空间第二型曲线积分

一投二代三计算

设 $\Gamma :\{\begin{array}{l}x=x(t),\\ y=y(t),t:\alpha \to \beta ,\text{则有}\\ z=z(t),\end{array}$ ${\int }_{\Gamma }P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z$ $={\int }_{\alpha }^{\beta }\{P[x(t),y(t),z(t)]x^{\prime }(t)+Q[x(t),y(t),z(t)]y^{\prime }(t)+R[x(t),y(t),z(t)]z^{\prime }(t)\}\mathrm{d}t.$

斯托克斯公式

• 方向与 $\sum$ 的法向量成右手系 $\begin{array}{rl}& {\oint }_{\Gamma }P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z={\iint }_{\Sigma }\left|\begin{array}{ccc}\mathrm{d}y\mathrm{d}z& \mathrm{d}z\mathrm{d}x& \mathrm{d}x\mathrm{d}y\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right|\text{(此为第二型曲面积分形式)}\\ & ={\iint }_{\Sigma }\left|\begin{array}{ccc}\cos \alpha & \cos \beta & \cos \gamma \\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right|\mathrm{d}S\text{(此为第一型曲面积分形式 }),\end{array}$ $\text{其中}{n}^{\circ }=(\cos \alpha ,\cos \beta ,\cos \gamma )\text{为}\Sigma \text{的单位外法向量}.$