# 外微分

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

## 定义

$d{\omega} = \sum_{i=1}^n \frac{\partial f_I}{\partial x_i} dx_i \wedge dx_I.$

（参看楔积）。

## 性质

$d(\omega \wedge \eta) = d\omega \wedge \eta+(-1)^{{\rm deg\,}\omega}(\omega \wedge d\eta)$
$d(d\omega)=0 \, \!$

d闭形式组成，而其恰当形式组成 （参看恰当微分）。

## 坐标不变公式

$d\omega(V_0,V_1,...V_k)=\sum_i(-1)^i V_i\omega(V_0,...,\hat V_i,...,V_k)$
$+\sum_{i

$d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y]).$

$\mathcal{L}_XY=[X,Y]$,

## 微积分中的外微分

### 梯度

$df = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\, dx_i.$

$df(V) = \langle \mbox{grad }f,V\rangle,$

### 旋度

$d \omega=\sum_{i,j}\frac{\partial f_i}{\partial x_j} dx_j\wedge dx_i,$

$d \omega = \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) dx \wedge dy + \left(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \right) dy \wedge dz + \left(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \right) dz \wedge dx.$

### 散度

$d \omega = \sum_{i,j,k} \frac{\partial h_{i,j}}{\partial x_k} dx_k \wedge dx_i \wedge dx_j.$

 $d \omega\,$ $= \left( \frac{\partial p}{\partial x} + \frac{\partial q}{\partial y} + \frac{\partial r}{\partial z} \right) dx \wedge dy \wedge dz$ $= \mbox{div}V\, dx \wedge dy \wedge dz,$

## 范例

$d \sigma = \left(\frac{\partial{v}}{\partial{x}} - \frac{\partial{u}}{\partial{y}}\right) dx \wedge dy$

$\nabla \times ( \nabla f ) = 0$

$\nabla \cdot ( \nabla \times \mathbf{F} ) = 0$