Multilinear form

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

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

where $V$ is a vector space over the field $K$ (and more generally, a module over a commutative ring), that is separately K-linear in each of its $k$ arguments. (The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.)

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

Tensor product

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

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

for all $v_{1},\ldots ,v_{k+\ell }\in V$. The tensor product of multilinear forms is not commutative; however it is distributive and associative:

$f\otimes (g_{1}+g_{2})=f\otimes g_{1}+f\otimes g_{2}$, $(f_{1}+f_{2})\otimes g=f_{1}\otimes g+f_{2}\otimes g$, and

$(f\otimes g)\otimes h=f\otimes (g\otimes h)$.

If $(v_{1},\ldots ,v_{n})$ forms a basis for n-dimensional vector space $V$ and $(\phi ^{1},\ldots ,\phi ^{n})$ is the corresponding dual basis for the dual space $V^{*}={\mathcal {T}}_{1}(V)$, then the products $\phi ^{i_{1}}\otimes \cdots \otimes \phi ^{i_{k}}$, with $1\leq i_{1},\ldots ,i_{k}\leq n$ form a basis for ${\mathcal {T}}_{k}(V)$. Consequently, ${\mathcal {T}}_{k}(V)$ has dimensionality $n^{k}$.

Examples

Bilinear forms

Main article: Bilinear forms

If $k=2$, $f:V\times V\to K$ 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

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

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

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

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

With the additional hypothesis that the characteristic of the field $K$ is not 2, setting $x_{p}=x_{q}=x$ implies as a corollary that $f(x_{1},\ldots ,x\ldots ,x,\ldots ,x_{k})=0$; that is, the form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors 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 $\mathrm {char} (K)\neq 2$.

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

The determinant on $n\times n$ matrices, viewed as an $n$ 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 ($\wedge$) on multicovectors can be defined, so that if $f\in {\mathcal {A}}_{k}(V)$ and $g\in {\mathcal {A}}_{\ell }(V)$, then $f\wedge g\in {\mathcal {A}}_{k+\ell }(V)$:

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

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

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

Differential forms

Main article: 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 as a modernization of 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 in these areas is the statement of the generalized Stokes' theorem, a sweeping generalization of the fundamental theorem of calculus to higher dimensions.

The synopsis below is primarily based on Spivak (1965) and Tu (2011).

Construction of differential forms

To define differential forms on open subsets $U\subset \mathbf {R} ^{n}$, we first need the notion of the tangent space of $\mathbf {R} ^{n}$at $p$, usually denoted $T_{p}\mathbf {R} ^{n}$ or $\mathbf {R} _{p}^{n}$. The vector space $\mathbf {R} _{p}^{n}$ can be defined most conveniently as the set of elements $v_{p}$ ($p\in \mathbf {R} ^{n}$) with vector addition and scalar multiplication defined by $v_{p}+w_{p}:=(v+w)_{p}$ and $a\cdot (v_{p}):=(av)_{p}$, respectively. Moreover, if $(e_{1},\ldots ,e_{n})$ is the standard basis for $\mathbf {R} ^{n}$, then $((e_{1})_{p},\ldots ,(e_{n})_{p})$ is the analogous standard basis for $\mathbf {R} _{p}^{n}$. In other words, each tangent space $\mathbf {R} _{p}^{n}$ can simply be regarded as a copy of $\mathbf {R} ^{n}$ (a set of tangent vectors) based at the point $p$. The collection (disjoint union) of tangent spaces of $\mathbf {R} ^{n}$ at all $p\in \mathbf {R} ^{n}$ is known as the tangent bundle of $\mathbf {R} ^{n}$ and is usually denoted ${\textstyle T\mathbf {R} ^{n}:=\bigcup _{p\in \mathbf {R} ^{n}}\mathbf {R} _{p}^{n}}$. While the definition given here provides a simple description of the tangent space of $\mathbf {R} ^{n}$, there are other, more sophisticated constructions that are better suited for defining the tangent spaces of smooth manifolds in general (see the page on tangent spaces for details).

A differential k-form on $U\subset \mathbf {R} ^{n}$ is defined as a function $\omega$ that assigns to every $p\in U$ a real-valued alternating multilinear form on the tangent space of $\mathbf {R} ^{n}$at $p$, usually denoted $\omega _{p}:=\omega (p)\in {\mathcal {A}}_{k}(\mathbf {R} _{p}^{n})$. The space of differential k-forms on $U$ is usually denoted $\Omega ^{k}(U)$; thus, $\omega \in \Omega ^{k}(U)$. In brief, a differential k-form is a k-covector field. By convention, a differential 0-form on $U$ is a continuous function: $f\in C^{0}(U)=\Omega ^{0}(U)$.

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 smooth differential forms constructed from infinitely differentiable ($C^{\infty }$, or smooth) functions. Let $f:\mathbf {R} ^{n}\to \mathbf {R}$ be a smooth function. We define the 1-form $df$ on $U$ for $p\in U$ and $v_{p}\in \mathbf {R} _{p}^{n}$ by $(df)_{p}(v_{p}):=Df|_{p}(v)$, where $Df|_{p}:\mathbf {R} ^{n}\to \mathbf {R}$ is the total derivative of $f$ at $p$. (Recall that the total derivative is a linear transformation.) Of particular interest are the projection maps (also known as coordinate functions) $\pi ^{i}:\mathbf {R} ^{n}\to \mathbf {R}$, defined by $x\mapsto x^{i}$, where $x^{i}$ is the ith standard coordinate of $x\in \mathbf {R} ^{n}$. The 1-forms $d\pi ^{i}$ are known as the basic 1-forms and are conventionally written as $dx^{i}$ (by conflating the function $\pi ^{i}$ and its value $x^{i}$). If the standard coordinates of $v_{p}\in \mathbf {R} _{p}^{n}$ are $(v^{1},\ldots ,v^{n})$, then application of the definition of $df$ yields $dx_{p}^{i}(v_{p})=v^{i}$, so that $dx_{p}^{i}((e_{j})_{p})=\delta _{j}^{i}$, where $\delta _{j}^{i}$ is the Kronecker delta. Thus, as the dual of the standard basis for $\mathbf {R} _{p}^{n}$, the 1-forms at $p$, $dx_{p}^{1},\ldots ,dx_{p}^{n}$, constitute a basis for ${\mathcal {A}}_{1}(\mathbf {R} _{p}^{n})=(\mathbf {R} _{p}^{n})^{*}$. As a consequence, if $\omega$ is a 1-form on $U$, then $\omega$ can be written as ${\textstyle \sum a_{i}\,dx^{i}}$ for smooth functions $a_{i}:U\to \mathbf {R}$. Furthermore, we can derive an expression for $df$ that coincides with the classical expression for a total differential:

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

[Comments on notation: In this article, we follow the convention from 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 of multivectors and multicovectors, which instead are written with upper and lower indices, respectively. For instance, we write the standard coordinates of vector $v\in \mathbf {R} ^{n}$ as $(v^{1},\ldots ,v^{n})$, so that ${\textstyle v=\sum _{i=1}^{n}v^{i}e_{i}}$ in terms of the standard basis $(e_{1},\ldots ,e_{n})$. 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. Applying these rules, the number of upper indices minus the number of lower indices in each term of a sum is conserved, both within the sum and across an equality sign, a feature that serves as a useful mnemonic device and helps to reduce computational errors.]

Basic operations on differential forms

The wedge product ($\wedge$) and exterior differentiation ($d$) are two fundamental operations that can be performed on differential forms. The wedge product $\wedge :\Omega ^{k}(U)\times \Omega ^{\ell }(U)\to \Omega ^{k+\ell }(U)$ of differential forms is defined as a special case of the wedge product of multicovectors in general (see above). As is true in general, the wedge product of differential forms is associative, distributive, and anticommutative.

More concretely, if $\omega =a_{i_{1}\ldots i_{k}}dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}$ and $\eta =a_{j_{1}\ldots i_{\ell }}dx^{j_{1}}\wedge \cdots \wedge dx^{j_{\ell }}$, then

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

with the stipulation that for any set of indices $\{\alpha _{1}\ldots ,\alpha _{m}\}$,

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

If $I=\{i_{1},\ldots ,i_{k}\}$, $J=\{j_{1},\ldots ,j_{\ell }\}$, and $I\cap J=\emptyset$, then the indices of $\omega \wedge \eta$ can be arranged in ascending order by a (finite) sequence of such swaps. If $I\cap J\neq \emptyset$, then $\omega \wedge \eta =0$. Finally, if $\omega$ and $\eta$ are the sums of several terms, the wedge product is defined so as to obey distributivity. In general, any differential k-form can be written as a sum of basic k-forms, which are, in turn, defined as the wedge product of basic 1-forms. With the indices of each term arranged in ascending order, we can write any $\omega \in \Omega ^{k}(U)$ as

$\omega =\sum _{i_{1}<\cdots <i_{k}}a_{i_{1}\ldots i_{k}}dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}$,

for smooth $a_{i_{1}\ldots i_{k}}:U\to \mathbf {R}$. This is known as the standard presentation of $\omega$. Used previously to define 1-forms $df$ from differentiable functions, i.e., 0-forms, the exterior derivative operator $d:\Omega ^{k}(U)\to \Omega ^{k+1}(U)$ can be generalized to operate on any k-form: with $\omega \in \Omega ^{k}(U)$ given in standard presentation as shown above, the $(k+1)$-form $d\omega$ is defined by

$d\omega :=\sum _{i_{1}<\ldots <i_{k}}da_{i_{1}\ldots i_{k}}\wedge dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}$.

Integration of differential forms and Stokes' theorem on chains

If $\omega =f\,dx^{1}\wedge \cdots \wedge dx^{n}$ is an n-form on $\mathbf {R} ^{n}$, we define its integral over a unit n-cell as

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

Given a differentiable function $f:\mathbf {R} ^{n}\to \mathbf {R} ^{m}$and k-form $\eta \in \Omega ^{k}(\mathbf {R} ^{m})$, we can define an induced k-form $f^{*}\eta \in \Omega ^{k}(\mathbf {R} ^{n})$, known as the pullback of $\eta$ by $f$, by

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

for $v_{1p},\ldots ,v_{kp}\in \mathbf {R} _{p}^{n}$, where $f_{*}:\mathbf {R} _{p}^{n}\to \mathbf {R} _{f(p)}^{m}$ is the map $v_{p}\mapsto (Df|_{p}(v))_{f(p)}$. If a domain of integration is parameterized by a differentiable function $c:[0,1]^{n}\to A\subset \mathbf {R} ^{m}$, known as an n-cube, we use the pullback to define the integral of $\omega$ over $c$ :

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

To integrate over more general domains, we can define an n-chain ${\textstyle C=\sum _{i}n_{i}c_{i}}$ as the formal sum of n-cubes and set

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

With an appropriate definition of $(n-1)$-chain $\partial C$, the manifold boundary of $C$ (omitted here, see Spivak (1965), pp. 98-99 for a brief discussion), we can now state the celebrated generalized Stokes' theorem (Stokes–Cartan theorem) on chains:

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

Using more sophisticated machinery (e.g., germs and derivations), the tangent space $T_{p}M$ of any smooth manifold $M$ (not necessarily embedded in euclidean space) can be defined. By analogy, a differential form $\omega \in \Omega ^{k}(M)$ on a general smooth manifold is a map $\omega :p\in M\mapsto \omega _{p}\in {\mathcal {A}}_{k}(T_{p}M)$, and Stokes' theorem can be further generalized to arbitrary smooth manifolds-with-boundary or even certain "rough" domains (see the page on Stokes' theorem for details).

See also

• Bilinear map
• Exterior algebra
• Homogeneous polynomial
• Multilinear map