问题
已知，$\Gamma$ 为分段光滑的空间有向闭曲线，$\Sigma$ 是以 $\Gamma$ 为边界的分片光滑的有向曲面，$\Gamma$ 的正向与 $\Sigma$ 的侧符合右手规则，且 $P$, $Q$, $R$ 在包含曲面 $\Sigma$ 在内的一个空间区域内具有一阶连续偏导数，则，根据斯托克斯公式，$\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $?$选项
[A]. $\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma}$ $($ $\frac{\partial R}{\partial y}$ $+$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d} y$ $\mathrm{~d} z$ $+$ $($ $\frac{\partial P}{\partial z}$ $+$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{~d} z \mathrm{~d} x$ $+$ $($ $\frac{\partial Q}{\partial x}$ $+$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{~d} x \mathrm{~d} y$[B]. $\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma}$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d} y$ $\mathrm{~d} z$ $+$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{~d} z \mathrm{~d} x$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{~d} x \mathrm{~d} y$
[C]. $\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $\times$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma}$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d} y$ $\mathrm{~d} z$ $\times$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{~d} z \mathrm{~d} x$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{~d} x \mathrm{~d} y$
[D]. $\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma}$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d} y$ $\mathrm{~d} z$ $-$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{~d} z \mathrm{~d} x$ $-$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{~d} x \mathrm{~d} y$
$\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma}$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d} y$ $\mathrm{~d} z$ $+$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{~d} z \mathrm{~d} x$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{~d} x \mathrm{~d} y$ $=$ $\iint_{\Sigma}\left|\begin{array}{ccc} \mathrm{d} y \mathrm{~d} z & \mathrm{~d} z \mathrm{~d} x & \mathrm{~d} x \mathrm{~d} y \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{array}\right|$ $=$ $\iint_{\Sigma}\left|\begin{array}{ccc}\cos \alpha & \cos \beta & \cos \gamma \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R \end{array}\right| \mathrm{d} S.$
其中，$n$ $=$ $($ $\cos \alpha$, $\cos \beta$, $\cos \gamma$ $)$ 为 $\Sigma$ 的单位法向量.