“Multilinear form”的意思、由来-开放百科全书

In abstract algebra and multilinear algebra, a multilinear form on

is a map of the type

A multilinear k-form on

over

is called a (covariant) k-tensor, and the vector space of such forms is usually denoted

.^[2]

Tensor product

Given k-tensor

and ℓ-tensor

, a product

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

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^[3]

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 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 wedge product (

) of multicovectors can be defined, so that 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

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.

Definition and construction of differential 1-forms

To define differential forms on open subsets

, we first need the notion of the tangent space of

, usually denoted

. 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

, 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

for

and

, where

is the total derivative of

. (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

, 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-forms

The wedge product (

) and exterior differentiation (

) are two fundamental operations on differential forms. The wedge product of a k-form and an ℓ-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.

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

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

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.

Given a differentiable function