Multilinear form

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


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.[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 .

Tensor product[edit]

Given multilinear forms and , 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 distributive: , and associative: .

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


Bilinear form[edit]

Main article: Bilinear forms

For , i.e. only two variables, 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 form[edit]

An important type of multilinear forms are alternating multilinear forms, which have the additional property that[2]


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 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[3] use this last property to define a form as being alternating. 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 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 ), . 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 product[edit]

The tensor product between two 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 () can be defined, so that if and , :


where the sum is taken over the set of all permutations over elements, . The wedge product is distributive, associative, and anti-commutative: if and then .

Given a basis for and its dual for dual vector space , the wedge products , with form a basis for . Hence, the dimensionality of for dimensional is .

Differential form[edit]

Main article: Differential forms

Differential forms are mathematical objects defined via tangent spaces and multilinear forms that behave, in many ways, like differentials in a classical sense, which are founded on nebulous and ill-defined notions of infinitesimal quantities. Thus, they allow the idea of the differential to be placed on mathematically rigorous foundations. 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 higher-dimensional analogues (differentiable manifolds) in multidimensional space. The synopsis below mainly follows Spivak (1965).[4]

Construction of differential forms[edit]

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 . If are the standard basis vectors for , then form an analogous standard basis for . 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). In this description, each tangent space can simply be regarded as a copy of based at the point (this is the set of tangent vectors at ). The (disjoint) union of tangent spaces at all is known as the tangent bundle and is usually denoted .

A differential k-form on is defined as a function that assigns to every an alternating multilinear form on the tangent space at : . In brief, 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 .

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 differential forms constructed from smooth () functions (called smooth forms). Let be a smooth function. We 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 , where is the ith standard coordinate of (the are also known as coordinate functions). We define the basic 1-forms by . If the standard coordinates of are , then application of the definition of yields , so that , where is the Kronecker delta.[5] Thus, as the dual of the standard basis for , the 1-forms constitute a basis for . As a consequence, if is a 1-form on , then can be written as for smooth functions . Furthermore, we can obtain an expression for in terms of the which matches the classical expression for a total differential, derived from the nonrigorous manipulation of infinitesimals:


In this article, we follow the convention from differential geometry in which vectors and covectors are indexed by subscripts (lower indices) and superscripts (upper indices), respectively. In particular, as covector fields, differential forms are indexed by upper indices, and continuous functions, as differential 0-forms, are also indexed this way.[2] However, the components of vectors (covectors, resp.) are indexed with superscripts (subscripts, resp.). For example, we generally denote the standard coordinates of as , so that for standard basis vectors . Also, an upper index in a denominator (e.g., ) is considered to be equivalent to a lower index. When these conventions are followed, the number of upper indices minus the number of lower indices in each term is conserved on two sides of an equal sign. We note that there is an inherent source of confusion when upper indices are used, and care must be taken to correctly interpret an expression in cases where a superscript could denote either an index or an exponent.

Basic operations on differential forms[edit]

The wedge product () and exterior differentiation () are two fundamental operations that can be performed on differential forms. The wedge product is defined as a special case of the wedge product of alternating multilinear forms in general (see above) and allows for the construction of general higher differential k-forms (), so that any k-form can be written in the standard presentation as


for smooth . As is the case for the wedge product on alternating forms in general, the wedge product on differential forms is associative, distributive, and anti-commutative. More concretely, if and , then , with the stipulation that for any set of indices ,


If , , and , then the indices of can be arranged in ascending order (the standard presentation) by a (finite) sequence of such swaps. If , then . Finally, if and are the sums of several terms, the wedge product is defined so as to obey distributivity. Used previously to define 1-forms from differentiable functions, i.e., 0-forms, the exterior derivative operator can be generalized to operate on an arbitrary k-form as given by (*): the -form is defined by


Integration of differential forms and Stokes' theorem on chains[edit]

If is a top-form (i.e., an n-form in ), we define


For a differentiable function , denotes the pullback of the differential form by , defined by


for , where is the map . For a continuous function (known as an n-cube), we can then define


To integrate over more general surfaces, we consider the formal sum of n-cubes into an n-chain and define


With an appropriate definition of the boundary of , , as an -chain (see Spivak, 1965, p. 98-99), we can now state the famous generalized Stokes' theorem on chains:

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

Lastly, we note that with more sophisticated machinery (e.g., using germs and derivations), 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 the Stokes–Cartan theorem can be further generalized to arbitrary smooth manifolds-with-boundary or even "manifolds-with-corners".[6]

See also[edit]


  1. ^ Weisstein, Eric W. "Multilinear Form". MathWorld. 
  2. ^ a b Tu, Loring W. (2011). An Introduction to Manifolds (PDF) (2nd ed.). New York: Springer. pp. 22–23. ISBN 978-1-4419-7399-3. 
  3. ^ Halmos, Paul R. (1958). Finite-Dimensional Vector Spaces (PDF) (2nd ed.). New York: Van Nostrand. p. 50. ISBN 0-387-90093-4. 
  4. ^ Spivak, Michael (1965). Calculus on Manifolds (PDF). New York: W. A. Benjamin, Inc. pp. 75–146. ISBN 0805390219. 
  5. ^ The Kronecker delta is usually denoted by and defined as . Here, the notation is used to parallel the use of sub- and superscripts on the left-hand side.
  6. ^ Loomis, Lynn Harold; Sternberg, Shlomo (1990). Advanced Calculus (PDF). Boston: Jones and Bartlett Publishers. pp. 442–452. ISBN 0-86720-122-3.