.nb 9
ToScalarProducts�a �Α m �
a Α1 � Α2 � � � Αm
a �Α m � a Α m
Hence the Clifford product of any number of scalars is just their underlying field product.
ToScalarProducts�a � b � c�
abc
� The Clifford product of 1-elements
12.4
The Clifford product of two 1-elements is just the sum of their interior (here scalar) and exterior
products. Hence it is of grade {0, 2}.
ToScalarProducts�x � y�
x � y � x � y
x � y � x � y � x ���� y
The Clifford product of any number of 1-elements can be computed.
ToScalarProducts�x � y � z�
z �x � y� � y �x � z� � x �y � z� � x � y � z
ToScalarProducts�w � x � y � z�
�w � z� �x � y� � �w � y� �x � z� � �w � x� �y � z� � �y � z� w � x � �x � z� w � y �
�x � y� w � z � �w � z� x � y � �w � y� x � z � �w � x� y � z � w � x � y � z
The Clifford product of an m-element and a 1-element
12.5
The Clifford product of an arbitrary simple m-element and a 1-element results in just two terms:
their exterior and interior products.
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x �Α m � x � Α m �Α m ���� x
12.6