Grassmann Algebra

TheInteriorProduct.nb 21
Unit elements
Α m ��ai�ei
m
� �Α� �
m ��������������
� ai 2
The unit m-element associated with the element Α m is denoted Α m
�
Α �
m
Α m
����������
�Α m �
� and may now be defined as:
6.49
6.50
The unit scalar a � corresponding to the scalar a, is equal to +1 if a is positive and is equal to -1
if a is negative.
� a
a � ���������� � Sign�a�
�a�
� �
1 � 1 �1 ��1
6.51
6.52
�
The unit n-element Α corresponding to the n-element Αn � ae1 � � � en is equal to the unit n-
n
element 1
����� if a is positive, and equal to � 1
����� if a is negative.
�
Α �
n
Α n
����������
�Α n �
�����
1
� � 1
�����
� ae1 � � � en
�����������������������������
�a�� �����
g
� Sign�a�� 1
�����
� 1
����� � �����
��1
�
The measure of a unit m-element Α is therefore always unity.
m
� Calculating measures
�
�Α� � 1
m
6.53
6.54
6.55
To calculate the measure of any element we can use the GrassmannAlgebra function Measure.
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