Grassmann Algebra

TheComplement.nb 37
Graphic of a vector bound through the origin perpendicular to a bivector x�y.
S � � � x � y � ��������� � x � y
One of the interesting properties of such an entity is that it is equal to its own complement. We
can show this by simple transformations using the formulae derived in the sections above. The
complement of each term turns into the other term.
����� ���������������
S � � � x
���������
� y
�
�
���������
x � y
� x � y � ����������
��������� � ��1� 2 �� � x � y � ���������
� x � y � � � x � y � ��������� � S
We can verify this by expressing x and y in basis form and using GrassmannSimplify.
�3; x� CreateVector�a�; y� CreateVector�b�;
��S � S
����� �
True
This entity is a simple case of a screw. We shall explore its properties further in Chapter 7:
Exploring Screw Algebra, and see applications to mechanics in Chapter 8: Exploring Mechanics.
5.11 Reciprocal Bases
Contravariant and covariant bases
Up to this point we have only considered one basis for �, which we now call the contravariant
1
basis. Introducing a second basis called the covariant basis enables us to write the formulae for
complements in a more symmetric way. Contravariant basis elements were denoted with
subscripted indices, for example ei . Following standard practice, we denote covariant basis
elements with superscripts, for example ei . The two bases are said to be reciprocal.
This section will summarize formulae relating basis elements and their cobases and
complements in terms of reciprocal bases. For simplicity, we adopt the Einstein summation
convention.
In � 1 the metric tensor gij forms the relationship between the reciprocal bases.
ei � gij�e j
This relationship induces a metric tensor g m
i,j on � m .
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