# 微积分笔记（53）——曲面积分（3）

Contents

## 曲面积分

### Gauss 公式

#### 再观察：Green 公式

$$(\mathrm{d} x, \mathrm{d} y) = \pmb{\tau} \, \mathrm{d} s = (\tau_1, \tau_2) \, \mathrm{d} s$$

$$(\mathrm{d} y, – \mathrm{d} x) = (\tau_2, -\tau_1) \, \mathrm{d} s = \pmb{n} \, \mathrm{d} s$$

$$\oint_{\partial D} (\pmb{F} \cdot \pmb{\tau}) \, \mathrm{d} s = \iint_D \left(\frac{\partial Q}{\partial x} – \frac{\partial P}{\partial y} \right) \, \mathrm{d} x \, \mathrm{d} y \\ \oint_{\partial D} (\pmb{F} \cdot \pmb{n}) \, \mathrm{d} s = \iint_D \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \right) \, \mathrm{d} x \, \mathrm{d} y$$

#### 即将的推广

1. 由平面区域 $D$ 推广到曲面 $S$——Stokes 公式。
2. 由 $2$ 维区域推广到 $3$ 维区域——Gauss 公式。

#### Gauss 公式（奥高公式，散度定理）

$$\oint_{\partial \Omega} (\pmb{F} \cdot \pmb{n}) \, \mathrm{d} \sigma = \int_\Omega \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \right) \, \mathrm{d} \mu$$

$$\def\oiint{\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}} \oiint\nolimits_{\partial \Omega} P \, \mathrm{d} y \, \mathrm{d} z + Q \, \mathrm{d} z \, \mathrm{d} x + R \, \mathrm{d} x \, \mathrm{d} y = \iiint_\Omega \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \right) \, \mathrm{d} x \, \mathrm{d} y \, \mathrm{d} z$$

#### Gauss 公式证明

$$\iint_{\partial \Omega} R \, \mathrm{d} x \, \mathrm{d} y = \iiint_\Omega \frac{\partial R}{\partial z} \, \mathrm{d} x \, \mathrm{d} y \, \mathrm{d} z$$

$$\Omega = \{(x, y, z) : z_1(x, y) \le z \le z_2(x, y), (x, y) \in D\}$$

$$S_1 : z = z_1(x, y), (x, y) \in D, 法向朝下 \\ S_2 : z = z_2(x, y), (x, y) \in D, 法向朝上$$

$$\iint_{S_3} R \, \mathrm{d} x \, \mathrm{d} y = 0$$

$$\iint_{\partial \Omega} R \, \mathrm{d} x \, \mathrm{d} y \\ = \iint_{S_1} R \, \mathrm{d} x \, \mathrm{d} y + \iint_{S_2} R \, \mathrm{d} x \, \mathrm{d} y \\ = -\iint_{D} R(x, y, z_1(x, y)) \, \mathrm{d} x \, \mathrm{d} y + \iint_{D} R(x, y, z_2(x, y)) \, \mathrm{d} x \, \mathrm{d} y$$

$$\iiint_\Omega \frac{\partial R}{\partial z} \, \mathrm{d} x \, \mathrm{d} y \, \mathrm{d} z \\ = \iint_D \, \mathrm{d} x \, \mathrm{d} y \int_{z_1(x, y)}^{z_2(x, y)} \frac{\partial R}{\partial z} \, \mathrm{d} z \\ = \iint_{D} [R(x, y, z_2(x, y)) – R(x, y, z_1(x, y))] \, \mathrm{d} x \, \mathrm{d} y \\ = \iint_{\partial \Omega} R \, \mathrm{d} x \, \mathrm{d} y$$

$$\Omega = \bigcup_{i = 1}^k \Omega_i$$

$$\iint_{\partial \Omega_i} R \, \mathrm{d} x \, \mathrm{d} y = \iiint_{\Omega_i} \frac{\partial R}{\partial z} \, \mathrm{d} x \, \mathrm{d} y \, \mathrm{d} z$$

$$\sum_{i = 1}^k \iint_{\partial \Omega_i} R \, \mathrm{d} x \, \mathrm{d} y = \sum_{i = 1}^k \iiint_{\Omega_i} \frac{\partial R}{\partial z} \, \mathrm{d} x \, \mathrm{d} y \, \mathrm{d} z = \iiint_\Omega \frac{\partial R}{\partial z} \, \mathrm{d} x \, \mathrm{d} y \, \mathrm{d} z$$

$$\therefore \sum_{i = 1}^k \iint_{\partial \Omega_i} R \, \mathrm{d} x \, \mathrm{d} y = \iint_{\partial \Omega} R \, \mathrm{d} x \, \mathrm{d} y = \iiint_\Omega \frac{\partial R}{\partial z} \, \mathrm{d} x \, \mathrm{d} y \, \mathrm{d} z \ \ \ \square$$

#### 理论应用

$$\pmb{F} \cdot \pmb{n} = \mathrm{grad} \, u \cdot \pmb{n} = D_{\pmb{n}} u$$

$$\oint_{\partial \Omega} \frac{\partial u}{\partial \pmb{n}} \, \mathrm{d} \sigma = \int_\Omega \Delta u \, \mathrm{d} \mu$$

$$\Delta := \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$$

$$\pmb{F} \cdot \pmb{n} = v \, \mathrm{grad} \, u \cdot \pmb{n} = v D_{\pmb{n}} u$$

$$\oint_{\partial D} v \frac{\partial u}{\partial \pmb{n}} \, \mathrm{d} s = \int_D \left(v \Delta u + \frac{\partial v}{\partial x} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \frac{\partial u}{\partial y} \right) \, \mathrm{d} \sigma$$

$$\oint_{\partial D} u \frac{\partial u}{\partial \pmb{n}} \, \mathrm{d} s = \int_D \| \mathrm{grad} \, u \|^2 \, \mathrm{d} \sigma$$

$$\oint_{\partial D} v \frac{\partial u}{\partial \pmb{n}} \, \mathrm{d} s = \int_D \left(v \Delta u + \frac{\partial v}{\partial x} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \frac{\partial u}{\partial y} \right) \, \mathrm{d} \sigma$$

$$\oint_{\partial D} u \frac{\partial v}{\partial \pmb{n}} \, \mathrm{d} s = \int_D \left(u \Delta v + \frac{\partial u}{\partial x} \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \frac{\partial v}{\partial y} \right) \, \mathrm{d} \sigma$$

$$\oint_{\partial D} \left(v \frac{\partial u}{\partial \pmb{n}} – u \frac{\partial v}{\partial \pmb{n}} \right) \, \mathrm{d} s = \int_D (v \Delta u – u \Delta v) \, \mathrm{d} \sigma$$