Multilinear form on arvau

Wikipedia / Multilinear form

In abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map

-linear arguments. 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.

are naturally identified with linear forms on the tensor product is called a (covariant) -tensor.

Tensor product of multilinear forms

, known as the tensor product, can be defined by the property

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

and

is the corresponding dual basis for the dual space.

Examples

Bilinear forms

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 forms

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

is a permutation denotes its sign (+1 if even, –1 if odd). As a consequence, alternating):

With the additional hypothesis that the characteristic of the field; that is, the form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors.

is called a or -covector, or, using the notation for the isomorphic kth exterior power(the dual space. Note that linear functionals.

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

Exterior product

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 exterior product, also known as the wedge product:

for n.

Application to differential forms

Differential forms are tensor fields of multilinear forms on tangent spaces that behave, in many ways, like differentials in the classical sense. Though conceptually and computationally useful, differentials, as originally conceived, were based 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 the Stokes–Cartan theorem, a sweeping generalization of the fundamental theorem of calculus to higher dimensions.

The elementary development for open subsets of Euclidean space outlined below is primarily based on Spivak (1965) and Tu (2011). For the full generality of differential forms on smooth manifolds, see the main article.

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

, we first need the notion of the tangent space fixed) with vector addition and scalar multiplication is the standard basis is known as the tangent bundle, 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 -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 is the total derivative is the i are known as the basic 1-forms is the Kronecker delta. that coincides with the classical expression for a total differential:

yielding formally:

[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. The opposite rule applies to the components. In addition, superscripts appearing in the denominator) 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. For a related convention, see: Einstein notation.]

Basic operations on differential k-forms

The exterior product) and exterior derivative-form, while the exterior derivative-form. Thus, both operations generate differential forms of higher degree from those of lower degree.

The exterior product 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 graded-alternating.

, then

are the sums of several terms, their exterior product obeys distributivity with respect to each of these terms.

constitutes a basis for the space of differential k can be written in the form

placed in ascending order, (*) is said to be the standard presentation of .

is defined by

and the equality of mixed second-order partial derivatives functions (see the article on closed and exact forms for details).

Integration of differential forms and Stokes' theorem for chains

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.

the pullback-form such that

-cell as the iterated Riemann integral:

, known as a (singular) n-cube to the unit n-cell:

To integrate over more general domains, we define an -cubes and set

-chain, we can state the celebrated Stokes' theorem: is a smooth is a smooth.Using more sophisticated machinery (e.g., germs and derivations. 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).