高数公式

$$|f\left(x\right)-A|<ℇ$$$$0<|x-x_0|<\delta$$$$ℇ$$$$x_0$$$$u\left(x\right)\in D,x\in D\text{'},D\text{'}$$$$K=\frac{|x\text{'}\left(t\right)y″\left(t\right)-x″\left(t\right)y\prime \left(t\right)|}{[x{\prime }^2\left(t\right)+y{\prime }^2\left(t\right){]}^{\frac{3}{2}}}$$$$\rho =\frac{1}{K}$$$$\int {\sin }^{n}xdx=-\frac{1}{n}{\sin }^{n-1}x\cos x+\frac{n-1}{n}\int {\sin }^{n-2}xdx$$$$$$$$\int {\cos }^{n}xdx=\frac{1}{n}{\cos }^{n-1}x\sin x+\frac{n-1}{n}\int {\cos }^{n-2}xdx$$$$\int \tan xdx=-\ln \left(\cos x\right)$$$$\int {\sec }^{2}xdx=\tan x$$$$\int \sec x\tan xdx=\sec x$$$$\int \cot xdx=\ln \left(\sin x\right)$$$$\int \sec xdx=\ln \left(\sec x+\tan x\right)$$$$\int cscxdx=-\ln \left(cscx+\cot x\right)$$$$\int \arcsin xdx=x\arcsin x+\sqrt{1-x^2}$$$$\int \arccos xdx=x\arccos x-\sqrt{1-x^2}$$$$\int \arctan xdx=x\arctan x-\frac{1}{2}\ln \left(x^2+1\right)$$$$\int \mathrm{arccot}xdx=x\mathrm{arccot}x+\frac{1}{2}\ln \left(x^2+1\right)$$$$\int \frac{1}{\sqrt{1-x^2}}dx=\arcsin x$$$$\int -\frac{1}{\sqrt{1-x^2}}dx=\arccos x$$$$\int \frac{1}{1+x^2}dx=\arctan x$$$$\int -\frac{1}{1+x^2}dx=\mathrm{arccot}x$$$$\int a^{x}dx=\frac{a^x}{\ln a}$$$${\int }_0^{+\infty }{e}^{-x^2}dx=\frac{\sqrt{\pi }}{2}$$$${\int }_0^{\frac{\pi }{2}}{\sin }^{n}xdx=\frac{\left(n-1\right)‼}{n‼}{\left(\frac{\pi }{2}\right)}^{\left(n+1\right)\%2}$$$$\sec \prime x=\sec x\tan x$$$$\tan \prime x={\sec }^{2}x$$$$\cot \prime x=-cs{c}^{2}x$$$$csc\prime x=-cscx\cot x$$$$\arcsin x\prime =\frac{1}{\sqrt{1-x^2}}$$$$\arccos x\prime =-\frac{1}{\sqrt{1-x^2}}$$$$\arctan x\prime =\frac{1}{1+x^2}$$$$\mathrm{arccot}x\prime =-\frac{1}{1+x^2}$$$$a^x\prime =a^x\ln a$$$$J=dx=\left|\begin{array}{c}{F}_y{F}_z\\ G_y{G}_z\end{array}\right|$$$$\frac{dz}{dx}=\frac{\left|\begin{array}{c}{F}_x{F}_y\\ G_x{G}_y\end{array}\right|}{J}$$$$\frac{dy}{dx}=\frac{\left|\begin{array}{c}{F}_z{F}_x\\ G_z{G}_x\end{array}\right|}{J}$$$$\overrightarrow{T}=\left(x_t,y_t,z_t\right)$$$${\overrightarrow{T}}_0=\left(\cos \alpha ,\cos \beta ,\cos \gamma \right)$$$$\frac{x-x_0}{x_t}=\frac{y-y_0}{y_t}=\frac{z-z_0}{z_t}$$$$\left(x-x_0\right)·1+\left(y-y_0\right)·\frac{d{y}_0}{d{x}_0}+\left(z-z_0\right)·\frac{d{z}_0}{d{x}_0}=0$$$$\phi \left(x\right),\psi \left(x\right),\omega \left(x\right)$$$$\phi \prime \left(x\right),\psi \prime \left(x\right),\omega \prime \left(x\right)$$$$\overrightarrow{n}=\left(F_x,F_y,F_z\right)$$$$\overrightarrow{T}·\overrightarrow{n}=0$$$$\frac{x-x_0}{F_x}=\frac{y-y_0}{F_y}=\frac{z-z_0}{F_z}$$$$\left(x-x_0\right)·F_x+\left(y-y_0\right)·F_y+\left(z-z_0\right)·F_z=0$$$$\frac{\partial f}{\partial l}{|}_{\left(x_{\mathrm{0,}}{y}_0\right)}=F_x\cos \alpha +F_y\cos \beta$$$$\overrightarrow{n}$$$$\mathrm{grad}f\left(x_{\mathrm{0,}}{y}_0\right)=\frac{\partial f}{\partial n}\overrightarrow{n}$$$$\frac{\partial f}{\partial l}=F_x\cos \alpha +F_y\cos \beta =\mathrm{grad}f\left(x_{\mathrm{0,}}{y}_0\right)·e_i=\left|\mathrm{grad}f\right|\cos \alpha$$$$\cos \alpha =\frac{-F_x}{\sqrt{1+F^{2}x+F^{2}y}}$$$$\cos \gamma =\frac{1}{\sqrt{1+F^{2}x+F^{2}y}}$$$$L\left(x\right)=f\left(x\right)+\lambda g\left(x\right)$$$$\cos \beta =\frac{-F_y}{\sqrt{1+F^{2}x+F^{2}y}}$$$$\sum \left(f_2-f_1\right)·\text{Δ}\sigma$$$$\underset{D}{\iint }fd\sigma ={\int }_{x_1}^{x_2}dx{\int }_{f_{x_1}}^{f_{x_2}}fdy$$$$A\left(x_0\right)={\int }_{{\phi }_1\left(x_0\right)}^{{\phi }_2\left(x_0\right)}f\left(x\right)dy$$$$d\sigma =dxdy$$$$\underset{D}{\iint }fd\sigma ={\int }_{{\theta }_1}^{{\theta }_2}d\theta {\int }_{{\phi }_1\left(\theta \right)}^{{\phi }_2\left(\theta \right)}f\rho d\rho$$$$d\sigma =\frac{1}{2}{\left(\rho +d\rho \right)}^{2}d\theta -\frac{1}{2}{\rho }^{2}d\theta =\rho d\rho d\theta$$$$\underset{D}{\iint }{e}^{-x^2-y^2}dxdy=\pi \left(1-e^{-a^2}\right)$$$${\left(x-a\right)}^2+y^2=a^2\Rightarrow y^2=2ax-x^2$$$$x^2+y^2={\rho }^{\mathrm{2,}}x=\rho \cos \theta \Rightarrow \rho =2a\cos \theta$$$$\int \rho d\rho =\frac{1}{2}\int d{\rho }^2$$$$y=x^2$$$$\rho =\tan \theta \sec \theta$$$$\sqrt{\frac{1-t}{1+t}}=\frac{1-t}{\sqrt{1-t^2}}$$$$\underset{\Omega }{\iiint }f\left(x,f,z\right)dv=\underset{D_{xy}}{\iint }F\left(x,y\right)d\sigma$$$$\Rightarrow F\left(x,y\right)=\underset{f\left(2x\right)}{\overset{f\left(x\right)}{\int }}f\left(x,y,z\right)dz$$$$A=\underset{D_{xy}}{\iint }\sqrt{1+Z^{2}x+Z^{2}y}dxdy$$$$\stackrel{-}{x}=\frac{M_y}{M}=\frac{{\Sigma }_{m_i{x}_i}}{{\Sigma }_{m_i}}=\frac{\underset{D}{\iint }x\mu d\sigma }{\underset{D}{\iint }\mu d\sigma }$$$$\stackrel{-}{x}=\underset{D}{\iint }xd\sigma$$$$\stackrel{-}{x}=\frac{\underset{\Omega }{\iiint }x\mu dv}{\underset{\Omega }{\iiint }\mu dv}$$$$I_x=\sum {y^2}_i{m}_i=\underset{D}{\iint }{y}^2\mu d\sigma$$$$F_x={\iiint }_{\Omega }\frac{G\rho \left(x-x_0\right)dv}{r^3}$$$${\int }_{L}fds={\int }_a^b\sqrt{{x^2\prime }_{\left(t\right)}+{y^2\prime }_{\left(t\right)}}dt$$$${\oint }_{L}fds$$$$ds\Rightarrow d\rho \Rightarrow d\sigma$$$$\text{Δ}W=F·\overrightarrow{M_{i-1}{M}_i}=P\text{Δ}{x}_i+Q\text{Δ}{y}_i$$$$F=P\overrightarrow{i}+Q\overrightarrow{j}$$$$·$$$${\int }_{L}Pdx=\underset{\lambda \to 0}{\lim }\sum P\text{Δ}{x}_i$$$${\int }_{L}Pdy=\underset{\lambda \to 0}{\lim }\sum P\text{Δ}{y}_i$$$${\int }_{L}F·dr$$$$\cos \alpha =\frac{x\prime \left(t\right)}{\sqrt{x{\prime }^2\left(t\right)+y{\prime }^2\left(t\right)}}$$$${\int }_{L}Pdx+Qdy$$$${\int }_{L}Pdx+Qdy={\int }_L\left(P\cos \alpha +Q\cos \beta \right)ds$$$$\frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}$$$$\underset{D}{\iint }\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy={\oint }_{P}dx+Qdy$$$$\frac{\partial Q}{\partial x}=\frac{\partial P}{\partial y}$$$$\underset{Dxy}{\iint }fdS=\underset{Dxy}{\iint }\sqrt{1+Z^{2}x+Z^{2}y}dxdy$$$$\frac{\partial u}{\partial x}=P,\frac{\partial u}{\partial y}=Q$$$$u=\int Pdx+\phi \left(y\right)$$$$=Q\Rightarrow \phi \prime \left(y\right)\Rightarrow \phi \left(y\right)$$$$\underset{\sum }{\iint }Pdydz+Qdzdx+Rdxdy$$$$\sum ={\sum }_1+{\sum }_2$$$${\sum }^{-}=-\sum$$$$\sum$$$$\underset{\sum }{\iint }Pdydz+Qdzdx+Rdxdy=\underset{\sum }{\iint }\left(P\cos \alpha +Q\cos \beta +R\cos \gamma \right)dS$$$$\cos \gamma =\frac{1}{\sqrt{1+{Z^2}_x+{Z^2}_y}}$$$$\underset{\Omega }{\iiint }\left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\right)dv=\underset{\sum }{∯}Pdydz+Qdzdx+Rdxdy$$$$\underset{\Omega }{\iiint }u\text{Δ}vdxdydz=\underset{\sum }{∯}u\frac{\partial v}{\partial n}dS-\underset{\Omega }{\iiint }\left(\frac{\partial u\partial v}{\partial x\partial x}+\frac{\partial u\partial v}{\partial y\partial y}+\frac{\partial u\partial v}{\partial z\partial z}\right)dxdydz$$$$\underset{\sum }{\iint }\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial x}\right)dydz+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)dzdx+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy$$$${\oint }_{\tau }Pdx+Qdy+Rdz$$$$\left|\begin{array}{c}dydzdzdxdxdy\\ \frac{\partial }{\partial x}\frac{\partial }{\partial y}\frac{\partial }{\partial z}\\ PQR\end{array}\right|$$$$\frac{dy}{dx}+p\left(x\right)y=q\left(x\right);y=e^{-\int qdx}[\int q{e}^{\int pdx}+C]$$$$y\prime +p\left(x\right)y=q\left(x\right)y^{\alpha }\Rightarrow z=y^{1-\alpha }\Rightarrow \frac{dz}{dx}+\left(1-\alpha \right)pz=\left(1-\alpha \right)q$$$$y″=f\left(x,y\prime \right)::p=y\prime \Rightarrow p\prime =f\left(x,p\right)$$$$y″=f\left(y,y\prime \right)::p=y\prime \Rightarrow y″=\frac{dp}{dx}=\frac{dp}{dy}·\frac{dy}{dx}=p\frac{dp}{dy}=f\left(y,p\right)$$$$y″+py\prime +qy=0::r^2+pr+q=0$$$$r_1\ne r_2$$$$y=c_1{e}^{r_{1}x}+c_2{e}^{r_{2}x}$$$$r^1=r^2=r$$$$y=\left(c_1+c_{2}x\right)e^{rx}$$$$r_{\mathrm{1,2}}=\alpha \pm i\beta$$$$y=\left(c_i\cos \beta x+c_2\sin \beta x\right)e^{\alpha x}$$$$y″+py\prime +qy=f\left(x\right)$$$$y=Y\left(x\right)+y^{\ast }\left(x\right)$$$$f\left(x\right)=e^{\lambda x}{P}_m\left(x\right)$$$$y^{\ast }=x^k{e}^{\lambda x}{Q}_m\left(x\right)$$$$f\left(x\right)=e^{\lambda x}[P_l\left(x\right)\cos \omega x+P_n\left(x\right)\sin \omega x]$$$$y^{\ast }=x^k{e}^{\lambda x}[Q_m\left(x\right)\cos \omega x+R_m\left(x\right)\sin \omega x]$$$$\lambda \pm i\omega$$$$\underset{\Omega }{\iiint }fdv=2\underset{{\Omega }_1}{\iiint }fdv$$$$\Omega ={\Omega }_1+{\Omega }_2$$$${\Omega }_1,{\Omega }_2$$$$\frac{1}{1\pm x}=\sum _{n=0}^{\infty }{\left(\mp x\right)}^n,x\in \left(-\mathrm{1,1}\right)$$$$$$$$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}x\in R$$$$\sin x=\sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{x^{2n+1}}{\left(2n+1\right)!},x\in R$$$$\cos x=\sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{x^{2n}}{\left(2n\right)!},x\in R$$$${\left(1+x\right)}^m=\sum _{n=0}^{\infty }{C}_m^n{x}^n$$$$m=\frac{1}{\mathrm{2,}}x\in \text{[ − 1,1]}$$$$m=-\frac{1}{\mathrm{2,}}x\in \text{(−1.1]}$$$$\sqrt{1+x}=\sum _{n=0}^{\infty }\frac{\left(2n-1\right)‼}{\left(2n\right)‼}{x}^n$$$$\ln \left(1+x\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^n\frac{x^{n+1}}{n+1},x\in \text{(−1¸1]}$$$$\underset{n=0}{\overset{\infty }{\lim }}\frac{{\left(-1\right)}^{n+n+1}}{n+1}$$$$y=e^{rx}$$$$r_1,r_2$$$$1Q″+\left(2\lambda +p\right)Q\prime +\left({\lambda }^2+p\lambda +q\right)Q=P_m$$$$Q=x^k{Q}_m$$$$y″=\frac{dp}{dx}=\frac{dp}{dy}·\frac{dy}{dx}=p\frac{dp}{dy}$$$$\frac{d^{2}y}{d{x}^2}=\frac{d\frac{dy}{dx}}{dx}$$$$\cos \theta =\frac{\overrightarrow{a}·\overrightarrow{b}}{|a\parallel b|}$$$$Pr{j}_{\overrightarrow{a}}\overrightarrow{b}=\frac{\overrightarrow{a}·\overrightarrow{b}}{\left|a\right|}$$$$d=\frac{|A{x}_0+B{y}_0+C{z}_0+D|}{\sqrt{A^2+B^2+C^2}}$$$$\frac{x-x_0}{m}=\frac{y-y_0}{n}=\frac{z-z_0}{p}$$$$T-T_0=0$$$$\cos \theta =\frac{\left|s_1{s}_2\right|}{|s_1\parallel s_2|}$$$$\cos \theta =\frac{\left|n_1{n}_2\right|}{|n_1\parallel n_2|}$$$$\sin \phi =\cos \theta =\frac{\left|ns\right|}{|n\parallel s|}$$$$\frac{x-x_1}{m}=\frac{y-y_1}{n}=\frac{z-z_1}{p}$$$$\frac{M_0{M}_1\times s}{\left|s\right|}$$$$$$