# 斯托克斯（Stokes）公式

Stokes 公式是空间曲线积分的基本定理。它的结论是空间上闭曲线的积分可以用一个曲面积分来计算，这个曲面以这个闭曲线为边界。

1，曲面的正向边界（右手法则）：我们的四个指头指向曲面边界前进的方向，那么拇指的指向就是曲面法向量的正向。

2，定理 （斯托克斯，Stokes公式）：设 $$S$$ 为有向曲面，$$L$$ 为其正向边界，$$P,Q,R$$ 在 $$S$$ 及其边界上有一阶连续偏导数，则

$\oint_LPdx+Qdy+Rdz=\iint_S\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)dydz+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)dzdx+\left(\frac{\partial Q}{\partial x}-\frac{\partial Q}{\partial z}\right)dxdy$

\begin{align*}\oint_LPdx+Qdy+Rdz&=\int_a^b\left(P,Q,R\right)\cdot\left(x'(t),y'(t),z'(t)\right)dt\\ &=\int_a^b\left[P\frac{dx}{dt}+Q\frac{dy}{dt}+R\left(\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}\right)\right]dt\\ &=\int_a^b\left[\left(P+\frac{\partial z}{\partial x}\right)\frac{dx}{dt}+\left(Q+\frac{\partial z}{\partial y}\right)\frac{dy}{dt}\right]dt\\ &=\oint_{L_1} \left(P+\frac{\partial z}{\partial x}\right)dx+\left(Q+\frac{\partial z}{\partial y}\right)dy\\ &=\iint_D\left[\frac{\partial}{\partial x}\left(Q+\frac{\partial z}{\partial y}\right)-\frac{\partial }{\partial y}\left(P+\frac{\partial z}{\partial y}\right)\right]dA\end{align*}

$\frac{\partial}{\partial x}\left(Q+\frac{\partial z}{\partial y}\right)=\frac{\partial Q}{\partial x}+\frac{\partial Q}{\partial z}\frac{\partial z}{\partial x}+\frac{\partial R}{\partial x}\frac{\partial z}{\partial y}+\frac{\partial R}{\partial z}\frac{\partial z}{\partial x}\frac{\partial z}{\partial y }+R\frac{\partial^2 z}{\partial x\partial y}$

$\frac{\partial }{\partial y}\left(P+\frac{\partial z}{\partial y}\right)=\frac{\partial P}{\partial y}+\frac{\partial P}{\partial z}\frac{\partial z}{\partial y}+\frac{\partial R}{\partial y}\frac{\partial z}{\partial x}+\frac{\partial R}{\partial z}\frac{\partial z}{\partial y}\frac{\partial z}{\partial x }+R\frac{\partial^2 z}{\partial y\partial x}$

\begin{align*}&\iint_D\left[\frac{\partial}{\partial x}\left(Q+\frac{\partial z}{\partial y}\right)-\frac{\partial }{\partial y}\left(P+\frac{\partial z}{\partial y}\right)\right]dA\\ &\qquad =\iint_D\left(\frac{\partial Q}{\partial x}+\frac{\partial Q}{\partial z}\frac{\partial z}{\partial x}+\frac{\partial R}{\partial x}\frac{\partial z}{\partial y}-\frac{\partial P}{\partial y}-\frac{\partial P}{\partial z}\frac{\partial z}{\partial x}-\frac{\partial R}{\partial y}\frac{\partial z}{\partial x}\right)dA\\ &\qquad =\iint_D\left[\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)+\left(\frac{\partial Q}{\partial z}-\frac{\partial R}{\partial y}\right)\frac{\partial z}{\partial y}+\left(\frac{\partial R}{\partial x}-\frac{\partial P}{\partial z}\right)\frac{\partial z}{\partial x}\right]dA \\ &\qquad =\iint_D\left[\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)+\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\left(-\frac{\partial z}{\partial y}\right)+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\left(-\frac{\partial z}{\partial x}\right)\right]dA \\ &\qquad =\iint_S\left[\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\right]\cdot \left(-\frac{\partial z}{\partial x},-\frac{\partial z}{\partial y}, 1\right)dxdy\\ &\qquad =\iint_S\text{curl} \vec{F}\cdot\vec{n}dS\end{align*}

\begin{align*}\oint_L\vec{F}\cdot d\vec{r}&=\iint_S\text{curl}\vec{F}\cdot\vec{n}dS\\ &=\iint_S\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)dydz+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)dzdx+\left(\frac{\partial Q}{\partial x}-\frac{\partial Q}{\partial z}\right)dxdy\end{align*}