# Multilinear form

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

${\displaystyle f:V^{n}\to K}$

where ${\displaystyle V}$ is a vector space over the field ${\displaystyle K}$ (and more generally, a module over a commutative ring), that is separately K-linear in each of its ${\displaystyle n}$ arguments.[1] A multilinear n-form on ${\displaystyle V}$ over ${\displaystyle \mathbf {R} }$ is called an n-tensor, and the vector space of such forms is often denoted ${\displaystyle {\mathcal {L}}^{n}(V)}$ or ${\displaystyle {\mathcal {T}}^{n}(V)}$.

## Examples

### Bilinear form

Main article: Bilinear forms

For ${\displaystyle n=2}$, i.e. only two variables, ${\displaystyle f}$ is referred to as a bilinear form.

### Alternating multilinear form

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

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

where ${\displaystyle \sigma :\mathbf {N} _{k}\to \mathbf {N} _{k}}$ is a permutation and ${\displaystyle \mathrm {sgn} (\sigma )}$ 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., ${\displaystyle \sigma (p)=q,\sigma (q)=p}$ and ${\displaystyle \sigma (i)=i,1\leq i\leq k,i\neq p,q}$):

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

With the additional hypothesis that ${\displaystyle \mathrm {char} (K)\neq 2}$, setting ${\displaystyle x_{p}=x_{q}=x}$ affords the corollary that ${\displaystyle f(x_{1},\ldots ,x\ldots ,x,\ldots ,x_{k})=0}$; that is, an alternating form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors[3] use this last property as the definition of alternating. This definition implies the one given in this article, but as noted above, the converse is true only when the characteristic of the field ${\displaystyle K}$ is not 2.

An alternating multilinear n-form on V over ${\displaystyle \mathbf {R} }$ is called an n-covector, and the vector space of such alternating forms is generally denoted ${\displaystyle {\mathcal {A}}^{n}(V)}$, ${\displaystyle \mathrm {Alt} ^{n}(V)}$, or using the notation for the isomorphic nth exterior power of ${\displaystyle V^{*}}$(the dual space of ${\displaystyle V}$), ${\textstyle \bigwedge ^{n}V^{*}}$.

### Differential form

Main article: Differential forms

Differential forms are mathematical objects rigorously defined via tangent spaces and multilinear forms that behave, in many ways, like differentials in a classical sense, founded on ill-defined notions of infinitesimal quantities. Furthermore, they turn out to have the correct transformation properties to allow them be integrated on curves, surfaces, and their higher-dimensional analogues (manifolds) in multidimensional space. The brief outline below mainly follows Spivak (1965).[4]

#### Construction of differential forms, basic properties and operations

To define differential forms on open subsets ${\displaystyle U\subset \mathbf {R} ^{n}}$, we need the notion of the tangent space ${\displaystyle T_{p}\mathbf {R} ^{n}(\equiv \mathbf {R} _{p}^{n})}$ of ${\displaystyle \mathbf {R} ^{n}}$ at ${\displaystyle p}$. The tangent space ${\displaystyle \mathbf {R} _{p}^{n}}$ can be defined most conveniently as the vector space of elements ${\displaystyle v_{p}}$ (${\displaystyle p\in \mathbf {R} ^{n}}$) with addition and scalar multiplication defined by ${\displaystyle (v+w)_{p}=v_{p}+w_{p}}$ and ${\displaystyle (av)_{p}=a(v_{p})}$. (This is the simplest description of the tangent space of ${\displaystyle \mathbf {R} ^{n}}$; other more sophisticated constructions are better suited for generalization to arbitrary smooth manifolds. See the page on tangent spaces for details.)

A differential k-form on ${\displaystyle U}$ is then defined to be a function ${\displaystyle \omega }$ that assigns to every ${\displaystyle p\in U}$ an alternating multilinear form ${\displaystyle \omega (p)=\omega _{p}\in {\mathcal {A}}^{k}(\mathbf {R} _{p}^{n})}$; in other words, a differential k-form is a k-covector field. The space of differential k-forms on ${\displaystyle U}$ is usually denoted ${\displaystyle \Omega ^{k}(U)}$, so that ${\displaystyle \omega \in \Omega ^{k}(U)}$. By convention, a differential 0-form on ${\displaystyle U}$ is a continuous function ${\displaystyle f\in C^{0}(U)=\Omega ^{0}(U)}$.

To simplify the discussion below, we will only consider differential forms constructed from smooth (${\displaystyle C^{\infty }}$) functions (called smooth forms). To construct a differential 1-form, let ${\displaystyle f:U\to \mathbf {R} }$ be a smooth function, and define the 1-form ${\displaystyle df}$ by ${\displaystyle (df)_{p}(v_{p}):=Df|_{p}(v)}$, where ${\displaystyle Df|_{p}:U\to \mathbf {R} }$ 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 ${\displaystyle \pi ^{i}:\mathbf {R} ^{n}\to \mathbf {R} ,(x^{1},\ldots ,x^{i},\ldots ,x^{n})\mapsto x^{i}}$. We define the 1-forms ${\displaystyle dx^{i}:=d\pi ^{i}}$. If ${\displaystyle v=(v^{1},\ldots ,v^{i},\ldots ,v^{n})}$, 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 ${\textstyle \delta _{j}^{i}={\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}}$. Thus, ${\displaystyle (dx_{p}^{1},\ldots ,dx_{p}^{n})}$ is the basis for ${\displaystyle {\mathcal {A}}^{1}(\mathbf {R} _{p}^{n})=(\mathbf {R} _{p}^{n})^{*}}$ and is the dual for the standard basis ${\displaystyle ((e_{1})_{p},\ldots ,(e_{n})_{p})}$ of ${\displaystyle \mathbf {R} _{p}^{n}}$. As a consequence, if ${\displaystyle \omega }$ is a 1-form on ${\displaystyle U}$, ${\displaystyle \omega }$ can be written as ${\textstyle \sum a_{i}\,dx^{i}}$ for smooth functions ${\displaystyle a_{i}:U\to \mathbf {R} }$. Further, we can obtain an expression for ${\displaystyle df}$ in terms of the ${\displaystyle dx^{i}}$ which matches the classical expression for a total differential derived non-rigorously by manipulation of infinitesimals:

${\displaystyle df={\partial f \over \partial x^{1}}dx^{1}+\cdots +{\partial f \over \partial x^{n}}dx^{n}}$.

The wedge product ${\displaystyle \wedge :\Omega ^{k}(U)\times \Omega ^{l}(U)\to \Omega ^{k+l}(U)}$ allows for the definition of higher differential k-forms (${\displaystyle k\geq 1}$), so that any k-form can be written as

${\displaystyle \omega =\sum _{i_{1}<\cdots (*),

for smooth ${\displaystyle a_{i_{1}\ldots i_{k}}:U\to \mathbf {R} }$. The wedge product is associative, distributive, and anti-commutative. Let ${\displaystyle \omega =a_{i_{1}\ldots i_{k}}dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}}}$ and ${\displaystyle \eta =a_{j_{1}\ldots i_{l}}dx_{i_{1}}\wedge \cdots \wedge dx_{i_{l}}}$, then ${\displaystyle \omega \wedge \eta =a_{i_{1}\ldots i_{k}}a_{j_{1}\ldots j_{l}}dx_{i_{1}}\wedge \cdots \wedge dx_{i_{k}}\wedge dx_{j_{1}}\wedge \cdots \wedge dx_{j_{l}}}$with the stipulation that

${\displaystyle dx_{i_{1}}\wedge \cdots \wedge dx_{i_{p}}\cdots \wedge \cdots dx_{i_{q}}\wedge \cdots \wedge dx_{i_{k}}=-dx_{i_{1}}\wedge \cdots \wedge dx_{i_{q}}\cdots \wedge \cdots dx_{i_{p}}\wedge \cdots \wedge dx_{i_{k}}}$.

If ${\displaystyle \omega }$ and ${\displaystyle \eta }$ are the sum of several terms, the wedge product is defined so as to obey distributivity. For ${\displaystyle \omega \in \Omega ^{k}(U)}$ as defined in (*), a generalized definition can also be given for the exterior derivative operator ${\displaystyle d:\Omega ^{k}(U)\to \Omega ^{k+1}(U)}$. The form ${\displaystyle d\omega \in \Omega ^{k+1}(U)}$ is given by

${\displaystyle d\omega =\sum _{i_{1}<\ldots .

#### Integration of differential forms

If ${\displaystyle \omega =f\,dx^{1}\wedge \cdots \wedge dx^{n}}$ is a top-form (i.e., an n-form in ${\displaystyle \mathbf {R} ^{n}}$), we define

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

For a differentiable function ${\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{m}}$, ${\displaystyle f^{*}\omega \in \Omega ^{k}(\mathbf {R} ^{n})}$ denotes the pullback of the differential form ${\displaystyle \omega \in \Omega ^{k}(\mathbf {R} ^{m})}$ by ${\displaystyle f}$, defined by

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

for ${\displaystyle v_{1p},\ldots ,v_{kp}\in \mathbf {R} _{p}^{n}}$, where ${\displaystyle f_{*}:\mathbf {R} _{p}^{n}\to \mathbf {R} _{f(p)}^{m}}$ is the map ${\displaystyle v_{p}\mapsto (Df|_{p}(v))_{f(p)}}$. For a continuous function ${\displaystyle c:[0,1]^{n}\to A\subset \mathbf {R} ^{m}}$ (known as an n-cube), we can then define

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

To integrate over more general surfaces, we consider the formal sum of n-cubes into an n-chain ${\displaystyle C=\sum _{i}n_{i}c_{i}}$ and define

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

With an appropriate definition of the boundary ${\displaystyle \partial C}$ of ${\displaystyle C}$ (see Spivak, 1965, p. 98-99), we can now state the famous generalized Stokes' theorem on chains:

If ${\displaystyle \omega }$ is a smooth ${\displaystyle (n-1)}$-form on an open set ${\displaystyle A\subset \mathbf {R} ^{m}}$ and ${\displaystyle C}$ is a smooth ${\displaystyle n}$-chain in ${\displaystyle A}$, then${\displaystyle \int _{C}d\omega =\int _{\partial C}\omega }$.

Finally we note that with more sophisticated machinery, the tangent space ${\displaystyle T_{p}M}$ of any smooth manifold ${\displaystyle M}$ can be defined. By analogy, a differential form ${\displaystyle \omega \in \Omega ^{k}(M)}$ on a general smooth manifold is then a map ${\displaystyle \omega :p\in M\mapsto \omega _{p}\in {\mathcal {A}}^{k}(T_{p}M)}$, and Stokes' theorem can be further generalized to arbitrary smooth manifolds-with-boundary.