Математика, страница 6

$z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$

$\nabla f(a,b)=(0,0)$

$D=f_{xx}(a,b)f_{yy}(a,b)-f_{xy}(a,b)^2; D>0, f_{xx}>0 \Rightarrow min; D>0, f_{xx}<0 \Rightarrow max; D<0 \Rightarrow saddle$

$J=\frac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix}x_u & x_v\\ y_u & y_v\end{vmatrix},\quad dA_{xy}=|J|dA_{uv}$

$\iint_D f(x,y)\,dA=\lim_{\max\Delta A_i\to0}\sum_i f(\xi_i,\eta_i)\,\Delta A_i$

$\iint_D f(x,y)\,dA=\int_a^b\int_{\alpha(x)}^{\beta(x)} f(x,y)\,dy\,dx=\int_c^d\int_{\gamma(y)}^{\delta(y)} f(x,y)\,dx\,dy$

$\iiint_G f(x,y,z)\,dV=\lim_{\max \Delta V_i\to0}\sum_i f(\xi_i,\eta_i,\zeta_i)\,\Delta V_i$

$x=r\cos\theta,\quad y=r\sin\theta,\quad dA=r\,dr\,d\theta$

$x=r\cos\theta,\quad y=r\sin\theta,\quad z=z,\quad dV=r\,dr\,d\theta\,dz$

$x=\rho\sin\varphi\cos\theta,\quad y=\rho\sin\varphi\sin\theta,\quad z=\rho\cos\varphi,\quad dV=\rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta$

$J=\frac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix}x_u & x_v\\ y_u & y_v\end{vmatrix},\qquad dA_{xy}=|J|\,dA_{uv}$

$S(D)=\iint_D 1\,dA$

$V(G)=\iiint_G 1\,dV$

$\bar x=\frac1M\iint_D x\rho(x,y)\,dA,\qquad \bar y=\frac1M\iint_D y\rho(x,y)\,dA,\qquad M=\iint_D \rho\,dA$

$\int_C f\,ds=\int_a^b f(\mathbf r(t))\,\|\mathbf r'(t)\|\,dt$

$\int_C \mathbf F\cdot d\mathbf r=\int_a^b \left(P\,x'(t)+Q\,y'(t)+R\,z'(t)\right)dt$

$\iint_S g\,dS=\iint_D g(\mathbf r(u,v))\,\|\mathbf r_u\times\mathbf r_v\|\,du\,dv$

$\Phi=\iint_S \mathbf F\cdot \mathbf n\,dS=\iint_S (P n_1+Q n_2+R n_3)\,dS$

$\nabla\cdot\mathbf F=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}$

$\nabla\times\mathbf F=\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z},\;\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x},\;\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)$

$\oint_{\partial D} P\,dx+Q\,dy=\iint_D \left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA$

$\iiint_V (\nabla\cdot\mathbf F)\,dV=\iint_{\partial V}\mathbf F\cdot\mathbf n\,dS$

$\iint_S (\nabla\times\mathbf F)\cdot \mathbf n\,dS=\oint_{\partial S}\mathbf F\cdot d\mathbf r$

$\mathbf F=\nabla\varphi \Rightarrow \int_C \mathbf F\cdot d\mathbf r=\varphi(B)-\varphi(A),\;\nabla\times\mathbf F=0\;\text{в односвязной области}$

$\sum_{n=0}^{\infty} a_n (x-a)^n$

$R=\frac{1}{\limsup_{n\to\infty}\sqrt[n]{|a_n|}},\quad (1/\limsup=\infty\text{ если }\limsup=0),\quad (1/\infty=0)$

$I=(a-R,a+R),\quad a\pm R\text{ — проверяются отдельно }$

$f(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k+R_n(x),\quad R_n(x)=\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}$

$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!},\quad x\in\mathbb R$

$\sin x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!},\quad x\in\mathbb R$

$\cos x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!},\quad x\in\mathbb R$

$\ln(1+x)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}x^n= x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots,\quad -1<x\le 1$

$(1+x)^\alpha=\sum_{n=0}^{\infty} \binom{\alpha}{n}x^n,\quad \binom{\alpha}{n}=\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!},\quad |x|<1\text{ (обычно)}$

$f(x)=\sum_{n=0}^{\infty}a_n(x-a)^n \Rightarrow f'(x)=\sum_{n=1}^{\infty} n a_n(x-a)^{n-1},\quad \int f(x)\,dx=C+\sum_{n=0}^{\infty}\frac{a_n}{n+1}(x-a)^{n+1}$

$\|x\|=\sqrt{x_1^2+x_2^2+\dots+x_n^2}$

$a\cdot b=\sum_{i=1}^{n}a_i b_i$

$\cos\varphi=\frac{a\cdot b}{\|a\|\,\|b\|}$

$(AB)_{ij}=\sum_{k=1}^{m}a_{ik}b_{kj}$

$\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$

$\det A=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}$

$A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$

$x=\frac{\Delta_x}{\Delta},\quad y=\frac{\Delta_y}{\Delta}$

$\operatorname{rank}A=\max\{r:\text{существует ненулевой минор порядка }r\}$

$\operatorname{tr}A=a_{11}+a_{22}+\dots+a_{nn}$

$p(\lambda)=\lambda^2-\operatorname{tr}(A)\lambda+\det(A)$

$Ax=b$

$\left[A\mid b\right]$

$R_i\leftrightarrow R_j,\quad R_i\leftarrow cR_i\ (c\ne0),\quad R_i\leftarrow R_i+cR_j$

$R_i\leftarrow R_i-\frac{a_{ik}}{a_{kk}}R_k$

$x_i=\frac{b'_i-\sum_{j=i+1}^{n}u_{ij}x_j}{u_{ii}}$

$p_1<p_2<\dots<p_r,\quad a_{ij}=0\ \text{ниже ведущих элементов}$

$\operatorname{rref}(A)$

$\left[A\mid b\right]\sim\left[I\mid x\right]$

$\operatorname{rank}[A\mid b]$

$\operatorname{rank}A=\operatorname{rank}[A\mid b]$

$\operatorname{rank}A<\operatorname{rank}[A\mid b]$

$\operatorname{rank}A=\operatorname{rank}[A\mid b]=n$

$\operatorname{rank}A=\operatorname{rank}[A\mid b]<n$

$k=n-\operatorname{rank}A$

$x=x_p+t_1v_1+\cdots+t_kv_k$