In abstract algebra and multilinear algebra, a multilinear form on is a map of the type

where is a vector space over the field (or more generally, a module over a commutative ring), that is separately K-linear in each of its arguments.[1] (The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.)

A multilinear k-form on over is called a (covariant) k-tensor, and the vector space of such forms is usually denoted or .[2]

Tensor productEdit

Given a k-tensor   and an -tensor  , a product  , known as the tensor product, can be defined by the property


for all  . The tensor product of multilinear forms is not commutative; however it is bilinear and associative:




If   forms a basis for an n-dimensional vector space   and   is the corresponding dual basis for the dual space  , then the products  , with   form a basis for  . Consequently,   has dimensionality  .


Bilinear formsEdit

Main article: Bilinear forms

If  ,   is referred to as a bilinear form. A familiar and important example of a (symmetric) bilinear form is the standard inner product (dot product) of vectors.

Alternating multilinear formsEdit

Main article: Alternating multilinear maps

An important class of multilinear forms are the alternating multilinear forms, which have the additional property that[3]


where   is a permutation and   denotes its sign (+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 the characteristic of the field   is not 2, setting   implies as a corollary that  ; that is, the form has a value of 0 whenever two of its arguments are equal. 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  .

An alternating multilinear k-form on   over   is called a multicovector of degree k or k-covector, and the vector space of such alternating forms, a subspace of  , is generally denoted  , or, using the notation for the isomorphic kth exterior power of  (the dual space of  ),  .[5] Note that linear functionals (multilinear 1-forms over  ) are trivially alternating, so that  , while, by convention, 0-forms are defined to be scalars:  .

The determinant on   matrices, viewed as an   argument function of the column vectors, is an important example of an alternating multilinear form.

Wedge productEdit

The tensor product of alternating multilinear forms is, in general, no longer alternating. However, by summing over all permutations of the tensor product, taking into account the parity of each term, the wedge product ( ) of multicovectors can be defined, so that if   and  , then  :


where the sum is taken over the set of all permutations over   elements,  . The wedge product is bilinear, associative, and anticommutative: if   and   then  .

Given a basis   for   and dual basis   for  , the wedge products  , with   form a basis for  . Hence, the dimensionality of   for n-dimensional   is  .

Differential formsEdit

Main article: Differential forms

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. Differential forms are especially useful in multivariable calculus (analysis) and differential geometry because they possess transformation properties that allow them be integrated on curves, surfaces, and their higher-dimensional analogues (differentiable manifolds). One far-reaching application is the modern statement of Stokes' theorem, a sweeping generalization of the fundamental theorem of calculus to higher dimensions.

The synopsis below is primarily based on Spivak (1965)[6] and Tu (2011).[3]

Definition of differential k-forms and construction of 1-formsEdit

To define differential forms on open subsets  , we first need the notion of the tangent space of  at  , usually denoted   or  . The vector space   can be defined most conveniently as the set of elements   ( , with   fixed) with vector addition and scalar multiplication defined by   and  , respectively. Moreover, if   is the standard basis for  , then   is the analogous standard basis for  . In other words, each tangent space   can simply be regarded as a copy of   (a set of tangent vectors) based at the point  . The collection (disjoint union) of tangent spaces of   at all   is known as the tangent bundle of   and is usually denoted  . 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).

A differential k-form on   is defined as a function   that assigns to every   a k-covector on the tangent space of  at  , usually denoted  . In brief, a differential k-form is a k-covector field. The space of k-forms on   is usually denoted  ; thus if   is a differential k-form, we write  . By convention, a continuous function on   is a differential 0-form:  .

We first construct differential 1-forms from 0-forms and deduce some of their basic properties. To simplify the discussion below, we will only consider smooth differential forms constructed from smooth ( ) functions. Let   be a smooth function. We define the 1-form   on   for   and   by  , where   is the total derivative of   at  . (Recall that the total derivative is a linear transformation.) Of particular interest are the projection maps (also known as coordinate functions)  , defined by  , where   is the ith standard coordinate of  . The 1-forms   are known as the basic 1-forms; they are conventionally denoted  . If the standard coordinates of   are  , then application of the definition of   yields  , so that  , where   is the Kronecker delta.[7] Thus, as the dual of the standard basis for  ,   forms a basis for  . As a consequence, if   is a 1-form on  , then   can be written as   for smooth functions  . Furthermore, we can derive an expression for   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. For instance, we represent the standard coordinates of vector   as  , so that   in terms of the standard basis  . In addition, superscripts appearing in the denominator of an expression (as in  ) are treated as lower indices in this convention. 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.]

Basic operations on differential k-formsEdit

The wedge product ( ) and exterior differentiation ( ) are two fundamental operations on differential forms. The wedge product of a k-form and an -form is a  -form, while the exterior derivative of a k-form is a  -form. Thus, both operations generate differential forms of higher degree from those of lower degree.

The wedge product   of differential forms is a special case of the wedge product of multicovectors in general (see above). As is true in general for the wedge product, the wedge product of differential forms is bilinear, associative, and anticommutative.

More concretely, if   and  , then


Furthermore, for any set of indices  ,


If  ,  , and  , then the indices of   can be arranged in ascending order by a (finite) sequence of such swaps. Since  ,   implies that  . Finally, as a consequence of bilinearity, if   and   are the sums of several terms, their wedge product obeys distributivity with respect to each of these terms.

The collection of the wedge products of basic 1-forms   constitutes a basis for the space of differential k-forms. Thus, any   can be written in the form


where   are smooth functions. With each set of indices   placed in ascending order, (*) is said to be the standard presentation of  .

In the previous section, the 1-form   was defined by taking the exterior derivative of the 0-form (continuous function)  . We now extend this by defining the exterior derivative operator   for  . If the standard presentation of k-form   is given by (*), the  -form   is defined by


A property of   that holds for all smooth forms is that the second exterior derivative of any   vanishes identically:  . This can be established directly from the definition of   and the equality of mixed second-order partial derivatives of   functions (see the article on closed and exact forms for details).

Integration of differential forms and Stokes' theorem for chainsEdit

To integrate a differential form over a parameterized domain, we first need to introduce the notion of the pullback of a differential form. Roughly speaking, when a differential form is integrated, applying the pullback transforms it in a way that correctly accounts for a change-of-coordinates.

Given a differentiable function   and k-form  , we call   the pullback of   by   and define it as the k-form such that


for  , where   is the map  .

If   is an n-form on   (i.e.,  ), we define its integral over the unit n-cell as the iterated Riemann integral of  :


Next, we consider a domain of integration parameterized by a differentiable function  , known as an n-cube. To define the integral of   over  , we "pull back" from   to the unit n-cell:


To integrate over more general domains, we define an n-chain   as the formal sum of n-cubes and set


An appropriate definition of the  -chain  , known as the boundary of  ,[8] allows us to state the celebrated Stokes' theorem (Stokes–Cartan theorem) for chains in a subset of  :

If   is a smooth  -form on an open set   and   is a smooth  -chain in  , then .

Using more sophisticated machinery (e.g., germs and derivations), the tangent space   of any smooth manifold   (not necessarily embedded in  ) can be defined. Analogously, a differential form   on a general smooth manifold is a map  . 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).

See alsoEdit


  1. ^ Weisstein, Eric W. "Multilinear Form". MathWorld.
  2. ^ Many authors use the opposite convention, writing   to denote the contravariant k-tensors on   and   to denote the covariant k-tensors on  .
  3. ^ a b c Tu, Loring W. (2011). An Introduction to Manifolds (2nd ed.). New York: Springer. pp. 22–23. ISBN 978-1-4419-7399-3.
  4. ^ Halmos, Paul R. (1958). Finite-Dimensional Vector Spaces (2nd ed.). New York: Van Nostrand. p. 50. ISBN 0-387-90093-4.
  5. ^ Spivak uses   for the space of k-covectors on  . However, this notation is more commonly reserved for the space of differential k-forms on  . In this article, we use   to mean the latter.
  6. ^ Spivak, Michael (1965). Calculus on Manifolds. New York: W. A. Benjamin, Inc. pp. 75–146. ISBN 0805390219.
  7. ^ The Kronecker delta is usually denoted by   and defined as  . Here, the notation   is used to conform to the tensor calculus convention on the use of upper and lower indices.
  8. ^ The formal definition of the boundary of a chain is somewhat involved and is omitted here (see Spivak (1965), pp. 98–99 for a discussion). Intuitively, if   maps to a square, then   is a linear combination of functions that maps to its edges in a counterclockwise manner. The boundary of a chain is distinct from the notion of a boundary in point-set topology.