where is a vector space over the field (and more generally, a module over a commutative ring), that is separately K-linear in each of its arguments. A multilinear n-form on over is called an n-tensor, and the vector space of such forms is often denoted or .
An important type of multilinear forms are alternating multilinear forms, which have the additional property that
where is a permutation and denotes its parity (+1 if even, –1 if odd). As a consequence, alternating multilinear forms are antisymmetric with respect to swapping of any two arguments (i.e., and ):
With the additional hypothesis that , setting affords the corollary that ; that is, an alternating form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors use this last property as the definition of alternating. This definition implies the one given in this article, but as noted above, the converse is true only when the characteristic of the field is not 2.
An alternating multilinear n-form on V over is called an n-covector, and the vector space of such alternating forms is generally denoted , , or using the notation for the isomorphic nth exterior power of (the dual space of ), .
Differential forms are mathematical objects rigorously defined via tangent spaces and multilinear forms that behave, in many ways, like differentials in a classical sense, founded on ill-defined notions of infinitesimal quantities. Furthermore, they turn out to have the correct transformation properties to allow them be integrated on curves, surfaces, and their higher-dimensional analogues (manifolds) in multidimensional space. The brief outline below mainly follows Spivak (1965).
Construction of differential forms, basic properties and operations
To define differential forms on open subsets , we need the notion of the tangent space of at . The tangent space can be defined most conveniently as the vector space of elements () with addition and scalar multiplication defined by and . (This is the simplest description of the tangent space of ; other more sophisticated constructions are better suited for generalization to arbitrary smooth manifolds. See the page on tangent spaces for details.)
A differential k-form on is then defined to be a function that assigns to every an alternating multilinear form ; in other words, a differential k-form is a k-covector field. The space of differential k-forms on is usually denoted , so that . By convention, a differential 0-form on is a continuous function .
To simplify the discussion below, we will only consider differential forms constructed from smooth () functions (called smooth forms). To construct a differential 1-form, let be a smooth function, and define the 1-form by , where is the total derivative of at . (Recall that the total derivative is a linear transformation.) Of particular interest are the projection maps . We define the 1-forms . If , then application of the definition of yields , so that , where . Thus, is the basis for and is the dual for the standard basis of . As a consequence, if is a 1-form on , can be written as for smooth functions . Further, we can obtain an expression for in terms of the which matches the classical expression for a total differential derived non-rigorously by manipulation of infinitesimals:
The wedge product allows for the definition of higher differential k-forms (), so that any k-form can be written as
for smooth . The wedge product is associative, distributive, and anti-commutative. Let and , then with the stipulation that
If and are the sum of several terms, the wedge product is defined so as to obey distributivity. For as defined in (*), a generalized definition can also be given for the exterior derivative operator . The form is given by
If is asmooth-form on an open set and is a smooth-chain in , then.
Finally we note that with more sophisticated machinery, the tangent space of any smooth manifold can be defined. By analogy, a differential form on a general smooth manifold is then a map , and Stokes' theorem can be further generalized to arbitrary smooth manifolds-with-boundary.