多元函数预备知识

2022.07.02

向量代数

• 数量积

  $$\begin{array}{c}\mathit{a}\cdot \mathit{b}=(a_1,a_2,a_3)\cdot (b_1,b_2,b_3)=a_1{b}_1+a_2{b}_2+a_3{b}_3\\ \mathit{a}\cdot \mathit{b}=|\mathit{a}||\mathit{b}|\cdot \cos \theta \to \cos \theta =\frac{\mathit{a}\cdot \mathit{b}}{|\mathit{a}||\mathit{b}|}=\frac{a_1{b}_1+a_2{b}_2+a_3{b}_3}{\sqrt{a_1^2+a_2^2+a_3^2}\cdot \sqrt{b_1^2+b_2^2+b_3^2}}\\ \mathit{a}\perp \mathit{b}\iff |\mathit{a}||\mathit{b}|\cdot \cos \theta =0\iff \theta =\frac{\pi }{2}\\ Pr{j}_b\mathit{a}=\frac{\mathit{a}\cdot \mathit{b}}{|\mathit{a}|},a\text{在}b\text{上}\text{的}\text{投}\text{影}\end{array}$$

• 向量积

  $$\begin{array}{c}\mathit{a}\times \mathit{b}=\left|\begin{array}{ccc}\mathit{i}& \mathit{j}& \mathit{k}\\ a_i& a_j& a_k\\ b_i& b_j& b_k\end{array}\right|\\ |\mathit{a}\times \mathit{b}|=|\mathit{a}|\cdot |\mathit{b}|\cdot \sin \theta \\ a\parallel b\iff \theta =0\text{或}\pi \end{array}$$

• 混合积

  $$\begin{array}{c}[\mathit{a}\mathit{b}\mathit{c}]=(\mathit{a}\times \mathit{b})\cdot \mathit{c}=\left|\begin{array}{ccc}{a}_i& a_j& a_k\\ b_i& b_j& b_k\\ c_i& c_j& c_k\end{array}\right|\\ \left|\begin{array}{ccc}{a}_i& a_j& a_k\\ b_i& b_j& b_k\\ c_i& c_j& c_k\end{array}\right|=0\iff \text{三}\text{向}\text{量}\text{共}\text{面}\end{array}$$

• 方向角和方向余弦：方向角—向量与x，y，z轴正向的夹角

  $$\begin{array}{c}\text{方}\text{向}\text{余}\text{弦}\\ \cos \alpha =\frac{a_x}{|\mathit{a}|}\\ \cos \beta =\frac{a_x}{|\mathit{a}|}\\ \cos \gamma =\frac{a_x}{|\mathit{a}|}\\ \text{单}\text{位}\text{向}\text{量}\\ {\mathit{a}}_{\mathbf{0}}=\frac{\mathit{a}}{|\mathit{a}|}=(\cos \alpha ,\cos \beta ,\cos \gamma )\\ \text{任}\text{意}\text{向}\text{量}\mathit{r}=|\mathit{r}|(\cos \alpha ,\cos \beta ,\cos \gamma )\end{array}$$

   

空间平面与直线

• 平面方程

  $$\begin{array}{cc}\text{法}\text{向}\text{量}:& \mathit{n}=(A,B,C)\\ \text{一}\text{般}\text{式}:& Ax+By+Cz+D=0\\ \text{点}\text{法}\text{式}:& A(x-x_0)+B(y-y_0)+C(z-z_0)=0\\ \text{三}\text{点}\text{式}:& \left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ x-x_2& y-y_2& z-z_2\\ x-x_3& y-y_3& z-z_3\end{array}\right|=0\\ \text{截}\text{距}\text{式}:& \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\end{array}$$

• 直线方程

  $$\begin{array}{cc}\text{方}\text{向}\text{向}\text{量}:& \tau =(l,m,n)\\ \text{一}\text{般}\text{式}:& \{\begin{array}{l}{A}_{1}x+B_{1}y+C_{1}z+D_1=0,{\mathit{n}}_1=(A_1,B_1,C_1)\\ A_{2}x+B_{2}y+C_{2}z+D_2=0,{\mathit{n}}_1=(A_2,B_2,C_2)\end{array}(n_1\text{不}\text{平}\text{行}\text{于}{n}_2)\\ \text{点}\text{向}\text{式}:& \frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}\\ \text{参}\text{数}\text{式}:& \{\begin{array}{l}x=x_0+lt\\ y=y_0+mt\\ z=z_0+nt\end{array}\\ \text{两}\text{点}\text{式}:& \frac{x-x_1}{x_0-x_1}=\frac{y-y_1}{y_0-y_1}=\frac{z-z_1}{z_0-z_1}\end{array}$$

• 位置关系

  $$\begin{array}{cc}P(x_0,y_0,z_0)\text{到}Ax+By+Cz+D=0\text{距}\text{离}:& \frac{|A{x}_0+B{y}_0+C{z}_0+D|}{\sqrt{A^2+B^2+C^2}}\\ \text{直}\text{线}\text{垂}\text{直}:& \text{方}\text{向}\text{向}\text{量}\text{内}\text{积}\text{为}0\\ \text{直}\text{线}\text{平}\text{行}:& \text{方}\text{向}\text{向}\text{量}\text{分}\text{量}\text{成}\text{比}\text{例}\\ \text{平}\text{面}\text{垂}\text{直}:& \text{法}\text{向}\text{量}\text{垂}\text{直}\\ \text{平}\text{面}\text{平}\text{行}:& \text{法}\text{向}\text{量}\text{平}\text{行}\end{array}$$

   

空间曲线与曲面

• 空间曲线

  $$\begin{array}{cc}\text{一}\text{般}\text{式}:& \{\begin{array}{l}F(x,y,z)=0\\ G(x,y,z)=0\end{array}\\ \text{参}\text{数}\text{式}:& \{\begin{array}{l}x={\varphi }_1(t)\\ y={\varphi }_2(t)\\ z={\varphi }_3(t)\end{array}\\ \text{空}\text{间}\text{曲}\text{线}\text{在}\text{坐}\text{标}\text{面}\text{上}\text{的}\text{投}\text{影}:& \text{如}\text{果}\text{投}\text{影}\text{到}XOY\text{平}\text{面}\\ & \text{消}\text{去}\{\begin{array}{l}F(x,y,z)=0\\ G(x,y,z)=0\end{array}\text{中}\text{的}z,\\ & \text{得}\text{到}\varphi (x,y)=0\\ & \text{投}\text{影}\text{为}：\{\begin{array}{l}\varphi (x,y)=0\\ z=0\end{array}\end{array}$$

• 空间曲面

  

  

• 柱面

  $$\begin{array}{cc}\text{椭}\text{圆}\text{柱}\text{面}:& \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\\ \text{双}\text{曲}\text{柱}\text{面}:& \frac{x^2}{a^2}-\frac{y^2}{b^2}=1\\ \text{抛}\text{物}\text{柱}\text{面}:& y=a{x}^2\end{array}$$

• 旋转曲面

  $$\begin{array}{cc}\text{曲}\text{线}\tau ：& \{\begin{array}{l}F(x,y,z)=0\\ G(x,y,z)=0\end{array}\\ \text{绕}\text{直}\text{线}：& \frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}\\ \text{原}\text{材}\text{料}:& \{\begin{array}{l}\text{直}\text{线}\text{上}\text{一}\text{点}:P_0(x_0,y_0,z_0)\\ \text{曲}\text{线}\text{上}\text{一}\text{点}:P_1(x_1,y_1,z_1)\\ \text{纬}\text{圆}\text{上}\text{一}\text{点}:P(x,y,z)\\ \text{直}\text{线}\text{方}\text{向}\text{向}\text{量}:(l,m,n)\end{array}\\ \text{旋}\text{转}\text{一}\text{周}\text{所}\text{组}\text{成}\text{的}\text{曲}\text{面}:& \{\begin{array}{l}【P{P}_1\text{与}\text{直}\text{线}\text{垂}\text{直}】\\ l(x-x_1)+m(y-y_1)+n(z-z_1)=0\\ 【|P_1{P}_0|=|P{P}_0|】\\ (x-x_0{)}^2+(y-y_0{)}^2+(z-z_0{)}^2=(x_1-x_0{)}^2+(y_1-y_0{)}^2+(z_1-z_0{)}^2\\ 【P_1\text{在}\text{曲}\text{线}\text{上}】\\ F(x_1,y_1,z_1)=0\\ G(x_1,y_1,z_1)=0\end{array}\end{array}$$

   

多元函数微分学的几何应用

• 空间曲线的切线与法平面

  $$\begin{array}{cc}\text{空}\text{间}\text{曲}\text{线}:& \{\begin{array}{l}x=\varphi (t)\\ y=\psi (t)\\ z=\omega (t)\end{array}\\ \text{空}\text{间}\text{曲}\text{线}\text{上}\text{一}\text{点}:& P_0(x_0,y_0,z_0)\\ P_0\text{处}\text{切}\text{向}\text{量}:& \tau =({\varphi }^{\prime }(t),{\psi }^{\prime }(t),{\omega }^{\prime }(t))\\ P_0\text{处}\text{切}\text{线}\text{方}\text{程}:& \frac{x-x_0}{{\varphi }^{\prime }(t)}=\frac{y-y_0}{{\psi }^{\prime }(t)}=\frac{z-z_0}{{\omega }^{\prime }(t)}\\ P_0\text{处}\text{法}\text{平}\text{面}:& (x-x_0){\varphi }^{\prime }(t)+(y-y_0){\psi }^{\prime }(t)+(z-z_0){\omega }^{\prime }(t)=0\end{array}$$
  $$\begin{array}{cc}\text{空}\text{间}\text{曲}\text{线}:& \{\begin{array}{l}F(x,y,z)=0\\ G(x,y,z)=0\end{array}\\ P_0\text{处}\text{切}\text{向}\text{量}:& \tau =({\left|\begin{array}{cc}{F}_y& F_z\\ G_y& G_z\end{array}\right|}_{P_0},{\left|\begin{array}{cc}{F}_z& F_x\\ G_z& G_x\end{array}\right|}_{P_0},{\left|\begin{array}{cc}{F}_x& F_y\\ G_x& G_y\end{array}\right|}_{P_0})\\ P_0\text{点}\text{线}:& \frac{x-x_0}{{\left|\begin{array}{cc}{F}_y& F_z\\ G_y& G_z\end{array}\right|}_{P_0}}=\frac{y-y_0}{{\left|\begin{array}{cc}{F}_z& F_x\\ G_z& G_x\end{array}\right|}_{P_0}}=\frac{z-z_0}{{\left|\begin{array}{cc}{F}_x& F_y\\ G_x& G_y\end{array}\right|}_{P_0}}\\ P_0\text{切}\text{平}\text{面}:& (x-x_0){\left|\begin{array}{cc}{F}_y& F_z\\ G_y& G_z\end{array}\right|}_{P_0}+(y-y_0){\left|\begin{array}{cc}{F}_z& F_x\\ G_z& G_x\end{array}\right|}_{P_0}+(z-z_0){\left|\begin{array}{cc}{F}_x& F_y\\ G_x& G_y\end{array}\right|}_{P_0}=0\end{array}$$

   

• 空间曲面的切平面与法线

  $$\begin{array}{cc}\text{空}\text{间}\text{曲}\text{面}:& F(x,y,z)=0\\ \text{平}\text{面}\text{上}\text{一}\text{点}:& P_0(x_0,y_0,z_0)\\ P_0\text{点}\text{的}\text{法}\text{线}:& n=(F_x^{\prime },F_y^{\prime },F_z^{\prime }{)}_{P_0}\\ P_0\text{点}\text{的}\text{切}\text{平}\text{面}:& F_{x_0}^{\prime }(x-x_0)+F_{y_0}^{\prime }(x-x_0)+F_{z_0}^{\prime }(z-z_0)=0\end{array}$$
  $$\text{空}\text{间}\text{曲}\text{面}:z=f(x,y)\to F(x,y,z)=f(x,y)-z$$

   

场论初步

• 方向导数

  $$\begin{array}{r}\text{定}\text{义}\text{法}\\ P(x,y,z)& =\{\begin{array}{l}x-x_0=\mathrm{\Delta }x=t\cdot \cos \alpha \\ y-y_0=\mathrm{\Delta }y=t\cdot \cos \beta \\ z-z_0=\mathrm{\Delta }z=t\cdot \cos \gamma \end{array}\\ t& =\sqrt{(\mathrm{\Delta }x{)}^2+(\mathrm{\Delta }y{)}^2+(\mathrm{\Delta }z{)}^2}\\ \underset{x\to 0^{+}}{lim}\frac{u(P)-u(P_0)}{t}& =\underset{x\to 0^{+}}{lim}\frac{u(x_0+t\cdot \cos \alpha ,y_0+t\cdot \cos \beta ,z_0+t\cdot \cos \gamma )-u(x_0,y_0,z_0)}{\sqrt{(\mathrm{\Delta }x{)}^2+(\mathrm{\Delta }y{)}^2+(\mathrm{\Delta }z{)}^2}}\\ & =u_x^{\prime }(P_0)\cos \alpha +u_y^{\prime }(P_0)\cos \beta +u_z^{\prime }(P_0)\cos \gamma \end{array}$$

• 梯度

  $$\begin{array}{rl}& [u_x^{\prime }(P_0)\cos \alpha +u_y^{\prime }(P_0)\cos \beta +u_z^{\prime }(P_0)\cos \gamma {]}_{max}\\ =& [u_x^{\prime }(P_0),u_y^{\prime }(P_0),u_z^{\prime }(P_0)]\cdot [\cos \alpha ,\cos \beta ,\cos \gamma {]}_{max}\\ =& [u_x^{\prime }(P_0),u_y^{\prime }(P_0),u_z^{\prime }(P_0)]\cdot [1,1,1]\\ \to & \text{梯}\text{度}\text{方}\text{向},\text{方}\text{向}\text{导}\text{数}\text{取}\text{的}\text{最}\text{大}\text{值}\end{array}$$

• 散度

  $$\begin{array}{c}A(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z))\\ divA=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\end{array}$$

• 旋度

  $$\begin{array}{c}A(x,y,z)=(P(x,y,z),Q(x,y,z),R(x,y,z))\\ rotA=\left|\begin{array}{ccc}i& j& k\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial z}{\partial }\\ P& Q& R\end{array}\right|\end{array}$$