积分关系公式汇总
XantC
格林公式
\[ 联系曲线积分与二重积分\colon \]
\[ \oint_LP\mathrm{d}x+Q\mathrm{d}y=\iint_D (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\mathrm{d}x\mathrm{d}y \]
高斯公式
\[ 联系曲面积分与三重积分\colon \]
\[ {\int\kern{-7pt}\int \kern{-24mu} \bigcirc}_{\Sigma}P\mathrm{d}y\mathrm{d}z+Q\mathrm{d}z\mathrm{d}x+R\mathrm{d}x\mathrm{d}y=\iiint_{\Omega}(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z})\mathrm{d}V \]
\[ {\int\kern{-7pt}\int \kern{-24mu} \bigcirc}_{\Sigma}\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{S}=\iiint_{\Omega}\text{div}\; \boldsymbol{A}\mathrm{d}V \]
通量与散度
\[ 通量是 {\int\kern{-7pt}\int \kern{-24mu} \bigcirc}_{\Sigma}\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{S}，则\text{div}\;\boldsymbol{A}为单位通量，即散度 \]
\[ \text{div}\;\boldsymbol{A}=\nabla\cdot\boldsymbol{A}=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z} \]
斯托克斯公式
\[ 联系曲线积分与曲面积分\colon \]
\[ \begin{align*} \oint_L P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z&=\iint_{\Sigma}\begin{vmatrix} \mathrm{d}y\mathrm{d}z&\mathrm{d}z\mathrm{d}x&\mathrm{d}x\mathrm{d}y\\ \displaystyle\frac{\partial}{\partial x}& \displaystyle\frac{\partial}{\partial y}& \displaystyle\frac{\partial}{\partial z}\\P&Q&R \end{vmatrix}\\ &=\iint_{\Sigma}\begin{vmatrix} \cos\alpha&\cos\beta&\cos\gamma\\ \displaystyle\frac{\partial}{\partial x}& \displaystyle\frac{\partial}{\partial y}& \displaystyle\frac{\partial}{\partial z}\\P&Q&R \end{vmatrix}\mathrm{d}S\\ \end{align*} \]
\[ \oint_{L}\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{r}=\iint_{\Sigma}\mathbf{rot}\;\boldsymbol{A}\cdot\boldsymbol{n}\mathrm{d}S \]
\[ 格林公式是斯托克斯公式在二维的弱化 \]
环流量与旋度
\[ 环流量是 \oint_{\Sigma}\boldsymbol{A}\cdot\mathrm{d}\boldsymbol{r}，则\begin{vmatrix} \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k}\\ \displaystyle\frac{\partial}{\partial x}& \displaystyle\frac{\partial}{\partial y}& \displaystyle\frac{\partial}{\partial z}\\P&Q&R \end{vmatrix}为单位环流量，即旋度 \]
\[ \mathbf{rot}\;\boldsymbol{A}=\nabla\times\boldsymbol{A}=\begin{vmatrix} \boldsymbol{i}&\boldsymbol{j}&\boldsymbol{k}\\ \displaystyle\frac{\partial}{\partial x}& \displaystyle\frac{\partial}{\partial y}& \displaystyle\frac{\partial}{\partial z}\\P&Q&R \end{vmatrix} \]
物理量
\[ 梯度\colon \mathbf{grad}\;\boldsymbol{A}= \nabla\boldsymbol{A} \]
\[ 散度\colon \text{div}\;\boldsymbol{A}=\nabla\cdot\boldsymbol{A} \]
\[ 旋度\colon \mathbf{rot}\;\boldsymbol{A}=\nabla\times\boldsymbol{A} \]
三个公式的统一
外积
\[ 为了让计算本身能够自动导出积分方向（正负），定义外积\colon \]
\[ \mathrm{d}x\wedge\mathrm{d}y=-\mathrm{d}y\wedge\mathrm{d}x \]
\[ \mathrm{d}x\wedge(\mathrm{d}y+\mathrm{d}z)=\mathrm{d}x\wedge\mathrm{d}y+\mathrm{d}x\wedge\mathrm{d}z \]
\[ (\mathrm{d}x\wedge\mathrm{d}y)\wedge\mathrm{d}z=\mathrm{d}x\wedge(\mathrm{d}y\wedge\mathrm{d}z) \]
\[ |\mathrm{d}x\wedge\mathrm{d}y|=\mathrm{d}x\mathrm{d}y \]
外微分
\[ \omega_0=f \]
\[ \omega_1=P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z \]
\[ \omega_2=A\mathrm{d}x\wedge\mathrm{d}y+B\mathrm{d}y\wedge\mathrm{d}z+C\mathrm{d}z\wedge\mathrm{d}x \]
\[ \omega_3=F\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z \]
计算案例
\[ 当做全微分计算\colon \]
\[ \begin{align*} \mathrm{d}\omega_1&=\mathrm{d}(P\mathrm{d}x+Q\mathrm{d}y)\\ &=\frac{\partial P}{\partial x}\mathrm{d}x\wedge\mathrm{d}x+\frac{\partial P}{\partial y}\mathrm{d}y\wedge\mathrm{d}x+\frac{\partial Q}{\partial x}\mathrm{d}x\wedge\mathrm{d}y+\frac{\partial Q}{\partial x}\mathrm{d}y\wedge\mathrm{d}y\\ &=(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\mathrm{d}x\wedge\mathrm{d}y \end{align*} \]
\[ \]
统一公式
\[ \int_{\partial D}\omega=\int_D \mathrm{d}\omega \]