微分形式的外微分

castelu

　　对三维空间中各种微分形式$w^k$，可以定义它们的外微分$dw^k$：
$$dw^0=dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy+\frac{\partial F}{\partial z}dz。$$
$$dw^1=dP \wedge dx+dQ \wedge dy+dR \wedge dz。$$
$$dw^2=dP \wedge dy \wedge dz+dQ \wedge dz \wedge dx+dR \wedge dx \wedge dy。$$
$$dw^3=dF \wedge dx \wedge dy \wedge dz。$$
　　运用外积的运算律与性质，上述各式可化简为
$$dw^1=(\frac{\partial P}{\partial x}dx+\frac{\partial P}{\partial y}dy+\frac{\partial P}{\partial z}dz) \wedge dx+(\frac{\partial Q}{\partial x}dx+\frac{\partial Q}{\partial y}dy+\frac{\partial Q}{\partial z}dz) \wedge dy+(\frac{\partial R}{\partial x}dx+\frac{\partial R}{\partial y}dy+\frac{\partial R}{\partial z}dz) \wedge dz$$
$$=\frac{\partial R}{\partial y}dy \wedge dz-\frac{\partial Q}{\partial z}dy \wedge dz+\frac{\partial P}{\partial z}dz \wedge dx-\frac{\partial R}{\partial x}dz \wedge dx+\frac{\partial Q}{\partial x}dx \wedge dy-\frac{\partial P}{\partial y}dx \wedge dy$$
$$=(\frac{\partial R}{\partial y}dy-\frac{\partial Q}{\partial z})dy \wedge dz+(\frac{\partial P}{\partial z}dy-\frac{\partial R}{\partial x})dz \wedge dx+(\frac{\partial Q}{\partial x}dy-\frac{\partial P}{\partial y})dx \wedge dy。$$
　　同样可推得
$$dw^2=(\frac{\partial P}{\partial x}dy+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z})dx \wedge dy \wedge dz，$$
$$dw^3=0。$$
　　由此可见$k$（$<3$）次微分形式的外微分是$k+1$次微分形式，但三次微分形式的外微分为零。