# 微积分笔记（54）——曲面积分（4）

Contents

## 曲面积分

### Stokes 公式

#### Stokes 公式

$$\oint_{\partial S} P \, \mathrm{d} x + Q \, \mathrm{d} y + R \, \mathrm{d} z = \iint_S \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \, \mathrm{d} y \, \mathrm{d} z + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \, \mathrm{d} z \, \mathrm{d} x + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, \mathrm{d} x \, \mathrm{d} y$$

#### Stokes 公式证明

$$\oint_{\partial S} R \, \mathrm{d} z = \iint_S \frac{\partial R}{\partial y} \, \mathrm{d} y \, \mathrm{d} z - \frac{\partial R}{\partial x} \, \mathrm{d} z \, \mathrm{d} x$$

\begin{align*} & \quad \iint_S \frac{\partial R}{\partial y} \, \mathrm{d} y \, \mathrm{d} z - \frac{\partial R}{\partial x} \, \mathrm{d} z \, \mathrm{d} x \\ & = \iint_D \left[\frac{\partial R}{\partial y} \frac{\partial(y, z)}{\partial(u, v)} - \frac{\partial R}{\partial x} \frac{\partial(z, x)}{\partial(u, v)} \right] \, \mathrm{d} u \, \mathrm{d} v \\ & = \iint_D \left[\frac{\partial R}{\partial y} \left( \frac{\partial y}{\partial u} \frac{\partial z}{\partial v} - \frac{\partial y}{\partial v} \frac{\partial z}{\partial u} \right) - \frac{\partial R}{\partial x} \left( \frac{\partial z}{\partial u} \frac{\partial x}{\partial v} - \frac{\partial z}{\partial v} \frac{\partial x}{\partial u} \right) \right] \, \mathrm{d} u \, \mathrm{d} v \\ & = \iint_D \left[\frac{\partial z}{\partial v} \left( \frac{\partial R}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial R}{\partial y} \frac{\partial y}{\partial u} \right) - \frac{\partial z}{\partial u} \left( \frac{\partial R}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial R}{\partial y} \frac{\partial y}{\partial v} \right) \right] \, \mathrm{d} u \, \mathrm{d} v \\ & = \iint_D \left[\frac{\partial z}{\partial v} \left( \frac{\partial R}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial R}{\partial y} \frac{\partial y}{\partial u} + \frac{\partial R}{\partial z} \frac{\partial z}{\partial u} \right) - \frac{\partial z}{\partial u} \left( \frac{\partial R}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial R}{\partial y} \frac{\partial y}{\partial v} + \frac{\partial R}{\partial z} \frac{\partial z}{\partial v} \right) \right] \, \mathrm{d} u \, \mathrm{d} v \\ & = \iint_D \left(\frac{\partial R}{\partial u} \frac{\partial z}{\partial v} - \frac{\partial R}{\partial v} \frac{\partial z}{\partial u} \right) \, \mathrm{d} u \, \mathrm{d} v \\ & = \iint_D\left(\frac{\partial R}{\partial u} \frac{\partial z}{\partial v} + R \frac{\partial^2 z}{\partial u \partial v} - \frac{\partial R}{\partial v} \frac{\partial z}{\partial u} - R \frac{\partial^2 z}{\partial v \partial v} \right) \, \mathrm{d} u \, \mathrm{d} v \\ & = \iint_D \left[\frac{\partial}{\partial u} \left(R \frac{\partial z}{\partial v} \right) - \frac{\partial}{\partial v} \left(R \frac{\partial z}{\partial u} \right) \right] \, \mathrm{d} u \, \mathrm{d} v \end{align*}

$$\iint_D \left[\frac{\partial}{\partial u} \left(R \frac{\partial z}{\partial v} \right) - \frac{\partial}{\partial v} \left(R \frac{\partial z}{\partial u} \right) \right] \, \mathrm{d} u \, \mathrm{d} v = \oint_{\partial D} R \frac{\partial z}{\partial v} \, \mathrm{d} u + R \frac{\partial z}{\partial u} \, \mathrm{d} v$$

$$u = u(t), v = v(t), t \in [a, b]$$
$t$ 增为 $\partial D$ 正向，得到：
$$\oint_{\partial D} R \frac{\partial z}{\partial v} \, \mathrm{d} u + R \frac{\partial z}{\partial u} \, \mathrm{d} v = \int_a^b \left[R \frac{\partial z}{\partial u} u^\prime(t) + R \frac{\partial z}{\partial v} v^\prime(t) \right] \, \mathrm{d} t$$

$$\pmb{r} = \pmb{r}(u(t), v(t)) \in \mathbb{R}^3, t \in [a, b]$$

$$\oint_{\partial S} R \, \mathrm{d} z = \int_a^b R \left[\frac{\partial z}{\partial u} u^\prime(t) + \frac{\partial z}{\partial v} v^\prime(t) \right] \, \mathrm{d} t$$

#### Stokes 公式的表示

$$\oint_{\partial S} P \, \mathrm{d} x + Q \, \mathrm{d} y + R \, \mathrm{d} z = \iint_S \begin{vmatrix} \mathrm{d} y \, \mathrm{d} z & \mathrm{d} z \, \mathrm{d} x & \mathrm{d} x \, \mathrm{d} y \\ D_x & D_y & D_z \\ P & Q & R \end{vmatrix}$$

$$\oint_{\partial S} P \, \mathrm{d} x + Q \, \mathrm{d} y + R \, \mathrm{d} z = \int_S \begin{vmatrix} \cos \alpha & \cos \beta & \cos \gamma \\ D_x & D_y & D_z \\ P & Q & R \end{vmatrix} \, \mathrm{d} \sigma$$

### 微分形式与外微分

#### 回忆

Newton-Leibniz 公式，Green 公式，Stokes 公式，Gauss 公式。

$\mathbb{R}^3$ 中是否还有类似的公式？

#### $\mathbb{R}^3$ 中的观察

$\mathrm{d} x, \mathrm{d} y, \mathrm{d} z$——一次微分（带方向的长度微元）。

$\mathrm{d} x \, \mathrm{d} y, \mathrm{d} y \, \mathrm{d} z, \mathrm{d} z \, \mathrm{d} x$——二次微分（带符号的面积微元）。

$P \, \mathrm{d} x + Q \, \mathrm{d} y + R \, \mathrm{d} z$——$1$ 次形式。

$P \, \mathrm{d} y \, \mathrm{d} z + Q \, \mathrm{d} z \, \mathrm{d} x + R \, \mathrm{d} x \, \mathrm{d} y$——$2$ 次形式。

$f(x, y, z) \, \mathrm{d} x \, \mathrm{d} y \, \mathrm{d} z$——$3$ 次形式。

1. 没有 $\mathrm{d} x \, \mathrm{d} x$ 微分形式——同方向长度微元张开面积为 $0$。
2. 面积微元带符号——$\mathrm{d} y \, \mathrm{d} x = -\mathrm{d} x \, \mathrm{d} y$。

#### $\mathbb{R}^3$ 中的微分形式

$f(x, y, z)$——$0$ 次形式。

$P \, \mathrm{d} x + Q \, \mathrm{d} y + R \, \mathrm{d} z$——$1$ 次形式。

$P \, \mathrm{d} y \land \mathrm{d} z + Q \, \mathrm{d} z \land \mathrm{d} x + R \, \mathrm{d} x \land \mathrm{d} y$——$2$ 次形式。

$f(x, y, z) \, \mathrm{d} x \land \mathrm{d} y \land \mathrm{d} z$——$3$ 次形式。

#### $\mathbb{R}^3$ 中微分形式的外积

1. $1$ 次形式与 $1$ 次形式的外积是 $2$ 次形式。
2. $1$ 次形式与 $2$ 次形式的外积是 $3$ 次形式。
3. $0$ 次形式与 $j$ 次形式的外积是 $j$ 次形式。

#### $\mathbb{R}^3$ 中微分形式的微分——外微分

$0$ 次形式微分：
$$\mathrm{d} f := \frac{\partial f}{\partial x} \, \mathrm{d} x + \frac{\partial f}{\partial y} \, \mathrm{d} y + \frac{\partial f}{\partial z} \, \mathrm{d} z$$
$2$ 次形式微分（略去中间计算）：
$$\omega = P \, \mathrm{d} y \land \mathrm{d} z + Q \, \mathrm{d} z \land \mathrm{d} x + R \, \mathrm{d} x \land \mathrm{d} y \\ \mathrm{d} \omega := \left(\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \right) \, \mathrm{d} x \land \mathrm{d} y \land \mathrm{d} z$$
$1$ 次形式微分（略去中间计算）：
$$\omega = P \, \mathrm{d} x + Q \, \mathrm{d} y + R \, \mathrm{d} z \\ \mathrm{d} \omega := \begin{vmatrix} \mathrm{d} y \land \mathrm{d} z & \mathrm{d} z \land \mathrm{d} x & \mathrm{d} x \land \mathrm{d} y \\ D_x & D_y & D_z \\ P & Q & R \end{vmatrix}$$

#### $\mathbb{R}^3$ 中微积分基本公式

$$\int_{\partial S} \omega = \int_S \, \mathrm{d} \omega$$

#### 进一步的推广

1. $\mathbb{R}^n$ 中的微分形式：外积，外微分，……
2. $\mathbb{R}^n$ 中微分形式的积分：$p$ 维参数“曲面”，多重积分，……
3. $\mathbb{R}^n$ 中微分形式的 Stokes 公式：形式是已知的！
4. 流形 $M^n$ 上的微积分：局部化（局部等价于 $\mathbb{R}^n$），……

$$\int_{\partial S} \omega = \int_S \, \mathrm{d} \omega$$