# 微积分笔记（47）——多重积分（3）

Contents

## 多重积分

### 二重积分的换元公式

#### 复习：一元函数积分换元公式

$$\int_\alpha^\beta f(x(t)) x^\prime(t) \, \mathrm{d} t = \int_{x(\alpha)}^{x(\beta)} f(x) \, \mathrm{d} x$$

$$\int_\alpha^\beta f(x(t)) |x^\prime(t)| \, \mathrm{d} t = \int_a^b f(x) \, \mathrm{d} x$$

#### 类比-猜想：二重积分换元公式

$$\int_D f \, \mathrm{d} \sigma = \int_{\tilde{D}} (f \circ T)(AT) \, \mathrm{d} \tilde{\sigma}$$

$$\iint_D f(x, y) \, \mathrm{d} x \, \mathrm{d} y = \iint_{\tilde{D}} f(x(u, v), y(u, v)) AT(u, v) \, \mathrm{d} u \, \mathrm{d} v$$

#### 二重积分换元分析

$$\Delta u = \frac{b - a}{n}, u_i = a + i \Delta u, i = 0, 1, 2, \cdots, n \\ \Delta v = \frac{d - c}{m}, v_j = c + j \Delta v, j = 0, 1, 2, \cdots, m$$

$$\tilde{D}_{ij} = [u_{i - 1}, u_i] \times [v_{j - 1}, v_j], i = 1, 2, \cdots, n, j = 1, 2, \cdots, n$$

$$T = T(u_i, v), c \le v \le d, i = 0, 1, \cdots, n$$

$$T = T(u, v_j), a \le u \le b, j = 0, 1, \cdots, m$$

$$D_{ij}, i = 1, 2, \cdots, n, j = 1, 2, \cdots, m$$

$$P_0 = (u, v), P_1 = (u + \Delta u, v) \\ P_2 = (u, v + \Delta v), P_3 = (u + \Delta u, v + \Delta v)$$

$$M_0 = T(P_0) = T(u, v) \\ M_1 = T(P_1) = T(u + \Delta u, v) \\ M_2 = T(P_2) = T(u, v + \Delta v) \\ M_3 = T(P_3) = T(u + \Delta u, v + \Delta v)$$

$$M_1 - M_0 = T(u + \Delta u, v) - T(u, v) = D_u T(u, v) \Delta u + o(\Delta u) \\ M_2 - M_0 = T(u, v + \Delta v) - T(u, v) = D_v T(u, v) \Delta v + o(\Delta v)$$

$$\sigma(\Delta D) \approx \| (M_1 - M_0) \times (M_2 - M_0) \| \\ = \| D_u T(u, v) \times D_v T(u, v) \| \Delta u \Delta v + o(\Delta u \Delta v)$$

$$D_u T(u, v) = (D_u x, D_u y, 0) \\ D_v T(u, v) = (D_v x, D_v y, 0)$$

$$\| D_u T(u, v) D_v T(u, v) \| = \left | \det \begin{pmatrix} D_u x & D_u y \\ D_v x & D_v y \end{pmatrix} \right| = | \det J T(u, v) |$$

$$\sum_{i, j} f(M_{i, j}) \sigma(D_{i, j}) \approx \sum_{i, j} f(T(P_{ij})) |J T(P_{ij})| \sigma(\tilde{D}_{ij})$$

$$\int_D f \, \mathrm{d} \sigma = \int_{\tilde{D}} f \circ T | \det(J T)| \, \mathrm{d} \tilde{\sigma}$$

#### 二重积分换元公式

$$\int_D f \, \mathrm{d} \sigma = \int_{\tilde{D}} f \circ T | \det(JT) | \, \mathrm{d} \tilde{\sigma}$$

$$\frac{\partial(x, y)}{\partial(u, v)} := \begin{vmatrix} D_u x & D_u y \\ D_v x & D_v y \end{vmatrix} = \det [JT(u, v)]$$

$$\iint_D f(x, y) \, \mathrm{d} x \, \mathrm{d} y = \iint_{\tilde{D}} f(x(u, v), y(u, v)) \left | \frac{\partial(x, y)}{\partial(u, v)} \right | \, \mathrm{d} u \, \mathrm{d} v$$

#### 应用

$$\int_D f \, \mathrm{d} \sigma = \int_{D_-} [f(x, y) + f(-x, y)] \, \mathrm{d} \sigma$$

$$\int_D f \, \mathrm{d} \sigma = 0$$

#### 特例——极坐标换元法

$$x = r \cos \theta, y = r \sin \theta$$

$$\frac{\partial(x, y)}{\partial(u, v)} = \begin{vmatrix} D_r x & D_r y \\ D_\theta x & D_\theta y \end{vmatrix} = \begin{vmatrix} \cos \theta & \sin \theta \\ -r \sin \theta & r \cos \theta \end{vmatrix} = r$$

$$\mathrm{d} x \, \mathrm{d} y = r \, \mathrm{d} r \, \mathrm{d} \theta$$

$$\iint_D f(x, y) \, \mathrm{d} \sigma = \iint_{\tilde{D}} f(r \cos \theta, r \sin \theta) r \, \mathrm{d} r \, \mathrm{d} \theta$$

#### Piosson 积分

$$I = \int_{-\infty}^{+\infty} e^{-x^2} \, \mathrm{d} x$$

$$I_R = \int_{-R}^R e^{-x^2} \, \mathrm{d} x \\ V(R) = \iint_{x^2 + y^2 \le R^2} e^{-(x^2 + y^2)} \, \mathrm{d} x \, \mathrm{d} y \\ = \int_0^{2\pi} \, \mathrm{d} \theta \int_0^R e^{-r^2} r \, \mathrm{d} r = \pi\left(1 - e^{-R^2}\right)$$

$$V(R) \le I_R^2 \le V(\sqrt{2} R)$$

$$I^2 = \lim_{R \to +\infty} I_R^2 = \lim_{R \to +\infty} V(R) = \pi \Rightarrow I = \sqrt{\pi}$$

### 三重积分的概念与性质

#### 三重积分概念（类比二重积分）

1. 将长方体做有限规则分割：$T = \pi_x \times \pi_y \times \pi_z$
$$\pi_x : a = x_0 < x_1 < \cdots < x_n = b \\ \pi_y : c = y_0 < y_1 < \cdots < y_m = d \\ \pi_z : g = z_0 < z_1 < \cdots < z_k = h$$ 将长方体分割为 $nmk$ 个小长方体，任意编号得 $\Omega = \bigcup\limits_{i = 1}^{nmk} \Omega_i$。

2. 构造 Riemann 和式：
$$\sum_{i = 1}^{nmk} f(\xi_i) \mu(\Omega_i)$$
其中 $\mu(\Omega_i)$ 表示长方体 $\Omega_i$ 的体积，$\xi_i \in \Omega_i$ 任取，$i = 1, 2, \cdots, nmk$。

3. 定义 $f$ 的三重积分：
$$\iiint_{\Omega} f(x, y, z) \, \mathrm{d} \mu := \lim_{\| T \| \to 0} \sum_{i = 1}^{nmk} f(\xi_i) \mu(D_i)$$
如果极限存在。

其中 $\| T \| = \| \pi_x \| + \| \pi_y \| + \| \pi_z \|$ 称为分割 $T$ 的直径。

如果上述极限存在，称函数 Riemann 可积，记为 $f \in R(\Omega)$。
$$\iiint_{\Omega} f(x, y, z) \, \mathrm{d} \mu = A$$
也记为：
$$\int_\Omega f \, \mathrm{d} \mu = A$$
称为积分值。

#### $3$ 维零测集

$E \subseteq \mathbb{R}^3$ 满足以下条件：$\forall \varepsilon > 0$，存在一列闭长方体 $\{\Omega_i\}_{i = 1}^\infty$，使得 $E \subseteq \bigcup\limits_{i = 1}^\infty \Omega_i$ 并且 $\sum\limits_{i = 1}^\infty \mu(\Omega_i) < \varepsilon$。

#### 零体积集

$E \subseteq \mathbb{R}^3$ 满足以下条件：$\forall \varepsilon > 0$，存在有限个长方体 $\{\Omega_i\}_{i = 1}^m$，使得 $E \subseteq \bigcup\limits_{i = 1}^m \Omega_i$ 并且 $\sum\limits_{i = 1}^m \mu(\Omega_i) < \varepsilon$。推论

1. 空间中任意有界平面（块）是零体积集；
2. 空间中任意有限块平面块是零体积集；
3. 空间中任意有限面积的光滑曲面是零体积集。

#### Fubini 定理

$$\int_{\Omega} f \, \mathrm{d} \mu = \int_a^b \, \mathrm{d} x \int_c^d \, \mathrm{d} y \int_g^h f(x, y, z) \, \mathrm{d} z$$

#### 一般有界区域上三重积分

$$f_\Omega(x, y, z) := \begin{cases} f(x, y, z) & (x, y, z) \in \Omega \\ 0 & (x, y, z) \not \in \Omega \end{cases}$$

$$\int_\Omega f \, \mathrm{d} \mu := \int_{\Omega_M} f_\Omega \, \mathrm{d} \mu$$