Grassmann Algebra

TheRegressiveProduct.nb 22
� Automating the application of the Common Factor Theorem
The Common Factor Theorem is automatically applied to regressive products by
GrassmannSimplify. In the example of the previous section we could therefore have used
GrassmannSimplify directly to determine the common factor.
X � ���3 e1 � e2 � 2e1 � e3 � 3e2 � e3���5�e2 � 7�e3��e1�
e1 � e2 � e3 ���11 e1 � 15 e2 � 21 e3�
If the resulting expression explicitly involves the basis n-element of the currently declared basis
we can use formula 3.10: 1 � � �e1 � e2 � � � en to eliminate any regressive products of
n
the basis n-element and replace them with scalar multiples of the congruence factor �.
�e1 � e2 � � � en��Α �
m 1
����� �1 � Α �
� n m 1
����� �Α
� m
We can automate this step by using the GrassmannAlgebra function ToCongruenceForm:
ToCongruenceForm�X�
�11 e1 � 15 e2 � 21 e3
�������������������������������� ������������������
�
Example: Multiple regressive products
Some expressions may simplify to a form in which there are regressive products of multiple
copies of the basis n-element. For example, the regressive product of three bivectors in a 3space
is a scalar. To compute this scalar we first create three general bivectors X, Y, and Z, by
using CreateBasisForm:
�X, Y, Z� � CreateBasisForm�2, �x, y, z��
�x1 e1 � e2 � x2 e1 � e3 � x3 e2 � e3,
y1 e1 � e2 � y2 e1 � e3 � y3 e2 � e3, z1 e1 � e2 � z2 e1 � e3 � z3 e2 � e3�
Next, we compute the regressive product R, say, of these three bivectors by using
GrassmannSimplify.
R � ��X � Y � Z�
�x3 ��y2 z1 � y1 z2� � x2 �y3 z1 � y1 z3� � x1 ��y3 z2 � y2 z3��
e1 � e2 � e3 � e1 � e2 � e3 � 1
We can convert this expression into one which more clearly portrays its scalar nature by using
ToCongruenceForm.
2001 4 5
ToCongruenceForm�R�
x3 ��y2 z1 � y1 z2� � x2 �y3 z1 � y1 z3� � x1 ��y3 z2 � y2 z3�
�������������������������������� �������������������������������� �������������������������������� �������������������������������� ����������
�2