高等数学（工本）

空间解析几何与向量代数

距离公式

$$\begin{array}{r}|P_1{P}_2|=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}\end{array}$$

加法

$$\begin{array}{r}\overrightarrow{OA}+\overrightarrow{OB}=\overrightarrow{OC}\\ \overrightarrow{OA}+\overrightarrow{AC}=\overrightarrow{OC}\\ \overrightarrow{OA}-\overrightarrow{OB}=\overrightarrow{BA}\end{array}$$

向量坐标

$$\begin{gathered} \overrightarrow{M_1{M}_2}=\{(x_2-x_1),(y_2-y_1),(z_2-z_1)\} \\ \alpha =\{a1,a2,a3\},\beta =\{b1,b2,b3\} \\ 若\alpha //\beta ，则\frac{a1}{b1}=\frac{a2}{b2}=\frac{a3}{b3} \end{gathered}$$

数量积

$$\begin{gathered} \alpha \cdot \beta =|\alpha ||\beta |\cos \phi \\ 若\alpha 与\beta 垂直，则\alpha \cdot \beta =0 \end{gathered}$$

数量积的坐标表示

$$\begin{array}{r}\alpha =\{a1,a2,a3\},\beta =\{b1,b2,b3\}\\ \alpha \cdot \beta =a_1{b}_1+a_2{b}_2+a_3{b}_3\end{array}$$

平面方程

点法式

$$\begin{array}{r}\text{点}{P}_0(x_0,y_0,z_0),\text{非零向量}n=\{A,B,C\}\\ n\cdot \overrightarrow{P_{0}P}=0\\ A(x-x_0)+B(y-y_0)+C(z-z_0)-0\end{array}$$

一般方程

$$Ax+By+Cz+D=0$$

截距式方程

$$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$$

两个平面夹角

$$\begin{array}{r}{n}_1=\{A_1,B_1,C_1\},n_2=\{A_2,B_2,C_2\}\\ \cos \theta =\frac{n_1\cdot n_2}{|n_1||n_2|}=\frac{|A_1{A}_2+B_1{B}_2+C_1{C}_2|}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\end{array}$$

点到平面距离

$$d=\frac{|A{x}_0+B{y}_0+C{z}_0+D|}{\sqrt{A^2+B^2+C^2}}$$

直线方程

对称式方程

$$\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$$

参数方程

$$\begin{array}{rl}& \frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}\\ & \{\begin{array}{l}x=x_0+lt,\\ y=y_0+mt,\\ z=z_0+nt,\end{array}\end{array}$$

一般方程

$$\{\begin{array}{l}{A}_{1}x+B_{1}y+C_{1}z+D_1=0,\\ A_{2}x+B_{2}y+C_{2}z+D_2=0,\end{array}$$

直线的夹角

$$\begin{array}{r}{v}_1=\{l_1,m_1,n_1\},v_2=\{l_2,m_2,n_2\}\\ \cos \theta =\frac{v_1\cdot v_2}{|v_1||v_2|}=\frac{|l_1{l}_2+m_1{m}_2+n_1{n}_2|}{\sqrt{l_1^2+m_1^2+n_1^2}\sqrt{l_2^2+m_2^2+n_2^2}}\end{array}$$

直线与平面夹角

$$\begin{array}{r}v=\{l,m,n\},n=\{A,B,C\}\\ \sin \theta =\frac{v\cdot n}{|v||n|}=\frac{|lA+mB+nC|}{\sqrt{l^2+m^2+n^2}\sqrt{A^2+B^2+C^2}}\end{array}$$

多元函数的微分学

偏导数

$$\begin{array}{r}\underset{\Delta x\to 0}{\lim }\frac{\Delta F}{\Delta x}=\underset{\Delta x\to 0}{\lim }\frac{f(x_0+\Delta x,y_0)-f(x_0,y_0)}{\Delta x}\end{array}$$

全微分

$$dz=f_x(x,y)dx+f_y(x,y)dy$$

复合函数偏导数

$$\frac{dz}{dx}=\frac{\partial z}{\partial u}\cdot \frac{du}{dx}+\frac{\partial z}{\partial v}\cdot \frac{dv}{dx}$$

隐函数偏导数

$$\frac{dy}{dx}=-\frac{F_x}{F_y}$$

偏导数应用

无条件极值

1. 解方程组

$$\{\begin{array}{l}{f}_x(x,y)=0\\ f_y(x,y)=0\end{array}$$

求出$f(x,y)$的全部驻点

2. 对每个驻点计算

$$A=f_{xx}(x_0,y_0),B=f_{xy}(x_0,y_0),C=f_{yy}(x_0,y_0),\Delta =B^2-AC$$

3. 判断

1. $\Delta <0$,点$(x_0,y_0)$是函数$f(x,y)$的极值点，当$A<0$时，$f(x_0,y_0)$是极大值，当$A>0$时，$f(x_0,y_0)$是极小值
2. $\Delta >0$,点$(x_0,y_0)$不是函数$f(x,y)$的极值点
3. $\Delta =0$,不能判断

条件极值

1. 构造拉格朗日函数

$$L(x,y)=f(x,y)+\lambda \phi (x,y)$$

2. 解方程组

$$\{\begin{array}{ll}\frac{\partial L}{\partial x}& =f_x(x,y)+\lambda {\phi }_x(x,y)=0\\ \frac{\partial L}{\partial y}& =f_y(x,y)+\lambda {\phi }_y(x,y)=0\\ \phi (x,y)& =0\end{array}$$

空间曲线的切线与法平

$$\begin{gathered} 空间区线L：\{\begin{array}{l}x=x(t)\\ y=y(t)\\ z=z(t)\end{array}，L上的点P_0(x_0,y_0,z_0) \\ L在P_0的切向量为：\{x^{\prime }(t_0),y^{\prime }(t_0),z^{\prime }(t_0)\} \end{gathered}$$

切线方程

$$\frac{x-x_0}{x^{\prime }(t_0)}=\frac{y-y_0}{y^{\prime }(t_0)}=\frac{z-z_0}{z^{\prime }(t_0)}$$

法平面方程

$$x^{\prime }(t_0)(x-x_0)+y^{\prime }(t_0)(y-y_0)+z^{\prime }(t_0)(z-z_0)=0$$

空间曲面的切平面与法线

$$\begin{gathered} 曲面\Sigma 的方程F(x,y,z)=0，\Sigma 上的点P_0(x_0,y_0,z_0) \\ \Sigma 在P_0上的法向量：\{F_x(x_0,y_0,z_0),F_y(x_0,y_0,z_0),F_z(x_0,y_0,z_0)\} \end{gathered}$$

法线方程

$$\frac{x-x_0}{F_x(x_0,y_0,z_0)}=\frac{y-y_0}{F_y(x_0,y_0,z_0)}=\frac{z-z_0}{F_y(x_0,y_0,z_0)}$$

切平面方程

$$F_x(x_0,y_0,z_0)(x-x_0)+F_y(x_0,y_0,z_0)(y-y_0)+F_z(x_0,y_0,z_0)(z-z_0)=0$$

方向导数

$$\frac{\partial z}{\partial l}=\frac{\partial z}{\partial x}\cos \alpha +\frac{\partial z}{\partial y}\cos \beta$$

梯度

$$gradf(x_0,y_0)=f_x(x_0,y_0)i+f_y(x_0,y_0)i=\{f_x(x_0,y_0),f_y(x_0,y_0)\}$$

重积分

二重积分性质

性质一

$${\iint }_{D}kf(x,y)d\sigma =k{\iint }_{D}f(x,y)d\sigma$$

性质二

$$\begin{gathered} {\iint }_D(f(x,y)\pm g(x,y))d\sigma ={\iint }_{D}f(x,y)d\sigma \pm {\iint }_{D}g(x,y)d\sigma \\ {\iint }_D(af(x,y)\pm bg(x,y))d\sigma =a{\iint }_{D}f(x,y)d\sigma \pm b{\iint }_{D}g(x,y)d\sigma \end{gathered}$$

性质三

$${\iint }_{D}f(x,y)d\sigma ={\iint }_{D_1}f(x,y)d\sigma +{\iint }_{D_2}f(x,y)d\sigma$$

性质四

$$\begin{gathered} 若f(x,y)\ge g(x,y) \\ 则{\iint }_{D}f(x,y)d\sigma \ge {\iint }_{D}g(x,y)d\sigma \end{gathered}$$ $$|{\iint }_{D}f(x,y)d\sigma |\le {\iint }_D|f(x,y)|d\sigma$$

性质五

$$\begin{gathered} 若m\le f(x,y)\le M \\ 则m|D|\le {\iint }_{D}f(x,y)d\sigma \le M|D|(|D|表示区域D的面积) \end{gathered}$$

性质六

$$\begin{gathered} {\iint }_{D}f(x,y)d\sigma =f(\xi ,\eta )|D| \\ 当f(x,y)=1时，有{\iint }_{D}f(x,y)d\sigma ={\iint }_{D}1d\sigma ={\iint }_{D}d\sigma =|D| \end{gathered}$$

直角坐标下的二重积分

$$\begin{array}{l}\begin{array}{lcl}D& & \text{有界闭区域,积分区域}\\ {\Delta }_{{\sigma }_i}& & \text{第i个小闭区域}\\ \lambda & & \text{所有小闭区域中直径最大的值}\\ f(x,y),g(x,y)& & \text{定义在D上的有界函数,被积函数}\\ \sigma & & \text{D的面积}\\ d\sigma ,dxdy& & \text{直角坐标系中的微元面积}\\ \alpha ,\beta & & \text{常数}\\ m,M& & \text{在D上f(x,y)的最小值和最大值}\end{array}\\ {\iint }_{D}f(x,y)d\sigma =\underset{\lambda \to 0}{\lim }\sum _{i=1}^{n}f({\xi }_i,{\eta }_i)\Delta {\sigma }_i\\ \underset{D}{\iint }f(x,y)dxdy={\int }_a^b[{\int }_{{\phi }_1(x)}^{{\phi }_2(x)}f(x,y)dy]dx\\ \underset{D}{\iint }f(x,y)dxdy={\int }_a^{b}dx{\int }_{{\phi }_1(x)}^{{\phi }_2(x)}f(x,y)dy\end{array}$$

奇偶性

设积分区域 D 关于 y 轴对称，$D=D_{\text{左}}+D_{\text{右}}$

1. 若被积函数 f(x,y)是关于 x 的奇函数，即$f(-x,y)=-f(x,y)$,则$I=\underset{D}{\iint }f(x,y)dxdy=0$
2. 若被积函数 f(x,y)是关于 x 的偶函数，即$f(-x,y)=f(-x,y)$,则$I=\underset{D}{\iint }f(x,y)dxdy=2\underset{D_{\text{左}}}{\iint }f(x,y)dxdy=2\underset{D_{\text{右}}}{\iint }f(x,y)dxdy$

极坐标下的二重积分

$$\underset{D}{\iint }f(x,y)dxdy=\underset{D}{\iint }f(r\cos \theta ,r\sin \theta )rdrd\theta ={\int }_{\alpha }^{\beta }d\theta {\int }_{{\phi }_1(\theta )}^{{\phi }_2(\theta )}f(r\cos \theta ,r\sin \theta )rdr$$

三重积分

直角坐标

$$\begin{array}{l}\begin{array}{lcl}\Omega & & \text{空间有界闭区域,积分区域}\\ \Delta {\upsilon }_i& & \text{第i个小闭区域}\\ f(x,y,z)& & \text{定义在}\Omega \text{上的有界函数,被积函数}\\ \upsilon & & \Omega \text{的体积}\\ d\upsilon ,dxdydz& & \text{直角坐标系中的微元面积}\end{array}\\ \underset{\Omega }{\iiint }f(x,y,z)d\upsilon =\underset{\lambda \to 0}{\lim }\sum _{i=1}^{n}f({\xi }_i,{\eta }_i,{\zeta }_i)\Delta \upsilon \\ \underset{\Omega }{\iiint }f(x,y,z)dxdydz={\int }_a^b({\int }_{{\phi }_1(x)}^{{\phi }_2(x)}({\int }_{z_1(x,y)}^{z_2(x,y)}f(x,y)dz)dy)dx\\ \underset{\Omega }{\iiint }f(x,y,z)dxdydz=\underset{D_{xy}}{\iint }dxdy{\int }_{z_1(x,y)}^{z_2(x,y)}f(x,y,z)dz\\ \underset{\Omega }{\iiint }f(x,y,z)dxdydz={\int }_a^{b}dx{\int }_{y_1(x)}^{y_2(x)}dy{\int }_{z_1(x,y)}^{z_2(x,y)}f(x,y,z)dz\end{array}$$

奇偶性

柱面坐标

$$\begin{array}{rl}& \{\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \\ z=z\end{array}\\ \underset{\Omega }{\iiint }f(x,y,z)dv& =\underset{\Omega }{\iiint }f(r\cos \theta ,r\sin \theta ,z)rdrd\theta dz\\ & =\underset{D}{\iint }dxdy{\int }_{z_1(x,y)}^{z_2(x,y)}f(x,y,z)dz\\ & =\underset{D}{\iint }drd\theta {\int }_{\stackrel{\sim }{z_1}(x,y)}^{\stackrel{\sim }{z_2}(x,y)}f(r\cos \theta ,r\sin \theta ,z)rdz\end{array}$$

曲线积分与曲面积分

对弧长的曲线积分

$$\{\begin{array}{l}x=\psi (t),\\ y=\phi (t)\end{array}\alpha \le t\le \beta$$ $${\int }_{L}f(x,y)ds={\int }_{\alpha }^{\beta }f(\psi (t),\phi (t))\sqrt{({\psi }^{\prime }(t){)}^2+{\phi }^{\prime }(t)}dt$$

极坐标

$$\begin{gathered} L:r=r(\theta ),{\theta }_0\le \theta \le {\theta }_1 \\ \{\begin{array}{l}x=r(\theta )\cos \theta \\ y=r(\theta )\sin \theta \end{array}({\theta }_0\le \theta \le {\theta }_1) \\ ds=\sqrt{(x^{{}^{\prime }}(\theta ){)}^2+(y^{{}^{\prime }}(\theta ){)}^2}d\theta \\ {\int }_{L}f(x,y)ds={\int }_{{\theta }_0}^{{\theta }_1}f(r(\theta )\cos \theta ,r(\theta )\sin \theta )\sqrt{r^2(\theta )+(r^{{}^{\prime }}(\theta ){)}^2}d\theta \end{gathered}$$

对坐标的曲线积分

$$\{\begin{array}{l}x=\psi (t),\\ y=\phi (t),\\ dx={\psi }^{\prime }(t)dt,\\ dy={\phi }^{\prime }(t)dt\end{array}$$ $${\int }_{L_{AB}}=P(x,y)dx+Q(x,y)dy={\int }_{\alpha }^{\beta }(P(\psi (t),\phi (t)){\psi }^{\prime }(t)+Q(\psi (t),\phi (t)){\phi }^{\prime }(t))dt$$

格林公式

$${\iint }_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy={\oint }_{L}Pdx+Qdy$$

对面积的曲面积分

$$\underset{\Sigma }{\iint }f(x,y,z)dS=\underset{D_{xy}}{\iint }f(x,y,z(x,y))\sqrt{1+z_x^2+z_y^2}dxdy$$

对坐标的曲面积分

$$\underset{\Sigma }{\iint }R(x,y,z)dxdy=\{\begin{array}{ll}+\underset{D_{xy}}{\iint }R(x,y,z(x,y))dxdy,& \text{若在}\Sigma \text{上恒有}0\le \gamma <\frac{\pi }{2}(\text{上侧})\\ -\underset{D_{xy}}{\iint }R(x,y,z(x,y))dxdy,& \text{若在}\Sigma \text{上恒有}\frac{\pi }{2}<\gamma \le \pi (\text{下侧})\\ 0,& \text{若在}\Sigma \text{上恒有}\gamma =\frac{\pi }{2}\end{array}$$

常微分方程

一阶微分方程

可分离变量的微分方程

$$\begin{array}{rl}& \frac{dy}{dx}=f(x)g(y)\\ & \frac{dy}{g(y)}=f(x)dx\end{array}$$

齐次方程

$$\begin{gathered} \frac{dy}{dx}=f(x,y) \\ \frac{dy}{dx}=\phi (\frac{y}{x}) \\ 令u=\frac{y}{x},则y=xu,\frac{dy}{dx}=u+x\frac{du}{dx} \\ u+x\frac{du}{dx}=\phi (u) \\ x\frac{du}{dx}=\phi (u)-u \end{gathered}$$

一阶线性微分方程

$$\begin{array}{rl}& \frac{dy}{dx}+P(x)y=Q(x)\\ & y=(\int Q(x)e^{\int P(x)dx}dx+C)e^{-\int P(x)dx}\end{array}$$

可降阶的二阶微分方程

y”=f(x)

求两次积分

二阶线性微分方程

齐次

$$y^{\prime \prime }=p(x)y^{\prime }=q(x)y=f(x)$$

非齐次

$$y=\overline{y}(x)+y^{\ast }(x)$$

二阶常系数线性微分方程

$$\begin{gathered} y^{\prime \prime }+p{y}^{\prime }+qy=0 \\ {r}^2+pr+q=0 \end{gathered}$$ $$\begin{array}{rl}& y=C_1{e}^{r_{1}x}+C_2{e}^{r_{2}x}\\ & y=e^{ax}(C_{1}cos\beta x+C_{2}sin\beta x)\\ & y=(C_1+C_{2}x)e^{rx}\end{array}$$

无穷级数

数项级数的概率

$$\begin{gathered} s_n=\sum _{k=1}^n{u}_k=u_1+u_2+u_3+\cdots +u_n \\ 若存在常数s使得:s=\lim s_n，则称该级数收敛，否则发散 \end{gathered}$$

数项级数的基本性质

性质一

$$\begin{gathered} \sum _{n=1}^{\infty }{u}_n和\sum _{n=1}^{\infty }C{u}_n都收敛,则 \\ \sum _{n=1}^{\infty }C{u}_n=C\sum _{n=1}^{\infty }{u}_n \end{gathered}$$

性质二

$$\begin{gathered} \sum _{n=1}^{\infty }{u}_n,\sum _{n=1}^{\infty }{v}_n分别收敛于和s,\sigma ,则 \\ \sum _{n=1}^{\infty }(u_n\pm v_n)也收敛于和s\pm \sigma \end{gathered}$$

性质三

在级数中去掉、增加或改变有限项，其敛散性不变

性质四

$$\begin{gathered} 若\sum _{n=1}^{\infty }{u}_n收敛，则对这个级数的项任意加括号所得的新级数 \\ (u_1+\cdots +u_{n_1})+(u_{n_1+1}+\cdots +u_{n_2})+\cdots +(u_{n_{k-1}+1}+\cdots +u_{n_k})+\cdots \\ 也收敛，且其和不变 \end{gathered}$$

性质五（收敛的必要条件）

$$若\sum _{n=1}^{\infty }{u}_n收敛,则必有\underset{n\to \infty }{\lim }{u}_n=0$$

常见级数

调和级数

$$\sum _{n=1}^{\infty }\frac{1}{n}发散$$

P 级数

$$\begin{gathered} \sum _{n=1}^{\infty }\frac{1}{n^p}(p>0) \\ p>1时收敛，0<p\le 1时发散 \end{gathered}$$

交错级数

$$\sum _{n=1}^{\infty }(-1{)}^{n-1}{u}_n和\sum _{n=1}^{\infty }(-1{)}^n{u}_n为交错级数，两者敛散性相同$$

绝对收敛和条件收敛

$$\begin{gathered} 若\sum _{n=1}^{\infty }|u_n|收敛，则称\sum _{n=1}^{\infty }{u}_n绝对收敛 \\ 若\sum _{n=1}^{\infty }{u}_n收敛,而\sum _{n=1}^{\infty }|u_n|发散，则称\sum _{n=1}^{\infty }{u}_n条件收敛 \end{gathered}$$

审敛法

定理 1

$$正项级数\sum _{n=1}^{\infty }{u}_n收敛的充要条件是:它的部分和数列\{s_n\}有上界$$

定理 2（比较审敛法）

$$\begin{gathered} 设\sum _{n=1}^{\infty }{u}_n和\sum _{n=1}^{\infty }{v}_n都是正项级数，且u_n\le v_n(n=1,2,\cdots ) \\ 若\sum _{n=1}^{\infty }{v}_n收敛，则\sum _{n=1}^{\infty }{u}_n收敛(大的收敛，小的也收敛) \\ 若\sum _{n=1}^{\infty }{u}_n发散，则\sum _{n=1}^{\infty }{v}_n发散(小的发散，大的也发散) \end{gathered}$$

定理 3（比较审敛法的极限形式）

$$\begin{gathered} 设\underset{n\to \infty }{\lim }\frac{u_n}{v_n}=l和\sum _{n=1}^{\infty }{v}_n都是正项级数 \\ 如果\underset{n\to \infty }{\lim }\frac{u_n}{v_n}=l(0<l<+\infty ) \\ 则\sum _{n=1}^{\infty }{u}_n和\sum _{n=1}^{\infty }{v}_n同收敛或同发散 \end{gathered}$$

定理 4【比值审敛法，达朗贝尔（D’Alembert）审敛法】

$$\begin{gathered} 有正项级数\sum _{n=1}^{\infty }{u}_n,若\underset{n\to \infty }{\lim }\frac{u_{n+1}}{u_n}=\rho \\ 则当\rho <1，该级数收敛，当\rho >1，该级数发散，当\rho =1时，不能判断 \end{gathered}$$

定理 5【根值审敛法，柯西（Cauchy）审敛法】

$$\begin{gathered} 有正项级数\sum _{n=1}^{\infty }{u}_n,若\underset{n\to \infty }{\lim }\sqrt[n]{u_n}=\rho \\ 则当\rho <1，该级数收敛，当\rho >1，该级数发散，当\rho =1时，不能判断 \end{gathered}$$

定理 6（布莱尼茨审敛法）

$$\begin{gathered} 若交错级数\sum _{n=1}^{\infty }(-1{)}^{n-1}{u}_n满足: \\ 1.数列\{u_n\}单调递减，对于一切正整数n,都有u_n\ge u_{n+1} \\ 2.\underset{n\to \infty }{\lim }{u}_n=0 \\ 则该级数收敛，且其和\le u_1 \end{gathered}$$

定理 7

$$若级数\sum _{n=1}^{\infty }{u}_n绝对收敛，则级数\sum _{n=1}^{\infty }{u}_n必收敛$$ $$\begin{array}{rl}& 1\end{array}$$

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