Exterior derivative - Knowlpedia

On a smooth manifold $M$ , the exterior derivative is the unique family of $\mathbb R$ -linear maps

d:\Omega^k(M)\to \Omega^{k+1}(M)\qquad (k\ge 0)

satisfying the following axioms:

(On functions) If $f\in\Omega^0(M)=C^\infty(M)$ , then $df$ is the usual differential (a 1-form) given by $df_p(v)=v(f)$ .
(Graded Leibniz rule) For $\alpha\in\Omega^k(M)$ and $\beta\in\Omega^\ell(M)$ , $d(\alpha\wedge\beta)=d\alpha\wedge\beta + (-1)^k\,\alpha\wedge d\beta.$
(Nilpotence) $d\circ d = 0$ .

Naturality

A key naturality property is that for any smooth map $F:M\to N$ , the pullback of differential forms satisfies

F^*(d\omega)=d(F^*\omega)

for every $\omega\in\Omega^k(N)$ .

The notions of closed and exact forms are defined using $d$ , and their quotient defines the de Rham cohomology groups.

Local coordinate formula

In a coordinate chart $(x^1,\dots,x^n)$ , write a $k$ -form as

\alpha=\sum_{I} a_I\, dx^{i_1}\wedge\cdots\wedge dx^{i_k},

where $I=(i_1<\cdots<i_k)$ and $a_I$ are smooth functions. Then

d\alpha=\sum_I \sum_{j=1}^n \frac{\partial a_I}{\partial x^j}\, dx^j \wedge dx^{i_1}\wedge\cdots\wedge dx^{i_k}.

Examples

A function on $\mathbb{R}^2$ . For $f(x,y)$ , the exterior derivative is $df = \frac{\partial f}{\partial x}\,dx + \frac{\partial f}{\partial y}\,dy.$

A 1-form on $\mathbb{R}^2$ . Let $\alpha = P(x,y)\,dx + Q(x,y)\,dy$ . Then $d\alpha = \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dx\wedge dy.$

A 2-form on $\mathbb{R}^3$ . If $\omega = f(x,y,z)\,dx\wedge dy$ , then $d\omega = df\wedge dx\wedge dy = \frac{\partial f}{\partial z}\,dz\wedge dx\wedge dy,$ since the terms involving $dx\wedge dx$ and $dy\wedge dy$ vanish by alternation.