Editing Multilinear form (section)

==== Basic operations on differential k-forms ====
The ''exterior product'' (<math>\wedge</math>) and ''exterior derivative'' (<math>d</math>) are two fundamental operations on differential forms.  The exterior product of a <math>k</math>-form and an <math>\ell</math>-form is a <math>(k+\ell)</math>-form, while the exterior derivative of a <math>k</math>-form is a <math>(k+1)</math>-form.  Thus, both operations generate differential forms of higher degree from those of lower degree.

The [[exterior product]] <math>\wedge:\Omega^k(U)\times\Omega^\ell(U)\to\Omega^{k+\ell}(U)</math> of differential forms is a special case of the exterior product of multicovectors in general (''see above'').  As is true in general for the exterior product, the exterior product of differential forms is bilinear, associative, and is [[Alternating algebra|graded-alternating]].

More concretely, if <math>\omega=a_{i_1\ldots i_k} \, dx^{i_1}\wedge\cdots\wedge dx^{i_k}</math> and <math>\eta=a_{j_1\ldots i_{\ell}} dx^{j_1}\wedge\cdots\wedge dx^{j_{\ell}}</math>, then

: <math>\omega\wedge\eta=a_{i_1\ldots i_k}a_{j_1\ldots j_\ell} \, dx^{i_1}\wedge\cdots\wedge dx^{i_k}\wedge dx^{j_1} \wedge \cdots\wedge dx^{j_\ell}.</math>

Furthermore, for any set of indices <math>\{\alpha_1\ldots,\alpha_m\}</math>,

: <math>dx^{\alpha_1} \wedge\cdots\wedge dx^{\alpha_p} \wedge \cdots \wedge dx^{\alpha_q} \wedge\cdots\wedge dx^{\alpha_m} = -dx^{\alpha_1} \wedge\cdots\wedge dx^{\alpha_q} \wedge \cdots\wedge dx^{\alpha_p}\wedge\cdots\wedge dx^{\alpha_m}.</math>

If <math>I=\{i_1,\ldots,i_k\}</math>, <math>J=\{j_1,\ldots,j_{\ell}\}</math>, and <math>I\cap J=\varnothing</math>, then the indices of <math>\omega\wedge\eta</math> can be arranged in ascending order by a (finite) sequence of such swaps.  Since <math>dx^\alpha\wedge dx^\alpha=0</math>, <math>I\cap J\neq\varnothing</math> implies that <math>\omega\wedge\eta=0</math>.  Finally, as a consequence of bilinearity, if <math>\omega</math> and <math>\eta</math> are the sums of several terms, their exterior product obeys distributivity with respect to each of these terms.  

The collection of the exterior products of basic 1-forms <math>\{dx^{i_1}\wedge\cdots\wedge dx^{i_k} \mid 1\leq i_1<\cdots< i_k\leq n\}</math> constitutes a basis for the space of differential ''k''-forms.  Thus, any <math>\omega\in\Omega^k(U)</math> can be written in the form

: <math>\omega=\sum_{i_1<\cdots<i_k} a_{i_1\ldots i_k} \, dx^{i_1}\wedge\cdots\wedge dx^{i_k}, \qquad (*)</math>

where <math>a_{i_1\ldots i_k}:U\to\R</math> are smooth functions.  With each set of indices <math>\{i_1,\ldots,i_k\}</math> placed in ascending order, (*) is said to be the '''standard presentation''' of '''<math>\omega</math>'''. <br>

In the previous section, the 1-form <math>df</math> was defined by taking the exterior derivative of the 0-form (continuous function) <math>f</math>.  We now extend this by defining the exterior derivative operator <math>d:\Omega^k(U)\to\Omega^{k+1}(U)</math> for <math>k\geq1</math>.  If the standard presentation of <math>k</math>-form <math>\omega</math> is given by (*), the <math>(k+1)</math>-form <math>d\omega</math> is defined by

: <math>d\omega:=\sum_{i_1<\ldots <i_k} da_{i_1\ldots i_k}\wedge dx^{i_1}\wedge\cdots\wedge dx^{i_k}.</math>

A property of <math>d</math> that holds for all smooth forms is that the second exterior derivative of any <math>\omega</math> vanishes identically: <math>d^2\omega=d(d\omega)\equiv 0</math>.  This can be established directly from the definition of <math>d</math> and the [[Symmetry of second derivatives|equality of mixed second-order partial derivatives]] of <math>C^2</math> functions (''see the article on [[Closed and exact differential forms|closed and exact forms]] for details'').