# Multilinear form

In abstract algebra and multilinear algebra, a multilinear form on ${\displaystyle V}$ is a map of the type

${\displaystyle f:V^{k}\to K}$,

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 ${\displaystyle k}$ arguments.[1] (The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.)

A multilinear k-form on ${\displaystyle V}$ over ${\displaystyle \mathbf {R} }$ is called a (covariant) k-tensor, and the vector space of such forms is usually denoted ${\displaystyle {\mathcal {T}}_{k}(V)}$ or ${\displaystyle {\mathcal {L}}_{k}(V)}$.

## Tensor product

Given multilinear forms ${\displaystyle f\in {\mathcal {T}}_{k}(V)}$ and ${\displaystyle g\in {\mathcal {T}}_{\ell }(V)}$, a product ${\displaystyle f\otimes g\in {\mathcal {T}}_{k+\ell }(V)}$, known as the tensor product, can be defined by the property

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

for all ${\displaystyle v_{1},\ldots ,v_{k+\ell }\in V}$. The tensor product of multilinear forms is not commutative; however it is distributive: ${\displaystyle f\otimes (g_{1}+g_{2})=f\otimes g_{1}+f\otimes g_{2}}$, ${\displaystyle (f_{1}+f_{2})\otimes g=f_{1}\otimes g+f_{2}\otimes g}$ and associative: ${\displaystyle (f\otimes g)\otimes h=f\otimes (g\otimes h)}$.

If ${\displaystyle (v_{1},\ldots ,v_{n})}$ forms a basis for 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 ${\displaystyle \phi ^{i_{1}}\otimes \cdots \otimes \phi ^{i_{k}}}$, ${\displaystyle 1\leq i_{1},\ldots ,i_{k}\leq n}$ form a basis for ${\displaystyle {\mathcal {T}}_{k}(V)}$. Consequently, ${\displaystyle {\mathcal {T}}_{k}(V)}$ has dimensionality ${\displaystyle n^{k}}$.

## Examples

### Bilinear form

Main article: Bilinear forms

For ${\displaystyle n=2}$, i.e. only two variables, ${\displaystyle f}$ 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

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 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[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 ${\displaystyle \mathrm {char} (K)\neq 2}$.

An alternating multilinear k-form on ${\displaystyle V}$ over ${\displaystyle \mathbf {R} }$ is called a k-covector, and the vector space of such alternating forms, a subspace of ${\displaystyle {\mathcal {T}}_{k}(V)}$, is generally denoted ${\displaystyle {\mathcal {A}}_{k}(V)}$, ${\displaystyle \mathrm {Alt} _{k}(V)}$, or, using the notation for the isomorphic kth exterior power of ${\displaystyle V^{*}}$(the dual space of ${\displaystyle V}$), ${\textstyle \bigwedge ^{k}V^{*}}$. Note that linear functionals (multilinear 1-forms over ${\displaystyle \mathbf {R} }$) 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: ${\displaystyle {\mathcal {A}}_{0}(V)={\mathcal {T}}_{0}(V)=\mathbf {R} }$.

The determinant on ${\displaystyle n\times n}$ matrices, viewed as an ${\displaystyle n}$ argument function of the column vectors, is an important example of an alternating multilinear form.

#### Wedge product

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 (${\displaystyle \wedge }$) can be defined, so that if ${\displaystyle f\in {\mathcal {A}}_{k}(V)}$ and ${\displaystyle g\in {\mathcal {A}}_{\ell }(V)}$, ${\displaystyle f\wedge g\in {\mathcal {A}}_{k+\ell }(V)}$:

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

where the sum is taken over the set of all permutations over ${\displaystyle k+\ell }$ elements, ${\displaystyle S_{k+\ell }}$. The wedge product is distributive, associative, and anti-commutative: if ${\displaystyle f\in {\mathcal {A}}_{k}(V)}$ and ${\displaystyle g\in {\mathcal {A}}_{\ell }(V)}$ then ${\displaystyle f\wedge g=(-1)^{k\ell }g\wedge f}$.

Given a basis ${\displaystyle (v_{1},\ldots ,v_{n})}$ for ${\displaystyle V}$ and its dual ${\displaystyle (\phi ^{1},\ldots ,\phi ^{n})}$ for dual vector space ${\displaystyle V^{*}={\mathcal {A}}_{1}(V)}$, the wedge products ${\displaystyle \phi ^{i_{1}}\wedge \cdots \wedge \phi ^{i_{k}}}$, with ${\displaystyle 1\leq i_{1}<\cdots form a basis for ${\displaystyle {\mathcal {A}}_{k}(V)}$. Hence, the dimensionality of ${\displaystyle {\mathcal {A}}_{k}(V)}$ for ${\displaystyle n}$ dimensional ${\displaystyle V}$ is ${\textstyle n \choose k}$.

### Differential form

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

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 ${\displaystyle (av)_{p}=a(v_{p})}$. If ${\displaystyle (e_{1},\ldots ,e_{n})}$ are the standard basis vectors for ${\displaystyle \mathbf {R} ^{n}}$, then ${\displaystyle ((e_{1})_{p},\ldots ,(e_{n})_{p})}$ form an analogous standard basis for ${\displaystyle \mathbf {R} _{p}^{n}}$. 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). In this description, each tangent space ${\displaystyle \mathbf {R} _{p}^{n}}$ can simply be regarded as a copy of ${\displaystyle \mathbf {R} ^{n}}$ based at the point ${\displaystyle p}$ (this is the set of tangent vectors at ${\displaystyle p}$). The (disjoint) union of tangent spaces at all ${\displaystyle p\in \mathbf {R} ^{n}}$ is known as the tangent bundle and is usually denoted ${\textstyle T\mathbf {R} ^{n}:=\bigcup _{p\in \mathbf {R} ^{n}}\mathbf {R} _{p}^{n}}$.

A differential k-form on ${\displaystyle U}$ is defined as a function ${\displaystyle \omega }$ that assigns to every ${\displaystyle p\in U}$ an alternating multilinear form on the tangent space at ${\displaystyle p}$: ${\displaystyle \omega (p)=\omega _{p}\in {\mathcal {A}}_{k}(\mathbf {R} _{p}^{n})}$. In brief, 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 ${\displaystyle f\in C^{0}(U)=\Omega ^{0}(U)}$.

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 (${\displaystyle C^{\infty }}$) functions (called smooth forms). Let ${\displaystyle f:U\to \mathbf {R} }$ be a smooth function. We 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 ${\displaystyle \pi ^{i}:\mathbf {R} ^{n}\to \mathbf {R} ,\ x\mapsto x^{i}}$, where ${\displaystyle x^{i}}$ is the ith standard coordinate of ${\displaystyle x}$ (the ${\displaystyle \pi ^{i}}$ are also known as coordinate functions). We define the basic 1-forms by ${\displaystyle dx^{i}:=d\pi ^{i}}$. If the standard coordinates of ${\displaystyle v_{p}\in \mathbf {R} _{p}^{n}}$ 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.[5] Thus, as the dual of the standard basis for ${\displaystyle \mathbf {R} _{p}^{n}}$, the 1-forms ${\displaystyle dx_{p}^{1},\ldots ,dx_{p}^{n}}$ constitute a basis for ${\displaystyle {\mathcal {A}}_{1}(\mathbf {R} _{p}^{n})=(\mathbf {R} _{p}^{n})^{*}}$. 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 ${\displaystyle a_{i}:U\to \mathbf {R} }$. Furthermore, 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 from the nonrigorous manipulation of infinitesimals:

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

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 ${\displaystyle v}$ as ${\displaystyle (v^{1},\ldots ,v^{n})}$, so that ${\textstyle v=\sum _{i=1}^{n}v^{i}e_{i}}$ for standard basis vectors ${\displaystyle (e_{1},\ldots ,e_{n})}$. Also, an upper index in a denominator (e.g., ${\displaystyle \partial /\partial x^{i}}$) 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

The wedge product (${\displaystyle \wedge }$) and exterior differentiation (${\displaystyle d}$) are two fundamental operations that can be performed on differential forms. The wedge product ${\displaystyle \wedge :\Omega ^{k}(U)\times \Omega ^{\ell }(U)\to \Omega ^{k+\ell }(U)}$ 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 (${\displaystyle k\geq 1}$), so that any k-form can be written in the standard presentation as

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

for smooth ${\displaystyle a_{i_{1}\ldots i_{k}}:U\to \mathbf {R} }$. 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 ${\displaystyle \omega =a_{i_{1}\ldots i_{k}}dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}}$ and ${\displaystyle \eta =a_{j_{1}\ldots i_{\ell }}dx^{j_{1}}\wedge \cdots \wedge dx^{j_{\ell }}}$, then ${\displaystyle \omega \wedge \eta =a_{i_{1}\ldots i_{k}}a_{j_{1}\ldots j_{\ell }}dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}\wedge dx^{j_{1}}\wedge \cdots \wedge dx^{j_{\ell }}}$, with the stipulation that for any set of indices ${\displaystyle \{\alpha _{1}\ldots ,\alpha _{m}\}}$,

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

If ${\displaystyle I=\{i_{1},\ldots ,i_{k}\}}$, ${\displaystyle J=\{j_{1},\ldots ,j_{\ell }\}}$, and ${\displaystyle I\cap J=\emptyset }$, then the indices of ${\displaystyle \omega \wedge \eta }$ can be arranged in ascending order (the standard presentation) by a (finite) sequence of such swaps. If ${\displaystyle I\cap J\neq \emptyset }$, then ${\displaystyle \omega \wedge \eta =0}$. Finally, if ${\displaystyle \omega }$ and ${\displaystyle \eta }$ are the sums of several terms, the wedge product is defined so as to obey distributivity. Used previously to define 1-forms ${\displaystyle df}$ from differentiable functions, i.e., 0-forms, the exterior derivative operator ${\displaystyle d:\Omega ^{k}(U)\to \Omega ^{k+1}(U)}$ can be generalized to operate on an arbitrary k-form ${\displaystyle \omega }$ as given by (*): the ${\displaystyle (k+1)}$-form ${\displaystyle d\omega }$ is defined by

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

#### Integration of differential forms and Stokes' theorem on chains

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 of ${\displaystyle C}$, ${\displaystyle \partial C}$, as an ${\displaystyle (n-1)}$-chain (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${\displaystyle \int _{C}d\omega =\int _{\partial C}\omega }$.

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

5. ^ The Kronecker delta is usually denoted by ${\displaystyle \delta _{ij}=\delta (i,j)}$ and defined as ${\textstyle \delta :X\times X\to \{0,1\},\ (i,j)\mapsto {\begin{cases}1,&i=j\\0,&i\neq j\end{cases}}}$. Here, the notation ${\displaystyle \delta _{j}^{i}}$ is used to parallel the use of sub- and superscripts on the left-hand side.