# 应用斯托克斯求空间曲线积分

1，定理（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

$\oint_L\vec{F}\cdot d\vec{r}=\iint_S\text{curl}\vec{F}\cdot\vec{n}dS$

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

$\vec{n}dS=(-g_x,-g_y,1)dxdy=(2,2,1)dxdy$

$$\vec{F}=(-y^3,x^3,-z^3)$$，它的旋度

$\text{curl}\vec{F}=\nabla\times \vec{F}=\begin{vmatrix}\vec{i}&\vec{j}&\vec{k}\\ \frac{\partial}{\partial x}& \frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\ -y^3&x^3&-z^3\end{vmatrix}=3x^2+3y^2\vec{k}$

\begin{align*}\oint_L\vec{F}\cdot d\vec{r}&=\oint_L-y^3dx+x^3dy-z^3dz\\ &=\iint_S(0,0,3x^2+3y^2)\cdot(2,2,1)dxdy\\ &=\iint_{x^2+y^2\le 1}(3x^2+3y^2)dxdy\\ &=\int_0^{2\pi}\int_0^13r^2\cdot rdrd\theta\\ &=\int_0^{2\pi}\frac{3}{4}r^4\Big|_0^1d\theta=\frac{3}{4}\theta\Big|_0^{2\pi}\\ &=\frac{3\pi}{2}\end{align*}

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