differential form

1 Notation and Preliminaries.

Let $M$ be an $n$-dimensional differential manifold. Let $TM$ denote the manifold’s tangent bundle, $C^{\mathrm{\infty }}(M)$ the algebra of smooth functions, and $V(M)$ the Lie algebra of smooth vector fields. The directional derivative makes $C^{\mathrm{\infty }}(M)$ into a $V(M)$ module. Using local coordinates, the directional derivative operation can be expressed as

$$v(f)=v^i{\partial }_{i}f,v\in V(M),f\in C^{\mathrm{\infty }}(M).$$

2 Definitions.

Differential forms.

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

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

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

$$\alpha (fu,\mathrm{\dots },v)=\mathrm{\cdots }=\alpha (u,\mathrm{\dots },fv)=f\alpha (u,\mathrm{\dots },v)$$

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

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

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

$$\mathrm{\Omega }(M)=\underset{k=0}{\overset{n}{\oplus }}{\mathrm{\Omega }}^k(M)$$

denote the vector space of all differential forms. There is a natural operator, called the exterior product, that endows $\mathrm{\Omega }(M)$ 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 $\alpha \in {\mathrm{\Omega }}^k(M)$ a differential form. We define ${\iota }_v(\omega )$, the interior product of $v$ and $\alpha$, to be the differential $k-1$ form given by

$${\iota }_v(\alpha )(u_1,\mathrm{\dots },u_{k-1})=\alpha (v,v_1,\mathrm{\dots },v_{k-1}),v_1,\mathrm{\dots },v_{k-1}\in V(M).$$

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

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

$${\iota }_v(\alpha \wedge \beta )={\iota }_v(\alpha )\wedge \beta +{(-1)}^k\alpha \wedge {\iota }_v(\beta )$$

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

$$(\alpha \wedge \beta )(v_1,\mathrm{\dots },v_{k+\mathrm{\ell }})=\sum _{\pi }\mathrm{sgn}(\pi )\alpha (v_{{\pi }_1},\mathrm{\dots },v_{{\pi }_k})\beta (v_{{\pi }_{k+1}},\mathrm{\dots },v_{{\pi }_{k+\mathrm{\ell }}}),$$

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

Exterior derivative.

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

$d(d\alpha )$ $=0,\alpha \in {\mathrm{\Omega }}^k(M);$

${\iota }_V(df)$ $=v(f),v\in V(M),f\in C^{\mathrm{\infty }}(M);$

$d(\alpha \wedge \beta )$ $=d(\alpha )\wedge \beta +{(-1)}^k\alpha \wedge d(\beta ),\alpha \in {\mathrm{\Omega }}^k(M),\beta \in {\mathrm{\Omega }}^{\mathrm{\ell }}(M).$

3 Local coordinates.

Let $(x^1,\mathrm{\dots },x^n)$ be a system of local coordinates on $M$, and let ${\partial }_1,\mathrm{\dots },{\partial }_n$ denote the corresponding frame of coordinate vector fields. In other words,

$${\partial }_i(x^j)={\delta }_i{}^j,$$

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

$${\iota }_{{\partial }_i}(d{x}^j)={\delta }_i{}^j;$$

In other words, the 1-forms $d{x}^1,\mathrm{\dots },d{x}^n$ form the dual coframe.

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

$$v=v^i{\partial }_i,$$

where the coordinate components $v^i$ are uniquely determined as

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

Similarly, locally the $d{x}^i$ freely generate ${\mathrm{\Omega }}^1(M)$. This means that every one-form $\alpha \in {\mathrm{\Omega }}^1(M)$ takes the form

$$\alpha ={\alpha }_{i}d{x}^i,$$

where

$${\alpha }_i={\iota }_{{\partial }_i}(\alpha ).$$

More generally, locally ${\mathrm{\Omega }}^k(M)$ is a freely generated by the differential $k$-forms

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

$\alpha$ (1)

$={\displaystyle \frac{1}{k!}}{\alpha }_{i_1\mathrm{\dots }{i}_k}d{x}^{i_1}\wedge \mathrm{\dots }\wedge d{x}^{i_k},$

where

$${\alpha }_{i_1\mathrm{\dots }{i}_k}=\alpha ({\partial }_{i_1},\mathrm{\dots },{\partial }_{i_k}).$$

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

$$\alpha (u,v,\mathrm{\dots },w)={\alpha }_{i_1{i}_2\mathrm{\dots }{i}_k}{u}^{i_1}{v}^{i_2}\mathrm{\cdots }{w}^{i_k}.$$

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

${({\iota }_v(\alpha ))}_{i_1\mathrm{\dots }{i}_k}$ $=v^j{\alpha }_{j{i}_1\mathrm{\dots }{i}_k},v\in V(M),\alpha \in {\mathrm{\Omega }}^{k+1}(M);$ (2)

${(\alpha \wedge \beta )}_{i_1\mathrm{\dots }{i}_{k+\mathrm{\ell }}}$ $=\left({\displaystyle \genfrac{}{}{0pt}{}{k+\mathrm{\ell }}{k}}\right){\alpha }_{[i_1\mathrm{\dots }{i}_k}{\beta }_{i_{k+1}\mathrm{\dots }{i}_{k+\mathrm{\ell }}]},\alpha \in {\mathrm{\Omega }}^k(M),\beta \in {\mathrm{\Omega }}^{\mathrm{\ell }}(M);$ (3)

${(d\alpha )}_{i_0{i}_1\mathrm{\dots }{i}_k}$ $=(k+1){\partial }_{[i_0}{\alpha }_{i_1\mathrm{\dots }{i}_k]},\alpha \in {\mathrm{\Omega }}^k(M).$ (4)

Note that some authors prefer a different definition of the components of a differential. According to this alternate convention, a factor of $k!$ placed before the summation sign in (1), and the leading factors are removed from (3) and (4).

Title	differential form
Canonical name	DifferentialForm
Date of creation	2013-03-22 12:44:46
Last modified on	2013-03-22 12:44:46
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	28
Author	rmilson (146)
Entry type	Definition
Classification	msc 58A10
Defines	exterior derivative
Defines	1-form
Defines	exterior product
Defines	wedge product
Defines	interior product
Defines	tensorial