# differential form

## 1 Notation and Preliminaries.

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

 $v(f)=v^{i}\partial_{i}f,\quad v\in V(M),\;f\in C^{\infty}(M).$

## 2 Definitions.

### Differential forms.

Let $A$ 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

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

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

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

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

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

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 $v$ and $\alpha$, to be the differential $k-1$ form given by

 $\iota_{v}(\alpha)(u_{1},\dots,u_{k-1})=\alpha(v,v_{1},\dots,v_{k-1}),\quad v_{% 1},\dots,v_{k-1}\in V(M).$

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

 $\iota_{v}(\alpha\wedge\beta)=\iota_{v}(\alpha)\wedge\beta+(-1)^{k}\alpha\wedge% \iota_{v}(\beta)$

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

 $(\alpha\wedge\beta)(v_{1},\dots,v_{k+\ell})=\sum_{\pi}\operatorname{sgn}(\pi)% \alpha(v_{\pi_{1}},\dots,v_{\pi_{k}})\beta(v_{\pi_{k+1}},\dots,v_{\pi_{k+\ell}% }),$

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

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

## 3 Local coordinates.

Let $(x^{1},\ldots,x^{n})$ be a system of local coordinates on $M$, and let $\partial_{1},\dots,\partial_{n}$ 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,

 $\iota_{\partial_{i}}(dx^{j})=\delta_{i}{}^{j};$

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

Locally, the $\partial_{i}$ freely generate $V(M)$, 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 $dx^{i}$ freely generate $\Omega^{1}(M)$. This means that every one-form $\alpha\in\Omega^{1}(M)$ takes the form

 $\alpha=\alpha_{i}dx^{i},$

where

 $\alpha_{i}=\iota_{\partial_{i}}(\alpha).$

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

 $dx^{i_{1}}\wedge\cdots\wedge dx^{i_{k}},\qquad 1\leq i_{1}

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

 $\displaystyle\alpha$ $\displaystyle=\!\!\!\sum_{i_{1}<\ldots (1) $\displaystyle=\frac{1}{k!}\;\alpha_{i_{1}\ldots i_{k}}\,dx^{i_{1}}\wedge\ldots% \wedge dx^{i_{k}},$

where

 $\alpha_{i_{1}\dots i_{k}}=\alpha(\partial_{i_{1}},\dots,\partial_{i_{k}}).$

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

 $\alpha(u,v,\dots,w)=\alpha_{i_{1}i_{2}\dots i_{k}}u^{i_{1}}v^{i_{2}}\cdots w^{% i_{k}}.$

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

 $\displaystyle(\iota_{v}(\alpha))_{i_{1}\dots i_{k}}$ $\displaystyle=v^{j}\alpha_{ji_{1}\dots i_{k}},\quad v\in V(M),\;\alpha\in% \Omega^{k+1}(M);$ (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) $\displaystyle(d\alpha)_{i_{0}i_{1}\dots i_{k}}$ $\displaystyle=(k+1)\,\partial_{[i_{0}}\alpha_{i_{1}\dots i_{k}]},\quad\alpha% \in\Omega^{k}(M).$ (4)

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