Multilinear form – Wikipedia

Map from multiple vectors to an underlying field of scalars, linear in each argument

In abstract algebra and multilinear algebra, a multilinear form on a vector space

${displaystyle V}$

over a field

${displaystyle K}$

is a map

${displaystyle fcolon V^{k}to K}$

that is separately

${displaystyle K}$

-linear in each of its

${displaystyle k}$

arguments.^[1] 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.

A multilinear

${displaystyle k}$

-form on

${displaystyle V}$

over

${displaystyle mathbb {R} }$

is called a (covariant)

${displaystyle {boldsymbol {k}}}$

-tensor, and the vector space of such forms is usually denoted

${displaystyle {mathcal {T}}^{k}(V)}$

or

${displaystyle {mathcal {L}}^{k}(V)}$

.^[2]

Tensor product[edit]

Given a

${displaystyle k}$

-tensor

${displaystyle fin {mathcal {T}}^{k}(V)}$

and an

${displaystyle ell }$

-tensor

${displaystyle gin {mathcal {T}}^{ell }(V)}$

, a product

${displaystyle fotimes gin {mathcal {T}}^{k+ell }(V)}$

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

${displaystyle (fotimes g)(v_{1},ldots ,v_{k},v_{k+1},ldots ,v_{k+ell })=f(v_{1},ldots ,v_{k})g(v_{k+1},ldots ,v_{k+ell }),}$

for all

${displaystyle v_{1},ldots ,v_{k+ell }in V}$

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

${displaystyle fotimes (ag_{1}+bg_{2})=a(fotimes g_{1})+b(fotimes g_{2})}$

,

${displaystyle (af_{1}+bf_{2})otimes g=a(f_{1}otimes g)+b(f_{2}otimes g),}$

and

${displaystyle (fotimes g)otimes h=fotimes (gotimes h).}$

If

${displaystyle (v_{1},ldots ,v_{n})}$

forms a basis for an

${displaystyle n}$

-dimensional vector space

${displaystyle V}$

and

${displaystyle (phi ^{1},ldots ,phi ^{n})}$

is the corresponding dual basis for the dual space

${displaystyle V^{*}={mathcal {T}}^{1}(V)}$

, then the products

${displaystyle phi ^{i_{1}}otimes cdots otimes phi ^{i_{k}}}$

, with

${displaystyle 1leq i_{1},ldots ,i_{k}leq n}$

form a basis for

${displaystyle {mathcal {T}}^{k}(V)}$

. Consequently,

${displaystyle {mathcal {T}}^{k}(V)}$

has dimensionality

${displaystyle n^{k}}$

.

Examples[edit]

Bilinear forms[edit]

If

${displaystyle k=2}$

,

${displaystyle f:Vtimes Vto K}$

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[edit]

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

${displaystyle f(x_{sigma (1)},ldots ,x_{sigma (k)})=operatorname {sgn}(sigma )f(x_{1},ldots ,x_{k}),}$

where

${displaystyle sigma :mathbf {N} _{k}to mathbf {N} _{k}}$

is a permutation and

${displaystyle operatorname {sgn}(sigma )}$

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.,

${displaystyle sigma (p)=q,sigma (q)=p}$

and

${displaystyle sigma (i)=i,1leq ileq k,ineq p,q}$

):

${displaystyle f(x_{1},ldots ,x_{p},ldots ,x_{q},ldots ,x_{k})=-f(x_{1},ldots ,x_{q},ldots ,x_{p},ldots ,x_{k}).}$

With the additional hypothesis that the characteristic of the field

${displaystyle K}$

is not 2, setting

${displaystyle x_{p}=x_{q}=x}$

implies as a corollary that

${displaystyle f(x_{1},ldots ,x,ldots ,x,ldots ,x_{k})=0}$

; 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

${displaystyle operatorname {char} (K)neq 2}$

.

An alternating multilinear

${displaystyle k}$

-form on

${displaystyle V}$

over

${displaystyle mathbb {R} }$

is called a multicovector of degree

${displaystyle {boldsymbol {k}}}$

or

${displaystyle {boldsymbol {k}}}$

-covector, and the vector space of such alternating forms, a subspace of

${displaystyle {mathcal {T}}^{k}(V)}$

, is generally denoted

${displaystyle {mathcal {A}}^{k}(V)}$

, or, using the notation for the isomorphic kth exterior power of

${displaystyle V^{*}}$

(the dual space of

${displaystyle V}$

),

${textstyle bigwedge ^{k}V^{*}}$

.^[5] Note that linear functionals (multilinear 1-forms over

${displaystyle mathbb {R} }$

) are trivially alternating, so that

${displaystyle {mathcal {A}}^{1}(V)={mathcal {T}}^{1}(V)=V^{*}}$

, while, by convention, 0-forms are defined to be scalars:

${displaystyle {mathcal {A}}^{0}(V)={mathcal {T}}^{0}(V)=mathbb {R} }$

.

The determinant on

${displaystyle ntimes n}$

matrices, viewed as an

${displaystyle n}$

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

Exterior product[edit]

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 (

${displaystyle wedge }$

, also known as the wedge product) of multicovectors can be defined, so that if

${displaystyle fin {mathcal {A}}^{k}(V)}$

and

${displaystyle gin {mathcal {A}}^{ell }(V)}$

, then

${displaystyle fwedge gin {mathcal {A}}^{k+ell }(V)}$

:

${displaystyle (fwedge g)(v_{1},ldots ,v_{k+ell })={frac {1}{k!ell !}}sum _{sigma in S_{k+ell }}(operatorname {sgn}(sigma ))f(v_{sigma (1)},ldots ,v_{sigma (k)})g(v_{sigma (k+1)},ldots ,v_{sigma (k+ell )}),}$

where the sum is taken over the set of all permutations over

${displaystyle k+ell }$

elements,

${displaystyle S_{k+ell }}$

. The exterior product is bilinear, associative, and graded-alternating: if

${displaystyle fin {mathcal {A}}^{k}(V)}$

and

${displaystyle gin {mathcal {A}}^{ell }(V)}$

then

${displaystyle fwedge g=(-1)^{kell }gwedge f}$

.

Given a basis

${displaystyle (v_{1},ldots ,v_{n})}$

for

${displaystyle V}$

and dual basis

${displaystyle (phi ^{1},ldots ,phi ^{n})}$

for

${displaystyle V^{*}={mathcal {A}}^{1}(V)}$

, the exterior products

${displaystyle phi ^{i_{1}}wedge cdots wedge phi ^{i_{k}}}$

, with

${displaystyle 1leq i_{1}$

form a basis for

${displaystyle {mathcal {A}}^{k}(V)}$

. Hence, the dimensionality of

${displaystyle {mathcal {A}}^{k}(V)}$

for n-dimensional

${displaystyle V}$

is

${textstyle {tbinom {n}{k}}={frac {n!}{(n-k)!,k!}}}$

.

Differential forms[edit]

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-forms[edit]

To define differential forms on open subsets

${displaystyle Usubset mathbb {R} ^{n}}$

, we first need the notion of the tangent space of

${displaystyle mathbb {R} ^{n}}$

at

${displaystyle p}$

, usually denoted

${displaystyle T_{p}mathbb {R} ^{n}}$

or

${displaystyle mathbb {R} _{p}^{n}}$

. The vector space

${displaystyle mathbb {R} _{p}^{n}}$

can be defined most conveniently as the set of elements

${displaystyle v_{p}}$

(

${displaystyle vin mathbb {R} ^{n}}$

, with

${displaystyle pin mathbb {R} ^{n}}$

fixed) with vector addition and scalar multiplication defined by

${displaystyle v_{p}+w_{p}:=(v+w)_{p}}$

and

${displaystyle acdot (v_{p}):=(acdot v)_{p}}$

, respectively. Moreover, if

${displaystyle (e_{1},ldots ,e_{n})}$

is the standard basis for

${displaystyle mathbb {R} ^{n}}$

, then

${displaystyle ((e_{1})_{p},ldots ,(e_{n})_{p})}$

is the analogous standard basis for

${displaystyle mathbb {R} _{p}^{n}}$

. In other words, each tangent space

${displaystyle mathbb {R} _{p}^{n}}$

can simply be regarded as a copy of

${displaystyle mathbb {R} ^{n}}$

(a set of tangent vectors) based at the point

${displaystyle p}$

. The collection (disjoint union) of tangent spaces of

${displaystyle mathbb {R} ^{n}}$

at all

${displaystyle pin mathbb {R} ^{n}}$

is known as the tangent bundle of

${displaystyle mathbb {R} ^{n}}$

and is usually denoted

${textstyle Tmathbb {R} ^{n}:=bigcup _{pin mathbb {R} ^{n}}mathbb {R} _{p}^{n}}$

. While the definition given here provides a simple description of the tangent space of

${displaystyle mathbb {R} ^{n}}$

, 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

${displaystyle {boldsymbol {k}}}$

-form on

${displaystyle Usubset mathbb {R} ^{n}}$

is defined as a function

${displaystyle omega }$

that assigns to every

${displaystyle pin U}$

a

${displaystyle k}$

-covector on the tangent space of

${displaystyle mathbb {R} ^{n}}$

at

${displaystyle p}$

, usually denoted

${displaystyle omega _{p}:=omega (p)in {mathcal {A}}^{k}(mathbb {R} _{p}^{n})}$

. In brief, a differential

${displaystyle k}$

-form is a

${displaystyle k}$

-covector field. The space of

${displaystyle k}$

-forms on

${displaystyle U}$

is usually denoted

${displaystyle Omega ^{k}(U)}$

; thus if

${displaystyle omega }$

is a differential

${displaystyle k}$

-form, we write

${displaystyle omega in Omega ^{k}(U)}$

. By convention, a continuous function on

${displaystyle U}$

is a differential 0-form:

${displaystyle fin C^{0}(U)=Omega ^{0}(U)}$

.

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 (

${displaystyle C^{infty }}$

) functions. Let

${displaystyle f:mathbb {R} ^{n}to mathbb {R} }$

be a smooth function. We define the 1-form

${displaystyle df}$

on

${displaystyle U}$

for

${displaystyle pin U}$

and

${displaystyle v_{p}in mathbb {R} _{p}^{n}}$

by

${displaystyle (df)_{p}(v_{p}):=Df|_{p}(v)}$

, where

${displaystyle Df|_{p}:mathbb {R} ^{n}to mathbb {R} }$

is the total derivative of

${displaystyle f}$

at

${displaystyle p}$

. (Recall that the total derivative is a linear transformation.) Of particular interest are the projection maps (also known as coordinate functions)

${displaystyle pi ^{i}:mathbb {R} ^{n}to mathbb {R} }$

, defined by

${displaystyle xmapsto x^{i}}$

, where

${displaystyle x^{i}}$

is the ith standard coordinate of

${displaystyle xin mathbb {R} ^{n}}$

. The 1-forms

${displaystyle dpi ^{i}}$

are known as the basic 1-forms; they are conventionally denoted

${displaystyle dx^{i}}$

. If the standard coordinates of

${displaystyle v_{p}in mathbb {R} _{p}^{n}}$

are

${displaystyle (v^{1},ldots ,v^{n})}$

, then application of the definition of

${displaystyle df}$

yields

${displaystyle dx_{p}^{i}(v_{p})=v^{i}}$

, so that

${displaystyle dx_{p}^{i}((e_{j})_{p})=delta _{j}^{i}}$

, where

${displaystyle delta _{j}^{i}}$

is the Kronecker delta.^[7] Thus, as the dual of the standard basis for

${displaystyle mathbb {R} _{p}^{n}}$

,

${displaystyle (dx_{p}^{1},ldots ,dx_{p}^{n})}$

forms a basis for

${displaystyle {mathcal {A}}^{1}(mathbb {R} _{p}^{n})=(mathbb {R} _{p}^{n})^{*}}$

. As a consequence, if

${displaystyle omega }$

is a 1-form on

${displaystyle U}$

, then

${displaystyle omega }$

can be written as

${textstyle sum a_{i},dx^{i}}$

for smooth functions

${displaystyle a_{i}:Uto mathbb {R} }$

. Furthermore, we can derive an expression for

${displaystyle df}$

that coincides with the classical expression for a total differential:

${displaystyle df=sum _{i=1}^{n}D_{i}f;dx^{i}={partial f over partial x^{1}},dx^{1}+cdots +{partial f over partial x^{n}},dx^{n}.}$

[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

${displaystyle vin mathbb {R} ^{n}}$

as

${displaystyle (v^{1},ldots ,v^{n})}$

, so that

${textstyle v=sum _{i=1}^{n}v^{i}e_{i}}$

in terms of the standard basis

${displaystyle (e_{1},ldots ,e_{n})}$

. In addition, superscripts appearing in the denominator of an expression (as in

${textstyle {frac {partial f}{partial x^{i}}}}$

) 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[edit]

The exterior product (

${displaystyle wedge }$

) and exterior derivative (

${displaystyle d}$

) are two fundamental operations on differential forms. The exterior product of a

${displaystyle k}$

-form and an

${displaystyle ell }$

-form is a

${displaystyle (k+ell )}$

-form, while the exterior derivative of a

${displaystyle k}$

-form is a

${displaystyle (k+1)}$

-form. Thus, both operations generate differential forms of higher degree from those of lower degree.

The exterior product

${displaystyle wedge :Omega ^{k}(U)times Omega ^{ell }(U)to Omega ^{k+ell }(U)}$

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.

More concretely, if

${displaystyle omega =a_{i_{1}ldots i_{k}},dx^{i_{1}}wedge cdots wedge dx^{i_{k}}}$

and

${displaystyle eta =a_{j_{1}ldots i_{ell }}dx^{j_{1}}wedge cdots wedge dx^{j_{ell }}}$

, then

${displaystyle omega wedge eta =a_{i_{1}ldots i_{k}}a_{j_{1}ldots j_{ell }},dx^{i_{1}}wedge cdots wedge dx^{i_{k}}wedge dx^{j_{1}}wedge cdots wedge dx^{j_{ell }}.}$

Furthermore, for any set of indices

${displaystyle {alpha _{1}ldots ,alpha _{m}}}$

,

${displaystyle dx^{alpha _{1}}wedge cdots wedge dx^{alpha _{p}}wedge cdots wedge dx^{alpha _{q}}wedge cdots wedge dx^{alpha _{m}}=-dx^{alpha _{1}}wedge cdots wedge dx^{alpha _{q}}wedge cdots wedge dx^{alpha _{p}}wedge cdots wedge dx^{alpha _{m}}.}$

If

${displaystyle I={i_{1},ldots ,i_{k}}}$

,

${displaystyle J={j_{1},ldots ,j_{ell }}}$

, and

${displaystyle Icap J=varnothing }$

, then the indices of

${displaystyle omega wedge eta }$

can be arranged in ascending order by a (finite) sequence of such swaps. Since

${displaystyle dx^{alpha }wedge dx^{alpha }=0}$

,

${displaystyle Icap Jneq varnothing }$

implies that

${displaystyle omega wedge eta =0}$

. Finally, as a consequence of bilinearity, if

${displaystyle omega }$

and

${displaystyle eta }$

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

The collection of the exterior products of basic 1-forms

${displaystyle {dx^{i_{1}}wedge cdots wedge dx^{i_{k}}mid 1leq i_{1}$

constitutes a basis for the space of differential k-forms. Thus, any

${displaystyle omega in Omega ^{k}(U)}$

can be written in the form

${displaystyle omega =sum _{i_{1}$

where

${displaystyle a_{i_{1}ldots i_{k}}:Uto mathbb {R} }$

are smooth functions. With each set of indices

${displaystyle {i_{1},ldots ,i_{k}}}$

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

${displaystyle omega }$

.

In the previous section, the 1-form

${displaystyle df}$

was defined by taking the exterior derivative of the 0-form (continuous function)

${displaystyle f}$

. We now extend this by defining the exterior derivative operator

${displaystyle d:Omega ^{k}(U)to Omega ^{k+1}(U)}$

for

${displaystyle kgeq 1}$

. If the standard presentation of

${displaystyle k}$

-form

${displaystyle omega }$

is given by (*), the

${displaystyle (k+1)}$

-form

${displaystyle domega }$

is defined by

${displaystyle domega :=sum _{i_{1}$

A property of

${displaystyle d}$

that holds for all smooth forms is that the second exterior derivative of any

${displaystyle omega }$

vanishes identically:

${displaystyle d^{2}omega =d(domega )equiv 0}$

. This can be established directly from the definition of

${displaystyle d}$

and the equality of mixed second-order partial derivatives of

${displaystyle C^{2}}$

functions (see the article on closed and exact forms for details).

Integration of differential forms and Stokes’ theorem for chains[edit]

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

${displaystyle f:mathbb {R} ^{n}to mathbb {R} ^{m}}$

and

${displaystyle k}$

-form

${displaystyle eta in Omega ^{k}(mathbb {R} ^{m})}$

, we call

${displaystyle f^{*}eta in Omega ^{k}(mathbb {R} ^{n})}$

the pullback of

${displaystyle eta }$

by

${displaystyle f}$

and define it as the

${displaystyle k}$

-form such that

${displaystyle (f^{*}eta )_{p}(v_{1p},ldots ,v_{kp}):=eta _{f(p)}(f_{*}(v_{1p}),ldots ,f_{*}(v_{kp})),}$

for

${displaystyle v_{1p},ldots ,v_{kp}in mathbb {R} _{p}^{n}}$

, where

${displaystyle f_{*}:mathbb {R} _{p}^{n}to mathbb {R} _{f(p)}^{m}}$

is the map

${displaystyle v_{p}mapsto (Df|_{p}(v))_{f(p)}}$

.

If

${displaystyle omega =f,dx^{1}wedge cdots wedge dx^{n}}$

is an

${displaystyle n}$

-form on

${displaystyle mathbb {R} ^{n}}$

(i.e.,

${displaystyle omega in Omega ^{n}(mathbb {R} ^{n})}$

), we define its integral over the unit

${displaystyle n}$

-cell as the iterated Riemann integral of

${displaystyle f}$

:

${displaystyle int _{[0,1]^{n}}omega =int _{[0,1]^{n}}f,dx^{1}wedge cdots wedge dx^{n}:=int _{0}^{1}cdots int _{0}^{1}f,dx^{1}cdots dx^{n}.}$

Next, we consider a domain of integration parameterized by a differentiable function

${displaystyle c:[0,1]^{n}to Asubset mathbb {R} ^{m}}$

, known as an n-cube. To define the integral of

${displaystyle omega in Omega ^{n}(A)}$

over

${displaystyle c}$

, we “pull back” from

${displaystyle A}$

to the unit n-cell:

${displaystyle int _{c}omega :=int _{[0,1]^{n}}c^{*}omega .}$

To integrate over more general domains, we define an

${displaystyle {boldsymbol {n}}}$

-chain

${textstyle C=sum _{i}n_{i}c_{i}}$

as the formal sum of

${displaystyle n}$

-cubes and set

${displaystyle int _{C}omega :=sum _{i}n_{i}int _{c_{i}}omega .}$

An appropriate definition of the

${displaystyle (n-1)}$

-chain

${displaystyle partial C}$

, known as the boundary of

${displaystyle C}$

,^[8] allows us to state the celebrated Stokes’ theorem (Stokes–Cartan theorem) for chains in a subset of

${displaystyle mathbb {R} ^{m}}$

:

If

${displaystyle omega }$

is a smooth

${displaystyle (n-1)}$

-form on an open set

${displaystyle Asubset mathbb {R} ^{m}}$

and

${displaystyle C}$

is a smooth

${displaystyle n}$

-chain in

${displaystyle A}$

, then

${displaystyle int _{C}domega =int _{partial C}omega }$

.

Using more sophisticated machinery (e.g., germs and derivations), the tangent space

${displaystyle T_{p}M}$

of any smooth manifold

${displaystyle M}$

(not necessarily embedded in

${displaystyle mathbb {R} ^{m}}$

) can be defined. Analogously, a differential form

${displaystyle omega in Omega ^{k}(M)}$

on a general smooth manifold is a map

${displaystyle omega :pin Mmapsto omega _{p}in {mathcal {A}}^{k}(T_{p}M)}$

. 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 also[edit]

References[edit]