# 第八讲 多元函数积分学

* 预备知识，三重积分，第一型积分（线，面），第二型积分（线Green公式，面Guass公式），stokes公式 

## 一、预备知识

### 1."切一刀，投下来，转一圈"

* 1 "切一刀"——曲面方程的切平面
  * $$设\sum : F(x,y,x)=0 切平面 P\_0为面上一点$$
  * $$1. \vec n = { F'*x|*{P\_0}, F'*y|*{P\_0}, F'*z|*{P\_0}, } = {A,B,C}$$
  * $$2. 写方程 A(x-x\_0)+B(y-y\_0)+C(z-z\_0) = 0$$
* 2."投下来"——曲线（曲面）在坐标面上的投影曲线（平面区域）
  * $$L 投下来\rightarrow L\_投$$ 
  * $$\Sigma 投下来\rightarrow D$$
  * $$投影时，先消去z,在补上z=0$$
* 3."转一圈"——旋转曲面方程
  * $$1.提法:将\Gamma: \left{ \begin{array}{ll}

    F(x,y,z)=0\\

    G(x,y,z)=0    \end{array} \right. 绕L: \ \frac {x-x\_0}{l} =  \frac {y-y\_0}{m}+\frac {z-z\_0}{n} 旋转一周所得曲面\Sigma$$ 
  * $$2.求法$$
    * $$M\_0(x\_0,y\_0,z\_0)在L上, \vec \tau (l,m,n)是L的法向量$$
    * $$M\_1(x\_1.y\_1,z\_1)在\Gamma上,M\_1绕L旋转一周得纬圆,P(x,y,z)为纬圆上一点$$
    * $$M\_0,M\_1,P构成圆锥$$
    * $$\left{ \begin{array}{ll}

      (x\_1-x\_0)^2+(y\_1-y\_0)^2+(z\_1-z\_0)^2=(x-x\_0)^2+(y-y\_0)^2+(z-z\_0)^2\l(x-x\_1)+m(y-y\_1)+n(z-z\_1)=0\\

      F(x\_1,y\_1,z\_1)=0 \G(x\_1,y\_1,z\_1)=0    \end{array} \right.  \ \Rightarrow \Sigma: f(x,y,z)= 0 ——空间曲面方程 $$

### 2.场论初步

* $$u=u(x,y) —— 数量场 field 没有向量方向，例如温度场$$
* $$\vec u = \vec u(x,y) —— 向量场，有方向，有大小 如重力场$$
* $$1.方向导数 u=u(x,y)$$
  * $$P\_0(x\_0,y\_0), P\_1(x\_0+t\cos\alpha,y\_0+t\sin \alpha),P\_0到P\_1为\vec l$$
  * 1.定义法 
    * $$\frac {\partial u}{\partial \vec l} | *{P\_0} \triangleq \lim*{t\to 0^+} \frac{u(x\_0+t\cos\alpha,y\_0+t\sin \alpha) - u(x\_0,y\_0}{t}$$
  * 2.公式法
    * $$若u(x,y)在P\_0处可微,\则\frac {\partial u}{\partial \vec l} | \_{P\_0}=u'*x|*{P\_0} \bullet \cos \alpha + u'*y|*{P\_0} \bullet \sin \alpha$$
* $$2.梯度gradient u=u(x,y)$$
  * $$\overrightarrow {grad}\ U |\_{P\_0} ={ u'*x|*{P\_0} , u'*y|*{P\_0} }$$
* $$3.散度 \vec u = \vec u(x,y,z) = {P,Q,R} 旋转$$
  * $$\vec u = P(x,y,z)\vec i + Q\vec j + R\vec k$$
  * $$散度 div\ \vec u = \frac {\partial P}{\partial x} + \frac {\partial Q}{\partial y} + \frac {\partial R}{\partial z}$$
* $$4.旋度, 三阶行列式$$
  * $$ \vec rot \quad \vec u = \left| \begin{array}{ccc}

    \vec i & \vec j & \vec k \\

    \frac{\partial }{\partial x} & \frac{\partial }{\partial y}  & \frac{\partial }{\partial z}  \\

    P & Q & R

    \end{array} \right| $$

## 二、三重积分

### 1.定义

* 类比二重积分
* $$\iint\_Df(x,y)d\sigma$$
* $$\iiint\_{\Omega}f(x,y,z)dv$$

### 2.计算

* 1.直角系与柱面系下
  * $$dv=dxdydz (直角系下)$$
  * $$dv=d\theta rdrdz (直角+极 = 柱)$$
  * 法一:先一后二法(先z后xy法，投影穿线法)
    * 后积先定限，限内画条线，先交写下限，后交写上限
    * 某个例子
    * $$I= \iiint\_{\Omega}zdv = \iint\_{D\_{*1xy}:x^2+y^2\le 1}d\sigma \bullet \int\_1^2 zdz \\+ \iint*{D\_{*2xy}:1\le x^2+y^2\le 4}d\sigma \bullet \int*{\sqrt{x^2+y^2}}^2 zdz$$
  * 法二:(先xy后z的方法，定限截面法)
    * $$\iiint\_{\Omega}zdv = \int\_1^2 dz \iint\_{D\_{xy}:x^2+y^2\le z^2}zd\sigma$$
* 2.球面系下
  * $$dv = r^2\sin\varphi d\theta d\varphi dr \qquad 记住即可$$
  * $$\left{ \begin{array}{ll}

    x = r\sin\varphi \bullet \cos\theta \\

    y = r\sin\varphi \bullet \sin\theta \ z = r\cos\varphi \end{array} \right.$$ 
  * $$I =  \iiint\_{\Omega}f(x,y,z)dv \ = \int\_{\theta\_1}^{\theta\_2}d\theta \int\_{\varphi\_1}^{\varphi\_2}d\varphi \int\_{r\_1}^{r\_2}f(x,y,z)r^2\sin\varphi dr \ = \int\_{\theta\_1}^{\theta\_2}d\theta \int\_{\varphi\_1}^{\varphi\_2}d\varphi \int\_{r\_1}^{r\_2}f(r\sin\varphi \cos\theta, r\sin\varphi\sin\theta, r\cos\varphi)r^2\sin\varphi dr$$

## 三、第一型积分

### 1.第一型曲线积分(和定积分，二重积分，三重积分本质一致都是面积)

* 1.定义
  * 对比定积分，底边由直线变曲线，dx变ds弧微分，L:y=y(x)
  * $$\int\_Lf(x,y)ds$$
  * $$ds =\sqrt{ (dx)^2 + (dy)^2}$$
  * $$ds =\sqrt{ 1 + (y'\_x)^2}dx$$
* 2.计算口诀:一投二代三计算
  * $$1.参数方程 L: \left{ \begin{array}{ll}

    x = x(t) \\

    y = y(t)  \end{array} \right. \qquad \alpha \le t \le \beta $$

    * $$\int\_Lf(x,y)ds = \int\_{\alpha}^{\beta}f(x(t),y(t))\sqrt {(x'(t))^2+(y'(t))^2}dt$$
  * $$2.显式方程 L: \left{ \begin{array}{ll}

    y = y(x) \\

    (x = x) \end{array} \right. \qquad a \le x \le b $$

    * $$\int\_Lf(x,y)ds = \int\_a^bf(x,y(x))\sqrt {1+(y'(x))^2}dx$$

### 2.第一型曲面积分

* 1.定义
  * $$对比二重积分，底面变为曲面，d\sigma 变为dS 面密度,曲面质量$$
  * $$\iint\_{\Sigma}f(x,y,z)dS$$
  * $$dS =\sqrt{1+(z'\_x)^2+(z'\_y)^2}dxdy$$
* 2.计算口诀:一投二代三计算
  * $$\iint\_{\Sigma:z=z(x,y)}f(x,y,z)dS \ = \iint\_{D\_{xy}}f(x,y,z(x,y))\sqrt{1+(z'\_x)^2+(z'\_y)^2}dxdy$$

## 四、第二型积分

### 1.第二型曲线积分——无几何背景

#### 1.概念

* $$给力场 \vec F=\vec F(x,y) = P(x,y)\vec i + Q(x,y)\vec j$$
* $$弧微分向量d\vec s = {dx,dy}=dx\vec i+dy\vec j$$
* $$dW=P(x,y)dx+Q(x,y)dy$$
* $$W = \int\_LdW = \int\_LP(x,y)dx+Q(x,y)dy ——物理意义做功$$

#### 2.计算

* 1.直接法:一投二代三计算
  * $$1.参数方程 L: \left{ \begin{array}{ll}

    x = x(t) \\

    y = y(t)  \end{array} \right. \qquad t:\alpha  \to \beta 起点到终点 $$
  * $$\int\_LP(x,y)dx+Q(x,y)dy = \ \int\_{\alpha}^{\beta}(P(x(t),y(t))x'(t)+Q(x(t),y(t))y'(t))dt$$
* 2.格林公式法(间接法,green公式)
  * 正方向，沿着正方向走，左手在D内
  * $$\oint\_{L^+}P(x,y)dx+Q(x,y)dy \\=\iint\_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy$$
  * $$成立要求: \left{ \begin{array}{ll}

    1.L封闭取正向 \\

    1. P,Q,\frac{\partial Q}{\partial x},\frac{\partial P}{\partial y}在D中连续  \end{array} \right.$$

### 2.第二型曲面积分——无几何背景

#### 1.概念

* $$给流场 \vec u = \vec u(x,y,z) = P(x,y,z)\vec i + Q(x,y,z)\vec j + R(x,y,z)\vec k$$
* $$d\vec S = dydz\vec i + dzdx\vec j+ dxdy\vec k \ \Rightarrow d\Phi = Pdydz+Qdzdx+Rdxdy$$
* $$\Phi = \iint\_{\Sigma}Pdydz+Qdzdx+Rdxdy ——通量$$

#### 2.计算

* 1.直接法:一投二代三计算
  * 一个一个算
  * $$\iint\_{\Sigma:z=z(x,y)}R(x,y,z)dxdy = \ \iint\_{D\_{xy}}R(x,y,z(x,y)(\pm dxdy)$$
  * $$\[注] 指定\vec n\_外,\vec k = 锐角 \Rightarrow +dxdy$$
  * $$\[注] 指定\vec n\_外,\vec k = 钝角 \Rightarrow -dxdy$$
* 2.高斯公式法
  * $$\oint\oint\_{\Sigma\_外}Pdydz+Qdzdx+Rdxdy = \ \iiint\_{\Omega}(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z})dv$$
  * $$成立要求: \left{ \begin{array}{ll}

    1.\Sigma封闭取外侧 \\

    1. P,Q,R,\frac{\partial P}{\partial x},\frac{\partial Q}{\partial y},\frac{\partial R}{\partial z}在\Omega中连续  \end{array} \right.$$

## 五、空间第二型曲线积分

### 1.斯托克斯公式

* $$设\Omega为空间某区域,  \Sigma为\Omega内分片光滑有向曲面片, \L为逐段光滑的\Sigma的边界,其方向与\Sigma外法向符合右手定则$$
* $$\oint*LPdx+Qdx+Rdz=\iint*{\Sigma}\left| \begin{array}{ccc}

  \cos\alpha & \cos\beta & \cos\gamma \\

  \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \\

  P & Q & R

  \end{array} \right|dS\其中\vec n\_0={\cos\alpha,\cos\beta,\cos\gamma}为\Sigma的单位法向量$$