Grassmann Algebra

TheComplement.nb 36
��������� ���������� ����
P � Α ��� � Α � x � Α
� Calculating with free complements
Entering a free complement
P � � � x
5.47
To enter a vector subspace complement of a Grassmann expression X in GrassmannAlgebra you
can either use the GrassmannAlgebra palette by selecting the expression X and clicking the
button � � ���
, or simply enter OverVector[X] directly.
OverVector�e1 � e2�
e1 � e2 � ���������������
Simplifying a free complement
If the basis of your currently declared space does not contain the origin �, then the vector space
complement (OverVector) operation is equivalent to the normal (OverBar) operation, and
GrassmannSimplify will treat them as the same.
�3; ���e1 � ��� ,x � ,e1
����� , x
����� ��
�e2 � e3, x � ,e2 � e3, x � �
If, on the other hand, the basis of your currently declared space does contain the origin �, then
GrassmannSimplify will convert any expressions containing OverVector complements
to their equivalent OverBar forms.
�3; ���e1 � ��� ,x � ,e1
����� , x
����� ��
���������
�e2 � e3, � � x,
��� � e2 � e3�, x � �
The free complement of the origin
Note that the free complement of the origin or any exterior product of elements involving the
origin is undefined, and will be left unevaluated by GrassmannSimplify.
�3; ���� � � ������������� � ���������� � �������������� , � � e1,
� � x, � � e1��
�� � , � � e1 � ������������� ��������� � �
, � � x � �, � � e2 � e3�
Example: The complement of a screw
Consider an interpreted 2-element S in a 3-plane which is the sum of a bivector x�y and a
vector bound through the origin perpendicular to the bivector. We can represent the vector as
congruent to the free complement of the bivector. In the simplest case when the vector is equal
to the free complement of the bivector we have:
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