[1] More generally, one can define multilinear forms on a module over a commutative ring.
The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.
The tensor product of multilinear forms is not commutative; however it is bilinear and associative: and If
As a consequence, alternating multilinear forms are antisymmetric with respect to swapping of any two arguments (i.e.,
Note, however, that some authors[4] use this last condition as the defining property of alternating forms.
This definition implies the property given at the beginning of the section, but as noted above, the converse implication holds only when
, or, using the notation for the isomorphic kth exterior power of
argument function of the column vectors, is an important example of an alternating multilinear form.
Differential forms are mathematical objects constructed via tangent spaces and multilinear forms that behave, in many ways, like differentials in the classical sense.
Though conceptually and computationally useful, differentials are founded on ill-defined notions of infinitesimal quantities developed early in the history of calculus.
Differential forms provide a mathematically rigorous and precise framework to modernize this long-standing idea.
One far-reaching application is the modern statement of Stokes' theorem, a sweeping generalization of the fundamental theorem of calculus to higher dimensions.
fixed) with vector addition and scalar multiplication defined by
(a set of tangent vectors) based at the point
While the definition given here provides a simple description of the tangent space of
, there are other, more sophisticated constructions that are better suited for defining the tangent spaces of smooth manifolds in general (see the article on tangent spaces for details).
(Recall that the total derivative is a linear transformation.)
Of particular interest are the projection maps (also known as coordinate functions)
that coincides with the classical expression for a total differential: [Comments on notation: In this article, we follow the convention from tensor calculus and differential geometry in which multivectors and multicovectors are written with lower and upper indices, respectively.
Since differential forms are multicovector fields, upper indices are employed to index them.
[3] The opposite rule applies to the components of multivectors and multicovectors, which instead are written with upper and lower indices, respectively.
When indices are applied and interpreted in this manner, the number of upper indices minus the number of lower indices in each term of an expression is conserved, both within the sum and across an equal sign, a feature that serves as a useful mnemonic device and helps pinpoint errors made during manual computation.]
of differential forms is a special case of the exterior product of multicovectors in general (see above).
can be arranged in ascending order by a (finite) sequence of such swaps.
was defined by taking the exterior derivative of the 0-form (continuous function)
functions (see the article on closed and exact forms for details).
Roughly speaking, when a differential form is integrated, applying the pullback transforms it in a way that correctly accounts for a change-of-coordinates.
to the unit n-cell: To integrate over more general domains, we define an
.Using more sophisticated machinery (e.g., germs and derivations), the tangent space
Stokes' theorem can be further generalized to arbitrary smooth manifolds-with-boundary and even certain "rough" domains (see the article on Stokes' theorem for details).