定理1
如果\(f\)在\(I = [a,b] \times [c,d]\)上可积，那么单变量函数\(\displaystyle \varphi(x) = \underline \int_c^d f(x, y) \mathrm{d} y\)和\(\displaystyle \psi(x) = \overline \int_c^d f(x, y) \mathrm{d} y\)在区间\([a,b]\)上可积，且
\[ \int_I f \mathrm{d} \sigma = \int_a^b \varphi(x) \mathrm{d} x = \int_a^b \psi(x) \mathrm{d} x \]
证：分别对\([a,b]\)和\([c,d]\)的分割
\[
\begin{aligned}
\pi_x: a = x_0 < x_1 < \cdots < x_n = b, \\
\pi_y: c = y_0 < y_1 < \cdots < y_m = d,
\end{aligned}
\]
令
\[
\begin{aligned}
& I_i = [x_{i-1}, x_i] \quad (i=1,2,\cdots,n) \\
& J_j = [y_{j-1}, y_j] \quad (j=1,2,\cdots,m)
\end{aligned}
\]
则子矩形
\[
I_i \times I_j \quad (i=1,2,\cdots,n; j=1,2,\cdots,m)
\]
形成了矩形\(I\)的分割\(\pi = \pi_x \times \pi_y\)。令\(\displaystyle A = \int_I f \mathrm{d}
\sigma\)，由积分存在的定义可知，对任意的\(\varepsilon > 0\)，存在\(\delta > 0\)，当\(I\)的分割\(\pi\)满足\(\Vert
\pi \Vert < \delta\)时，必有
\[
A - \varepsilon < \sum_{i=1}^n \sum_{j=1}^m f(\xi_i, \eta_j)
\Delta x_i \Delta y_j < A + \varepsilon
\]
其中\(\xi_i \in I_i, \eta_j \in J_j
(i=1,2,\cdots,n;j=1,2,\cdots,m)\)，现取分割\(\pi_x, \pi_y\)满足\(\Vert \pi_x \Vert < \delta / \sqrt 2, \Vert
\pi_y \Vert < \delta / \sqrt 2\)，那么\(\Vert \pi \Vert <
\delta\)，从而上式成立，所以有
\[
\begin{aligned}
A - \varepsilon &\le \sum_{i=1}^n \sum_{j=1}^m \inf f(\xi_i,
\eta_j) \Delta x_i \Delta y_j \\
& \le \sum_{i=1}^n \sum_{j=1}^m \sup f(\xi_i, \eta_j) \Delta x_i
\Delta y_j \\
& \le A + \varepsilon
\end{aligned}
\]
而\(\displaystyle \sum_{j=1}^m \inf f(\xi_i,
\eta_j) \Delta y_j\)表示函数\(f(\xi_i)\)在\([c,d]\)上的下和，所以
\[
\sum_{j=1}^m \inf f(\xi_i, \eta_j) \Delta y_j \le \underline
\int_c^d f(\xi_i, y) \mathrm{d} y = \varphi(\xi_i)
\]
同理
\[
\sum_{j=1}^m \sup f(\xi_i, \eta_j) \Delta y_j \le \overline \int_c^d
f(\xi_i, y) \mathrm{d} y = \psi(\xi_i)
\]
从而
\[
A - \varepsilon \le \sum_{i=1}^n \varphi(\xi_i) \Delta x_i \le
\sum_{i=1}^n \psi(\xi_i) \Delta x_i \le A + \varepsilon
\]
即
\[
\lim_{\Vert \pi_x \Vert \to 0} \sum_{i=1}^n \varphi(\xi_i) \Delta
x_i = \lim_{\Vert \pi \Vert \to 0} \sum_{i=1}^n \psi(\xi_i) \Delta x_i =
A
\]
成立
Q.E.D.
定理2
设\(f\)在\(I = [a,b] \times [c, d]\)上可积，如果对每一个\(x \in [a,b]\)，函数\(f(x, y)\)在\([c, d]\)上可积，则
\[ \int_I f \mathrm{d} \sigma = \int_a^b \left(\int_c^d f(x, y) \mathrm{d} y \right) \mathrm{d} x \]
上面等式右边称为累次积分，也可以记为
\[ \int_a^b \mathrm{d} x \int_c^d f(x, y) \mathrm{d} y \]
同样，如果对于每一个\(y \in [c, d]\)，函数\(f(x, y)\)在\([a,b]\)上可积，那么有
\[ \int_I f \mathrm{d} \sigma = \int_c^d \left(\int_a^b f(x, y) \mathrm{d} x \right) \mathrm{d} y \]
上面等式右边也可以记为
\[ \int_c^d \mathrm{d} y \int_a^b f(x, y) \mathrm{d} x \]
证：由定理1可知，
\[
\varphi(x) = \psi(x) = \int_c^d f(x, y) \mathrm{d} y
\]
所以
\[
\int_I f \mathrm{d} \sigma = \int_a^b \varphi(x) \mathrm{d} x =
\int_a^b \mathrm{d} x \int_c^d f(x, y) \mathrm{d} y
\]
后半部分同样的证明方法。
Q.E.D.
定理3
设\(f\)是\([a,b] \times [c, d]\)上的连续函数，则有
\[ \int_c^d \mathrm{d} y \int_a^b f(x, y) \mathrm{d} x = \int_a^b \mathrm{d} x \int_c^d f(x, y) \mathrm{d} y \]
证：由于\(f\)连续，从而\(\varphi(x)\)与\(\psi(x)\)都可积，再由定理2易证。
Q.E.D.
定理4
设点集
\[ B = \{(x, y): y_1(x) \le y \le y_2(x), a \le x \le b\} \]
其中函数\(y_1,y_2\)在\([a,b]\)上连续，函数\(f\)在\(B\)上可积。如果对任意的\(x \in [a,b]\)，单变量积分
\[ \int_{y_1(x)}^{y_2(x)} f(x, y) \mathrm{d} y \]
存在，那么
\[ \int_B f \mathrm{d} \sigma = \int_a^b \mathrm{d} x \int_{y_1(x)}^{y_2(x)} f(x, y) \mathrm{d} y \]
证：令\(c = \inf y_1([a, b]), d = \sup
y_2([a, b])\)，从而\(I = [a, b] \times
[c, d] \supset B\)，由于\(f\)在\(B\)上可积，从而\(f_B\)在\(I\)上可积，而且
\[
\int_B f \mathrm{d} \sigma = \int_I f_B \mathrm{d} \sigma
\]
显然易知，对每一个\(x \in
[a,b]\)，\(f_B(x, y)\)在\([c, d]\)上可积，所以由定理2可知，
\[
\int_I f_B \mathrm{d} \sigma = \int_a^b \mathrm{d} x \int_c^d f_B(x,
y) \mathrm{d} y = \int_a^b \mathrm{d} x \int_{y_1(x)}^{y_2(x)} f_B
\mathrm{d} y = \int_a^b \mathrm{d} x\int_{y_1(x)}^{y_2(x)} f(x, y)
\mathrm{d} y
\]
Q.E.D.
定义1：正则映射
设有界闭域\(D \subset \mathbb{R}^2\)，连续函数\(F: D \to \mathbb{R}\)，映射\(\boldsymbol{\varphi}\)由公式
\[ x = x(u, v), \quad y = y(u, v) \quad ((u, v) \in \Delta) \]
定义，其中\(\Delta\)是\(uv\)平面上的有界闭区域，设映射\(\boldsymbol{\varphi}\)是正则的，即\(\boldsymbol{\varphi}\)是从\(\Delta\)到\(D\)上的一对一的映射，\(\boldsymbol{\varphi}\)在\(\Delta\)上连续可导，并且
\[ \frac{\partial (x, y) }{\partial (u, v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} \neq 0 \]
在\(\Delta\)上成立。
定理5
设正则映射\(\boldsymbol{\varphi}\)把\(\mathbb{R}^2\)中以
\[ (u_0, v_0), \quad (u_0 + h, v_0), \quad (u_0+h, v_0 +h), \quad (u_0, v_0 +h) \]
为顶点的的矩形\(A_{hk}\)一对一的映射为\(\boldsymbol{\varphi}(A_{hk})\)，那么
\[ \lim \limits_{ h\to 0, k \to 0} \frac{\sigma(\boldsymbol{\varphi}(A_{hk}))}{\sigma(A_{hk})} = \left| \frac{\partial(x, y)}{\partial(u, v)} \right|_{(u_0, v_0)} \]
证：由映射后的图像有
\[
\sigma(\boldsymbol{\varphi}(A_{hk})) \approx \Vert
(\boldsymbol{\varphi}(u_0 + h, v_0) - \boldsymbol{\varphi}(u_0, v_0))
\times (\boldsymbol{\varphi}(u_0, v_0 + k) - \boldsymbol{\varphi}(u_0,
v_0)) \Vert
\]
由于
\[
\begin{aligned}
\boldsymbol{\varphi}(u_0 + h, v_0) - \boldsymbol{\varphi}(u_0, v_0)
= \frac{\partial \boldsymbol{\varphi}}{\partial u} (u_0, v_0) h +
\boldsymbol{\xi} \\
\boldsymbol{\varphi}(u_0, v_0+h) - \boldsymbol{\varphi}(u_0, v_0) =
\frac{\partial \boldsymbol{\varphi}}{\partial v} (u_0, v_0) h +
\boldsymbol{\eta}
\end{aligned}
\]
其中
\[
\Vert \boldsymbol{\xi} \Vert = o(h), \quad \Vert \boldsymbol{\eta}
\Vert = o(k)
\]
所以
\[
\sigma(\boldsymbol{\varphi}(A_{hk})) \approx \Vert \frac{\partial
\boldsymbol{\varphi}}{\partial u}(u_0, v_0) \times \frac{\partial
\boldsymbol{\varphi}}{\partial v}(u_0, v_0) \Vert hk + o(hk)
\]
而
\[
hk = \sigma(A_{hk})
\]
所以上式可以写为
\[
\frac{\sigma (\boldsymbol{\varphi}(A_{hk}))}{\sigma(A_{hk})} \approx
\Vert \frac{\partial \boldsymbol{\varphi}}{\partial u}(u_0, v_0) \times
\frac{\partial \boldsymbol{\varphi}}{\partial v}(u_0, v_0) \Vert + o(1)
\]
又因为
\[
\frac{\partial \boldsymbol{\varphi}}{\partial u} \times
\frac{\partial \boldsymbol{\varphi}}{\partial v} = \left|
\begin{matrix}
\boldsymbol{i} & \boldsymbol{j} & \boldsymbol{k}\\
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}
& 0 \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
& 0 \\
\end{matrix}
\right| = \frac{\partial(x, y)}{\partial(u, v)} \boldsymbol{k}
\]
所以
\[
\Vert \frac{\partial \boldsymbol{\varphi}}{\partial u} \times
\frac{\partial \boldsymbol{\varphi}}{\partial v} \Vert = \left|
\frac{\partial (x, y)}{\partial(u, v)} \right|
\]
从而
\[
\frac{\sigma (\boldsymbol{\varphi}(A_{hk}))}{\sigma(A_{hk})} \approx
\left| \frac{\partial(x, y)}{\partial(u, v)} \right|_{(u_0, v_0)} +
o(hk)
\]
所以当\(h \to 0, k \to 0\)时，有
\[
\frac{\sigma (\boldsymbol{\varphi}(A_{hk}))}{\sigma(A_{hk})} =
\left| \frac{\partial(x, y)}{\partial(u, v)} \right|_{(u_0, v_0)}
\]
笔者注：感觉该证明方法不太好，约等于记号不应该用在这里，就算使用也应该讲明原因。
Q.E.D.
定理6
设\(\mathbb{R}^2\)中的有界闭区域\(D\)有面积，函数\(F: \to \mathbb{R}\)，映射
\[ \boldsymbol{\varphi}: \left\{ \begin{aligned} x = x(u, v) \\ y = y(u, v) \end{aligned} \right. \quad (u, v) \in \Delta \]
是从\(\Delta\)到\(D\)上的正则映射，那么
\[ \iint \limits_D F(x, y) \mathrm{d} x \mathrm{d} y = \iint \limits_\Delta F \circ \boldsymbol{\varphi}(u, v) \left| \frac{\partial(x, y)}{\partial(u, v)} \right| \mathrm{d} u \mathrm{d} v \]
证：用矩形\(I\)把\(uv\)平面上的闭区域\(\Delta\)覆盖起来，用两族平行直线\(u =u_i (i=0,1,2, \cdots, m)\)和\(v = v_j (j=0,1,2, \cdots,n)\)分割\(I\)，其中
\[
u_0 < u_1 < \cdots < u_{m-1} < u_m, \quad v_0 < v_1
< \cdots < v_{n-1} < v_n
\]
令\(\Delta u_i = u_i -
u_{i-1}(i=1,2,\cdots,m), \Delta v_j = v_j -
v_{j-1}(j=1,2,\cdots,n)\)。分割后可以得到\(mn\)个矩形，在映射\(\boldsymbol{\varphi}\)的作用下，它们变成了\(xy\)平面下的\(mn\)个曲边平行四边形，这时只需要考虑那些完全被包含在\(D\)中的曲边平行四边形，将它们记为\(D_i(i=1,2,\cdots,k)\)，即\(D_i=\boldsymbol{\varphi}(\Delta_i)\)，其中\(\Delta_i(i=1,2,\cdots,k)\)是完全被包含在\(\Delta\)中的矩形，任取一点\(\boldsymbol{\eta}_i \in D_i\) ，并设\(\Delta_i\)中唯一的点\(\xi_i\)，使得\(\boldsymbol{\varphi} (\xi_i) =
\eta_i(i=1,2,\cdots,k)\)，作积分和
\[
\sum_{i=1}^k F(\boldsymbol{\eta}_i) \sigma(D_i)
\]
由定理5可知
\[
\sigma(D_i) \sim |det J \boldsymbol{\varphi}(\xi_i)|
\sigma(\Delta_i) \quad (i=1,2,\cdots, k)
\]
所以得到
\[
\sum_{i=1}^k F(\boldsymbol{\eta}_i) \sigma(D_i) \sim \sum_{i=1}^k F
\circ \boldsymbol{\varphi} (\boldsymbol{\xi}_i) |det J
\boldsymbol{\varphi} (\boldsymbol{\xi}_i)| \sigma(\Delta_i)
\]
当分割无限细时，由函数积分九的定理7可得，
\[
\int_D F \mathrm{d} \sigma = \int_\Delta F \circ
\boldsymbol{\varphi} |J \boldsymbol{\varphi}| \mathrm{d} \sigma
\]
Q.E.D.
定理7：极坐标换元
令\(x = r \cos \theta, y= r \sin \theta\)，则
\[ \iint \limits_D F(x, y) \mathrm{d} x \mathrm{d} y = \iint \limits_\Delta F(r \cos \theta, r \sin \theta) r \mathrm{d} r \mathrm{d} \theta \]
证：这时
\[
\frac{\partial(x, y)}{\partial (u, v)} = \left| \begin{matrix}
\cos \theta & -r\sin \theta \\
\sin \theta & r \cos \theta
\end{matrix}
\right| = r
\]
代入定理6的结果即证。
Q.E.D.