Editing Exterior algebra (section)

=== Index notation ===
Suppose that ''V'' has finite dimension ''n'', and that a basis {{nowrap|'''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>}} of ''V'' is given.  Then any alternating tensor {{nowrap|''t'' ∈ A<sup>''r''</sup>(''V'') ⊂ ''T''<sup>''r''</sup>(''V'')}} can be written in [[index notation]] with the [[Einstein summation convention]] as
: <math>t = t^{i_1i_2\cdots i_r}\, {\mathbf e}_{i_1} \otimes {\mathbf e}_{i_2} \otimes \cdots \otimes {\mathbf e}_{i_r},</math>
where ''t''<sup>''i''<sub>1</sub>⋅⋅⋅''i''<sub>''r''</sub></sup> is [[antisymmetric tensor|completely antisymmetric]] in its indices.

The exterior product of two alternating tensors ''t'' and ''s'' of ranks ''r'' and ''p'' is given by
: <math>
t~\widehat{\otimes}~s
= \frac{1}{(r+p)!}\sum_{\sigma \in {\mathfrak S}_{r+p}}\operatorname{sgn}(\sigma)t^{i_{\sigma(1)} \cdots i_{\sigma(r)}} s^{i_{\sigma(r+1)} \cdots i_{\sigma(r+p)}} {\mathbf e}_{i_1} \otimes {\mathbf e}_{i_2} \otimes \cdots \otimes {\mathbf e}_{i_{r+p}}.
</math>

The components of this tensor are precisely the skew part of the components of the tensor product {{nowrap|''s'' ⊗ ''t''}}, denoted by square brackets on the indices:
: <math>
(t~\widehat{\otimes}~s)^{i_1\cdots i_{r+p}}
= t^{[i_1\cdots i_r}s^{i_{r+1}\cdots i_{r+p}]}.
</math>

<!--For the interior product-->
The [[#Interior product|interior product]] may also be described in index notation as follows.  Let <math> t = t^{i_0i_1\cdots i_{r-1}} </math> be an antisymmetric tensor of rank {{tmath|r}}.  Then, for {{nowrap|''α'' ∈ ''V''<sup>∗</sup>}}, {{tmath|\iota_\alpha t}} is an alternating tensor of rank {{tmath|r - 1}}, given by
: <math>
(\iota_\alpha t)^{i_1\cdots i_{r-1}}
= r\sum_{j=0}^n\alpha_j t^{ji_1\cdots i_{r-1}}.
</math>
where ''n'' is the dimension of ''V''.