Calculus

二元函数泰勒公式

$$f(x_0+\delta x,y_0+\delta y)=\sum _{k=0}^n{(\delta x\frac{\partial }{\partial x}+\delta y\frac{\partial }{\partial y})}^{k}f(x_0,y_0)+\mathcal{O}({\delta }^{n+1})$$

高斯公式和斯托克斯公式

• Gauss

$${\iint }_{\mathrm{\Sigma }}{A}_n\mathrm{d}S={\iiint }_{\mathrm{\Omega }}\mathrm{\nabla }\cdot \overrightarrow{A}\mathrm{d}V$$$$\begin{array}{rl}& {\iint }_{\mathrm{\Sigma }}P\mathrm{d}y\mathrm{d}z+Q\mathrm{d}z\mathrm{d}x+R\mathrm{d}x\mathrm{d}y=\\ & {\iint }_{\mathrm{\Sigma }}(P\cos \alpha +Q\cos \beta +R\cos \gamma )\mathrm{d}S={\iiint }_{\mathrm{\Omega }}(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z})\mathrm{d}V\end{array}$$

• Stokes

$${\oint }_{\mathrm{\Gamma }}\overrightarrow{A}\cdot \mathrm{d}\overrightarrow{l}={\iint }_{\mathrm{\Sigma }}\mathrm{\nabla }\times \overrightarrow{A}\mathrm{d}S$$$$\begin{array}{rl}{\oint }_{\mathrm{\Gamma }}P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z& ={\iint }_{\mathrm{\Sigma }}\left|\begin{array}{ccc}\mathrm{d}y\mathrm{d}z& \mathrm{d}z\mathrm{d}x& \mathrm{d}x\mathrm{d}y\\ {\displaystyle \frac{\partial }{\partial x}}& {\displaystyle \frac{\partial }{\partial y}}& {\displaystyle \frac{\partial }{\partial z}}\\ P& Q& R\end{array}\right|\\ & ={\iint }_{\mathrm{\Sigma }}\left|\begin{array}{ccc}\cos \alpha & \cos \beta & \cos \gamma \\ {\displaystyle \frac{\partial }{\partial x}}& {\displaystyle \frac{\partial }{\partial y}}& {\displaystyle \frac{\partial }{\partial z}}\\ P& Q& R\end{array}\right|\end{array}$$

二重积分换元 & Jacobi matrix

作变换 $x=x(u,v),y=y(u,v)$

Jacobi matrix $J(u,v)$

$$J(u,v)=\frac{\partial (x,y)}{\partial (u,v)}\equiv \left[\begin{array}{cc}{\displaystyle \frac{\partial x}{\partial u}}& {\displaystyle \frac{\partial x}{\partial v}}\\ {\displaystyle \frac{\partial y}{\partial u}}& {\displaystyle \frac{\partial y}{\partial v}}\end{array}\right]$$$${\iint }_{D}f(x,y)\mathrm{d}x\mathrm{d}y={\iint }_{D^{\prime }}f[x(u,v),y(u,v)]\text{det}J(u,v)\mathrm{d}u\mathrm{d}v$$