multilinear form

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

: $(f\backslash otimes\; g)(v\_1,\backslash ldots,v\_k,v\_\{k+1\},\backslash ldots,\; v\_\{k+\backslash ell\})=f(v\_1,\backslash ldots,v\_k)g(v\_\{k+1\},\backslash ldots,\; v\_\{k+\backslash ell\}),$

for all $v\_1,\backslash ldots,v\_\{k+\backslash ell\}\backslash in\; V$. The tensor product of multilinear forms is not commutative; however it is bilinear and associative:

: $f\backslash otimes(ag\_1+bg\_2)=a(f\backslash otimes\; g\_1)+b(f\backslash otimes\; g\_2)$, $(af\_1+bf\_2)\backslash otimes\; g=a(f\_1\backslash otimes\; g)+b(f\_2\backslash otimes\; g),$

and

: $(f\backslash otimes\; g)\backslash otimes\; h=f\backslash otimes\; (g\backslash otimes\; h).$

If $(v\_1,\backslash ldots,\; v\_n)$ forms a basis for an $n$-dimensional vector space $V$ and $(\backslash phi^1,\backslash ldots,\backslash phi^n)$ is the corresponding dual basis for the dual space $V^*=\backslash mathcal\{T\}^1(V)$, then the products $\backslash phi^\{i\_1\}\backslash otimes\backslash cdots\backslash otimes\backslash phi^\{i\_k\}$, with $1\backslash le\; i\_1,\backslash ldots,i\_k\backslash le\; n$ form a basis for $\backslash mathcal\{T\}^k(V)$. Consequently, $\backslash mathcal\{T\}^k(V)$ has dimension $n^k$.

{{main|Bilinear form}}

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

{{main|Alternating multilinear map}}

An important class of multilinear forms are the alternating multilinear forms, which have the additional property that{{Cite book|title=An Introduction to Manifolds|url=https://archive.org/details/introductiontoma00lwtu_506|url-access=limited|last=Tu|first=Loring W.|authorlink = Loring W. Tu|publisher=Springer|year=2011|isbn=978-1-4419-7399-3|edition=2nd |pages=[https://archive.org/details/introductiontoma00lwtu_506/page/n40 22]–23}}

: $f(x\_\{\backslash sigma(1)\},\backslash ldots,\; x\_\{\backslash sigma(k)\})\; =\; \backslash sgn(\backslash sigma)f(x\_1,\backslash ldots,\; x\_k),$

where $\backslash sigma:\backslash mathbf\{N\}\_k\backslash to\backslash mathbf\{N\}\_k$ is a permutation and $\backslash sgn(\backslash sigma)$ 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., $\backslash sigma(p)=q,\backslash sigma(q)=p$ and $\backslash sigma(i)=i,\; 1\backslash le\; i\backslash le\; k,\; i\backslash neq\; p,q$):

: $f(x\_1,\backslash ldots,\; x\_p,\backslash ldots,\; x\_q,\backslash ldots,\; x\_k)\; =\; -f(x\_1,\backslash ldots,\; x\_q,\backslash ldots,\; x\_p,\backslash ldots,\; x\_k).$

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 $f(x\_1,\backslash ldots,\; x,\backslash ldots,\; x,\backslash ldots,\; x\_k)\; =\; 0$; that is, the form has a value of 0 whenever two of its arguments are equal. Note, however, that some authors{{Cite book|title=Finite-Dimensional Vector Spaces|last=Halmos|first=Paul R.|authorlink = Paul R. Halmos|publisher=Van Nostrand|year=1958|isbn=0-387-90093-4|edition=2nd |pages=50}} 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 $\backslash operatorname\{char\}(K)\backslash neq\; 2$.

An alternating multilinear $k$-form on $V$ over $\backslash R$ is called a multicovector of degree $\backslash boldsymbol\{k\}$ or $\backslash boldsymbol\{k\}$-covector, and the vector space of such alternating forms, a subspace of $\backslash mathcal\{T\}^k(V)$, is generally denoted $\backslash mathcal\{A\}^k(V)$, or, using the notation for the isomorphic kth exterior power of $V^*$(the dual space of $V$), $\backslash bigwedge^k\; V^*$.Spivak uses $\backslash Omega^k(V)$ for the space of $k$-covectors on $V$. However, this notation is more commonly reserved for the space of differential $k$-forms on $V$. In this article, we use $\backslash Omega^k(V)$ to mean the latter. Note that linear functionals (multilinear 1-forms over $\backslash R$) are trivially alternating, so that $\backslash mathcal\{A\}^1(V)=\backslash mathcal\{T\}^1(V)=V^*$, while, by convention, 0-forms are defined to be scalars: $\backslash mathcal\{A\}^0(V)=\backslash mathcal\{T\}^0(V)=\backslash R$.

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

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 ($\backslash wedge$, also known as the wedge product) of multicovectors can be defined, so that if $f\backslash in\backslash mathcal\{A\}^k(V)$ and $g\backslash in\backslash mathcal\{A\}^\backslash ell(V)$, then $f\backslash wedge\; g\backslash in\backslash mathcal\{A\}^\{k+\backslash ell\}(V)$:

: $(f\backslash wedge\; g)(v\_1,\backslash ldots,\; v\_\{k+\backslash ell\})=\backslash frac\{1\}\{k!\backslash ell!\}\backslash sum\_\{\backslash sigma\backslash in\; S\_\{k+\backslash ell\}\}\; (\backslash sgn(\backslash sigma))\; f(v\_\{\backslash sigma(1)\},\; \backslash ldots,\; v\_\{\backslash sigma(k)\})g(v\_\{\backslash sigma(k+1)\}$

,\ldots,v_{\sigma(k+\ell)}),

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

Given a basis $(v\_1,\backslash ldots,\; v\_n)$ for $V$ and dual basis $(\backslash phi^1,\backslash ldots,\backslash phi^n)$ for $V^*=\backslash mathcal\{A\}^1(V)$, the exterior products $\backslash phi^\{i\_1\}\backslash wedge\backslash cdots\backslash wedge\backslash phi^\{i\_k\}$, with n-dimensional $V$ is $\backslash tbinom\{n\}\{k\}=\backslash frac\{n!\}\{(n-k)!\backslash ,k!\}$.

{{main|Differential form}}

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){{Cite book|url=https://archive.org/details/SpivakM.CalculusOnManifoldsPerseus2006Reprint|title=Calculus on Manifolds|last=Spivak|first=Michael|authorlink = Michael Spivak|publisher=W. A. Benjamin, Inc.|year=1965|isbn=0805390219 |pages=75–146}} and Tu (2011).

To define differential forms on open subsets $U\backslash subset\backslash R^n$, we first need the notion of the tangent space of $\backslash R^n$at $p$, usually denoted $T\_p\backslash R^n$ or $\backslash R^n\_p$. The vector space $\backslash R^n\_p$ can be defined most conveniently as the set of elements $v\_p$ ($v\backslash in\backslash R^n$, with $p\backslash in\backslash R^n$ fixed) with vector addition and scalar multiplication defined by $v\_p+w\_p:=(v+w)\_p$ and $a\backslash cdot(v\_p):=(a\backslash cdot\; v)\_p$, respectively. Moreover, if $(e\_1,\backslash ldots,e\_n)$ is the standard basis for $\backslash R^n$, then $((e\_1)\_p,\backslash ldots,(e\_n)\_p)$ is the analogous standard basis for $\backslash R^n\_p$. In other words, each tangent space $\backslash R^n\_p$ can simply be regarded as a copy of $\backslash R^n$ (a set of tangent vectors) based at the point $p$. The collection (disjoint union) of tangent spaces of $\backslash R^n$ at all $p\backslash in\backslash R^n$ is known as the tangent bundle of $\backslash R^n$ and is usually denoted $T\backslash R^n:=\backslash bigcup\_\{p\backslash in\backslash R^n\}\backslash R^n\_p$. While the definition given here provides a simple description of the tangent space of $\backslash R^n$, 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 $\backslash boldsymbol\{k\}$-form on $U\backslash subset\backslash R^n$ is defined as a function $\backslash omega$ that assigns to every $p\backslash in\; U$ a $k$-covector on the tangent space of $\backslash R^n$at $p$, usually denoted $\backslash omega\_p:=\backslash omega(p)\backslash in\backslash mathcal\{A\}^k(\backslash R^n\_p)$. In brief, a differential $k$-form is a $k$-covector field. The space of $k$-forms on $U$ is usually denoted $\backslash Omega^k(U)$; thus if $\backslash omega$ is a differential $k$-form, we write $\backslash omega\backslash in\backslash Omega^k(U)$. By convention, a continuous function on $U$ is a differential 0-form: $f\backslash in\; C^0(U)=\backslash Omega^0(U)$.

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 ($C^\backslash infty$) functions. Let $f:\backslash R^n\backslash to\backslash R$ be a smooth function. We define the 1-form $df$ on $U$ for $p\backslash in\; U$ and $v\_p\backslash in\backslash R^n\_p$ by $(df)\_p(v\_p):=Df|\_p(v)$, where $Df|\_p:\backslash R^n\backslash to\backslash R$ 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) $\backslash pi^i:\backslash R^n\backslash to\backslash R$, defined by $x\backslash mapsto\; x^i$, where $x^i$ is the ith standard coordinate of $x\backslash in\backslash R^n$. The 1-forms $d\backslash pi^i$ are known as the basic 1-forms; they are conventionally denoted $dx^i$. If the standard coordinates of $v\_p\backslash in\backslash R^n\_p$ are $(v^1,\backslash ldots,\; v^n)$, then application of the definition of $df$ yields $dx^i\_p(v\_p)=v^i$, so that $dx^i\_p((e\_j)\_p)=\backslash delta\_j^i$, where $\backslash delta^i\_j$ is the Kronecker delta.The Kronecker delta is usually denoted by $\backslash delta\_\{ij\}=\backslash delta(i,j)$ and defined as . Here, the notation $\backslash delta^i\_j$ is used to conform to the tensor calculus convention on the use of upper and lower indices. Thus, as the dual of the standard basis for $\backslash R^n\_p$, $(dx^1\_p,\backslash ldots,dx^n\_p)$ forms a basis for $\backslash mathcal\{A\}^1(\backslash R^n\_p)=(\backslash R^n\_p)^*$. As a consequence, if $\backslash omega$ is a 1-form on $U$, then $\backslash omega$ can be written as $\backslash sum\; a\_i\backslash ,dx^i$ for smooth functions $a\_i:U\backslash to\backslash R$. Furthermore, we can derive an expression for $df$ that coincides with the classical expression for a total differential:

: $df=\backslash sum\_\{i=1\}^n\; D\_i\; f\backslash ;\; dx^i=\{\backslash partial\; f\backslash over\backslash partial\; x^1\}\; \backslash ,\; dx^1+\backslash cdots+\{\backslash partial\; f\backslash over\backslash partial\; x^n\}\; \backslash ,\; dx^n.$

[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. 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 $v\backslash in\backslash R^n$ as $(v^1,\backslash ldots,v^n)$, so that $v=\backslash sum\_\{i=1\}^n\; v^ie\_i$ in terms of the standard basis $(e\_1,\backslash ldots,e\_n)$. In addition, superscripts appearing in the denominator of an expression (as in $\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x^i\}$) 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.]

The exterior product ($\backslash wedge$) and exterior derivative ($d$) are two fundamental operations on differential forms. The exterior product of a $k$-form and an $\backslash ell$-form is a $(k+\backslash ell)$-form, while the exterior derivative of a $k$-form is a $(k+1)$-form. Thus, both operations generate differential forms of higher degree from those of lower degree.

The exterior product $\backslash wedge:\backslash Omega^k(U)\backslash times\backslash Omega^\backslash ell(U)\backslash to\backslash Omega^\{k+\backslash ell\}(U)$ 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 $\backslash omega=a\_\{i\_1\backslash ldots\; i\_k\}\; \backslash ,\; dx^\{i\_1\}\backslash wedge\backslash cdots\backslash wedge\; dx^\{i\_k\}$ and $\backslash eta=a\_\{j\_1\backslash ldots\; i\_\{\backslash ell\}\}\; dx^\{j\_1\}\backslash wedge\backslash cdots\backslash wedge\; dx^\{j\_\{\backslash ell\}\}$, then

: $\backslash omega\backslash wedge\backslash eta=a\_\{i\_1\backslash ldots\; i\_k\}a\_\{j\_1\backslash ldots\; j\_\backslash ell\}\; \backslash ,\; dx^\{i\_1\}\backslash wedge\backslash cdots\backslash wedge\; dx^\{i\_k\}\backslash wedge\; dx^\{j\_1\}\; \backslash wedge\; \backslash cdots\backslash wedge\; dx^\{j\_\backslash ell\}.$

Furthermore, for any set of indices $\backslash \{\backslash alpha\_1\backslash ldots,\backslash alpha\_m\backslash \}$,

: $dx^\{\backslash alpha\_1\}\; \backslash wedge\backslash cdots\backslash wedge\; dx^\{\backslash alpha\_p\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^\{\backslash alpha\_q\}\; \backslash wedge\backslash cdots\backslash wedge\; dx^\{\backslash alpha\_m\}\; =\; -dx^\{\backslash alpha\_1\}\; \backslash wedge\backslash cdots\backslash wedge\; dx^\{\backslash alpha\_q\}\; \backslash wedge\; \backslash cdots\backslash wedge\; dx^\{\backslash alpha\_p\}\backslash wedge\backslash cdots\backslash wedge\; dx^\{\backslash alpha\_m\}.$

If $I=\backslash \{i\_1,\backslash ldots,i\_k\backslash \}$, $J=\backslash \{j\_1,\backslash ldots,j\_\{\backslash ell\}\backslash \}$, and $I\backslash cap\; J=\backslash varnothing$, then the indices of $\backslash omega\backslash wedge\backslash eta$ can be arranged in ascending order by a (finite) sequence of such swaps. Since $dx^\backslash alpha\backslash wedge\; dx^\backslash alpha=0$, $I\backslash cap\; J\backslash neq\backslash varnothing$ implies that $\backslash omega\backslash wedge\backslash eta=0$. Finally, as a consequence of bilinearity, if $\backslash omega$ and $\backslash eta$ 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 constitutes a basis for the space of differential k-forms. Thus, any $\backslash omega\backslash in\backslash Omega^k(U)$ can be written in the form

:

where $a\_\{i\_1\backslash ldots\; i\_k\}:U\backslash to\backslash R$ are smooth functions. With each set of indices $\backslash \{i\_1,\backslash ldots,i\_k\backslash \}$ placed in ascending order, (*) is said to be the standard presentation of $\backslash omega$.

In the previous section, the 1-form $df$ was defined by taking the exterior derivative of the 0-form (continuous function) $f$. We now extend this by defining the exterior derivative operator $d:\backslash Omega^k(U)\backslash to\backslash Omega^\{k+1\}(U)$ for $k\backslash geq1$. If the standard presentation of $k$-form $\backslash omega$ is given by (*), the $(k+1)$-form $d\backslash omega$ is defined by

:

A property of $d$ that holds for all smooth forms is that the second exterior derivative of any $\backslash omega$ vanishes identically: $d^2\backslash omega=d(d\backslash omega)\backslash equiv\; 0$. This can be established directly from the definition of $d$ and the equality of mixed second-order partial derivatives of $C^2$ functions (see the article on closed and exact forms for details).

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 $f:\backslash R^n\backslash to\backslash R^m$ and $k$-form $\backslash eta\backslash in\backslash Omega^k(\backslash R^m)$, we call $f^*\backslash eta\backslash in\backslash Omega^k(\backslash R^n)$ the pullback of $\backslash eta$ by $f$ and define it as the $k$-form such that

: $(f^*\backslash eta)\_p(v\_\{1p\},\backslash ldots,\; v\_\{kp\}):=\backslash eta\_\{f(p)\}(f\_*(v\_\{1p\}),\backslash ldots,f\_*(v\_\{kp\})),$

for $v\_\{1p\},\backslash ldots,v\_\{kp\}\backslash in\backslash R^n\_p$, where $f\_*:\backslash R^n\_p\backslash to\backslash R^m\_\{f(p)\}$ is the map $v\_p\backslash mapsto(Df|\_p(v))\_\{f(p)\}$.

If $\backslash omega=f\backslash ,\; dx^1\backslash wedge\backslash cdots\backslash wedge\; dx^n$ is an $n$-form on $\backslash R^n$ (i.e., $\backslash omega\backslash in\backslash Omega^n(\backslash R^n)$), we define its integral over the unit $n$-cell as the iterated Riemann integral of $f$:

: $\backslash int\_\{[0,1]^n\}\; \backslash omega\; =\; \backslash int\_\{[0,1]^n\}\; f\backslash ,dx^1\backslash wedge\backslash cdots\; \backslash wedge\; dx^n:=\; \backslash int\_0^1\backslash cdots\backslash int\_0^1\; f\backslash ,\; dx^1\backslash cdots\; dx^n.$

Next, we consider a domain of integration parameterized by a differentiable function $c:[0,1]^n\backslash to\; A\backslash subset\backslash R^m$, known as an n-cube. To define the integral of $\backslash omega\backslash in\backslash Omega^n(A)$ over $c$, we "pull back" from $A$ to the unit n-cell:

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

To integrate over more general domains, we define an $\backslash boldsymbol\{n\}$-chain $C=\backslash sum\_i\; n\_ic\_i$ as the formal sum of $n$-cubes and set

: $\backslash int\_C\; \backslash omega\; :=\backslash sum\_i\; n\_i\backslash int\_\{c\_i\}\; \backslash omega.$

An appropriate definition of the $(n-1)$-chain $\backslash partial\; C$, known as the boundary of $C$,The formal definition of the boundary of a chain is somewhat involved and is omitted here (see {{harvnb|Spivak|1965|pp=98–99}} for a discussion). Intuitively, if $C$ maps to a square, then $\backslash partial\; C$ is a linear combination of functions that maps to its edges in a counterclockwise manner. The boundary of a chain is distinct from the notion of a boundary in point-set topology. allows us to state the celebrated Stokes' theorem (Stokes–Cartan theorem) for chains in a subset of $\backslash R^m$:

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

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

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Category:Multilinear algebra