Grassmann Algebra

TheExteriorProduct.nb 20
That is:
ei � ei ����� � e1 � e2 � � � en
ei
����� � ��1�i�1 �e1 � � � ���� i � � � en
where ���� i means that ei is missing from the product.
The choice of the underbar notation to denote the cobasis may be viewed as a mnemonic to
indicate that the element ei
����� is the basis element of � n with ei 'struck out' from it.
Cobasis elements have similar properties to Euclidean complements, which are denoted with
overbars. However, it should be noted that the underbar denotes an element, and is not an
operation. For example: aei �������� is not defined.
More generally, the cobasis element of the basis m-element ei � ei1 m � � � eim of � m is
denoted ei and is defined as the product of the remaining basis elements such that:
����� m
That is:
m
where �m � �Γ�1 ei � ei
m ����� m
� e1 � e2 � � � en
ei
����� m
� ��1��m �e1 � � � ���� i1 � � � ���� im � � � en
�iΓ � 1
����
2 �m��m � 1�.
From the above definition it can be seen that the exterior product of a basis element with the
cobasis element of another basis element is zero. Hence we can write:
The cobasis of unity
ei � ej
m
����� m
�∆ij�e1 � e2 � � � en
The natural basis of � 0 is 1. The cobasis 1
����� of this basis element is defined by formula 2.24 as
the product of the remaining basis elements such that:
Thus:
2001 4 5
1 � 1
������ e1 � e2 � � � en
1
������ e1 � e2 � � � en � e n
2.21
2.22
2.23
2.24
2.25