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differential form

(Definition)

Notation and Preliminaries.

Let

be an

-dimensional differential manifold. Let

denote the manifold's tangent bundle, $C^\infty(M)$ the algebra of smooth functions, and

the Lie algebra of smooth vector fields. The directional derivative makes $C^\infty(M)$ into a

module. Using local coordinates, the directional derivative operation can be expressed as

$\displaystyle v(f) = v^i \partial_i f,\quad v\in V(M),\; f\in C^\infty(M).$

Definitions.

Differential forms.

Let

be a $C^\infty(M)$ module. An $\mathbb{R}$ -linear mapping $\alpha: V(M)\to A$ is said to be tensorial if it is a $C^\infty(M)$ -homomorphism, in other words, if it satisfies

$\displaystyle \alpha(f v) = f \alpha(v)$

for all for all vector fields $v\in V(M)$ and functions $f\in C^\infty(M)$ . More generally, a multilinear map $\alpha: V(M)\times\dots\times V(M)\to A$ is called tensorial if it satisfies

$\displaystyle \alpha(f u,\dots,v) = \cdots = \alpha(u,\dots, fv) = f \alpha(u,\dots, v)$

for all vector fields $u,\dots,v$ and all functions $f\in C^\infty(M)$ .

We now define a differential 1-form to be a tensorial linear mapping from to $C^\infty(M)$ . More generally, for $k=0,1,2,\ldots,$ we define a differential -form to be a tensorial multilinear, antisymmetric, mapping from $V(M)\times \cdots \times V(M)$ ( times) to $C^\infty(M)$ . Using slightly fancier language, the above amounts to saying that a -form is a section of the cotangent bundle $T^*M = \operatorname{Hom}(TM,\mathbb{R})$ , while a differential -form as a section of $\operatorname{Hom}(\Lambda^k TM,\mathbb{R})$ .

Henceforth, we let $\Omega^k(M)$ denote the $C^\infty(M)$ -module of differential -forms. In particular, a differential 0-form is the same thing as a function. Since the tangent spaces of are -dimensional vector spaces, we also have $\Omega^k(M)=0$ for . We let

$\displaystyle \Omega(M) = \bigoplus_{k=0}^n \Omega^k(M)$

denote the vector space of all differential forms. There is a natural operator, called the exterior product, that endows $\Omega(M)$ with the structure of a graded algebra. We describe this operation below.

Exterior and Interior Product.

Let $v\in V(M)$ be a vector field and $\alpha\in \Omega^k(M)$ a differential form. We define $\iota_v(\omega)$ , the interior product of

and $\alpha$ , to be the differential

form given by

$\displaystyle \iota_v(\alpha)(u_1,\dots,u_{k-1}) = \alpha(v,v_1,\dots,v_{k-1}),\quad v_1,\dots,v_{k-1}\in V(M).$

The interior product of a vector field with a 0-form is defined to be zero.

Let $\alpha\in \Omega^k(M)$ and $\beta\in\Omega^\ell(M)$ be differential forms. We define the exterior, or wedge product $\alpha\wedge\beta\in\Omega^{k+\ell}(M)$ to be the unique differential form such that

$\displaystyle \iota_v(\alpha\wedge\beta) = \iota_v(\alpha)\wedge \beta + (-1)^k \alpha \wedge \iota_v(\beta)$

for all vector fields $v\in V(M)$ . Equivalently, we could have defined

$\displaystyle (\alpha\wedge\beta)(v_1,\dots, v_{k+\ell}) = \sum_{\pi}\operatorn... ... \alpha(v_{\pi_1},\dots,v_{\pi_k}) \beta(v_{\pi_{k+1}},\dots,v_{\pi_{k+\ell}}),$

where the sum is taken over all permutations $\pi$ of $\{1,2,\dots, k+\ell\}$ such that $\pi_1< \pi_2 < \cdots \pi_k$ and $\pi_{k+1} < \cdots < \pi_{k+\ell}$ , and where $\operatorname{sgn} \pi=\pm 1$ according to whether $\pi$ is an even or odd permutation.

Exterior derivative.

The exterior derivative is a first-order differential operator $d:\Omega^*(M)\rightarrow \Omega^*(M)$ , that can be defined as the unique linear mapping satisfying

$\displaystyle d(d \alpha)$	$\displaystyle =0, \qquad \alpha\in\Omega^k(M);$
$\displaystyle \iota_V(df)$	$\displaystyle =v(f),\qquad v\in V(M),\;f\in C^\infty(M);$
$\displaystyle d(\alpha \wedge \beta)$	$\displaystyle = d(\alpha)\wedge \beta + (-1)^k \alpha\wedge d(\beta), \qquad \alpha\in\Omega^k(M),\;\beta\in \Omega^\ell(M).$

Local coordinates.

Let $(x^1,\ldots,x^n)$ be a system of local coordinates on

, and let $\partial_1,\dots,\partial_n$ denote the corresponding frame of coordinate vector fields. In other words,

$\displaystyle \partial_i(x^j) = \delta_i{}^j,$

where the right hand side is the usual Kronecker delta symbol. By the definition of the exterior derivative,

$\displaystyle \iota_{\partial_i} (dx^j) = \delta_i{}^j;$

In other words, the 1-forms $dx^1,\dots,dx^n$ form the dual coframe.

Locally, the $\partial_i$ freely generate , meaning that every vector field $v\in V(M)$ has the form

$\displaystyle v= v^i \partial_i,$

where the coordinate components

are uniquely determined as

$\displaystyle v^i=v(x^i).$

Similarly, locally the

freely generate $\Omega^1(M)$ . This means that every one-form $\alpha\in\Omega^1(M)$ takes the form

$\displaystyle \alpha=\alpha_i dx^i,$

where

$\displaystyle \alpha_i=\iota_{\partial_i} (\alpha).$

More generally, locally $\Omega^k(M)$ is a freely generated by the differential

-forms

$\displaystyle dx^{i_1}\wedge\cdots\wedge dx^{i_k},\qquad 1\leq i_1<i_2<\cdots<i_k\leq n.$

Thus, a differential form $\alpha\in \Omega^k(M)$ is given by

$\displaystyle \alpha$	$\displaystyle = \!\!\!\sum_{i_1< \ldots< i_k} \!\!\!\alpha_{i_1\ldots i_k}\, d x^{i_1} \wedge\ldots\wedge d x^ {i_k},$	(1)
	$\displaystyle = \frac{1}{k!} \; \alpha_{i_1\ldots i_k}\, d x^{i_1} \wedge\ldots\wedge d x^ {i_k},$

where

$\displaystyle \alpha_{i_1\dots i_k} = \alpha(\partial_{i_1},\dots, \partial_{i_k}).$

Consequently, for vector fields $u,v,\dots, w\in V(M)$ , we have

$\displaystyle \alpha(u,v,\dots,w) = \alpha_{i_1i_2\dots i_k} u^{i_1} v^{i_2}\cdots w^{i_k}.$

In terms of local coordinates and the skew-symmetrization index notation, the interior and exterior product, and the exterior derivative take the following expressions:

$\displaystyle (\iota_v (\alpha))_{i_1\dots i_k}$	$\displaystyle = v^j \alpha_{j i_1\dots i_k},\quad v\in V(M),\; \alpha\in\Omega^{k+1}(M);$	(2)
$\displaystyle (\alpha\wedge\beta)_{i_1\dots i_{k+\ell}}$	$\displaystyle = \binom{k+\ell}{k}\, \alpha_{[i_1\dots i_k} \beta_{i_{k+1}\dots i_{k+\ell}]},\quad \alpha\in\Omega^k(M),\; \beta\in\Omega^\ell(M);$	(3)
$\displaystyle (d\alpha)_{i_0i_1\dots i_k}$	$\displaystyle = (k+1) \,\partial_{[i_0} \alpha_{i_1\dots i_k]},\quad \alpha\in\Omega^k(M).$	(4)

Note that some authors prefer a different definition of the components of a differential. According to this alternate convention, a factor of

placed before the summation sign in (1), and the leading factors are removed from (3) and (4).

"differential form" is owned by rmilson. [ full author list (2) | owner history (3) ]

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Also defines:

exterior derivative, 1-form, exterior product, wedge product, interior product, tensorial

Attachments:: formulas for differential forms of small valence (Theorem) by rmilson
coboundary definition of exterior derivative (Definition) by rmilson

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Cross-references: factor, expressions, interior, index, terms, freely generated, components, freely generate, coframe, Kronecker delta, right hand side, coordinate, frame, differential operator, odd permutation, even, permutations, sum, exterior, graded algebra, structure, operator, vector spaces, tangent spaces, cotangent bundle, section, language, antisymmetric, linear mapping, multilinear, functions, mapping, operation, local coordinates, module, directional derivative, vector fields, smooth, Lie algebra, smooth functions, algebra, tangent bundle, differential manifold
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This is version 24 of differential form, born on 2002-06-05, modified 2006-09-25.
Object id is 3050, canonical name is DifferentialForms.
Accessed 25156 times total.

Classification:

AMS MSC:

58A10 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Differential forms)

Pending Errata and Addenda

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