外微分

定义

${\displaystyle d{\omega }=\sum _{i=1}^{n}{\frac {\partial f_{I}}{\partial x_{i}}}dx_{i}\wedge dx_{I}.}$

（参看楔积）。

性质

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

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

${\displaystyle d\omega (V_{0},V_{1},...V_{k})=\sum _{i}(-1)^{i}V_{i}\omega (V_{0},...,{\hat {V}}_{i},...,V_{k})}$
${\displaystyle +\sum _{i

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

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

微积分中的外微分

梯度

${\displaystyle df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\,dx_{i}.}$

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

旋度

${\displaystyle d\omega =\sum _{i,j}{\frac {\partial f_{i}}{\partial x_{j}}}dx_{j}\wedge dx_{i},}$

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

散度

${\displaystyle d\omega =\sum _{i,j,k}{\frac {\partial h_{i,j}}{\partial x_{k}}}dx_{k}\wedge dx_{i}\wedge dx_{j}.}$

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

范例

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

${\displaystyle \nabla \times (\nabla f)=0}$

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