多元的积分
二重积分
性质
有界性：当$f(x,y)$在$D$上可积，则$f(x,y)$在$D$上必有界
线性性质： $\underset{D}{\iint}[k_1f(x,y)+k_2g(x,y)]d\sigma = k_1\underset{D}{\iint}f(x,y)d\sigma+k_2\underset{D}{\iint}g(x,y)d\sigma$
可加性：$\underset{D}{\iint}f(x,y)d\sigma =\underset{D_1}{\iint}f(x,y)d\sigma+\underset{D_2}{\iint}f(x,y)d\sigma$
保号性：
若$f(x,y)≤g(x,y)$，则$\underset{D}{\iint}f(x,y)d\sigma≤\underset{D}{\iint}g(x,y)d\sigma\\\left| \left. \underset{D}{\iint}f(x,y)d\sigma \right| \right. ≤\underset{D}{\iint}\left| \left. f(x,y)d\sigma \right| \right.$
估值定理：$mA \le \underset{D}{\iint}f(x,y)d\sigma \le MA$
对称性：
普通对称性：$\underset{D}{\iint}f(x,y)d\sigma=\left\{ \begin{array}{l} 2\underset{D_1}{\iint{f\left( x,y \right) d\sigma}},\ f\left( -x,y \right) =f\left( x,y \right)\\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f\left( -x,y \right) \ne f\left( x,y \right)\\ \end{array} \right.$
轮换对称性：$\underset{D}{\iint}f(x,y)dxdy=\underset{D}{\iint}f(y,x)dydx$
计算
直角坐标系
X型区域（上下型）积分：$\underset{D}{\iint}f(x,y)d\sigma=\int^b_adx\int_{\varphi_1(x)}^{\varphi_2(x)}f(x,y)dy$
Y型区域（左右型）积分：$\underset{D}{\iint}f(x,y)d\sigma=\int^d_cdy\int_{\varphi_1(y)}^{\varphi_2(y)}f(x,y)dx$
极坐标系
$$d\sigma=rdrd\theta\\ \underset{D}{\iint}f(x,y)d\sigma=\int_\alpha^\beta d\theta \int_{r_1(\theta)}^{r_2(\theta)}f(r\cos \theta,r\sin \theta)rdr$$- 若被积函数$f(x^2+y^2),f(\dfrac{y}{x})$且区域$D$为圆部分，优先考虑极坐标
极坐标转换
- $\theta$为扫描范围
- $r$为射线入点出点的极坐标方程
无法积分（换积分次序）
$\left\{ \begin{array}{l} \int{\frac{\sin x}{x}dx\ \ \int{\frac{\cos x}{x}dx\ \ \int{\frac{\tan x}{x}dx}}}\\ \int{\sin\dfrac{1}{x}}\ \ \int{\cos\dfrac{1}{x}} \\ \int{\sin x^2}dx\ \ \int{\cos x^2}dx\ \ \int{\tan x^2}dx\\ \int{\text{e}^{ax^{2+bx+c}}dx}\ \ \int{\text{e}^{x^2}dx}\ \ \ \int{\text{e}^{-x^2}dx}\ \ \int{\frac{\text{e}^x}{x}dx}\\ \int{\frac{dx}{\ln x}}\\ \end{array} \right.$三重积分
直角坐标系：
先一后二：$\iiint_\varOmega f(x,y,z)\text dz=\iint_{D_{xy}}\text d\sigma \int_{z_1(x,y)}^{z_2(x,y)}f(x,y,z)\text dz$
先二后一：$\iiint_\varOmega f(x,y,z)\text dz=\int_{c_1}^{c_2}\text dz\iint_{D_z}f(x,y,z)\text dx\text dy$
柱坐标：$\left\{ \begin{array}{l} x=r\cos \theta \ \ \ \ \ \ \ \ \ \ \ 0\le r<+\infty\\
y=r\sin \theta \ \ \ \ \ \ \ \ \ \ \ \ 0\le \theta \le 2\pi\\ z=z\ \ \ \ \ \ \ \ \ \ \ \ \ -\infty < z<+\infty\\\end{array} \right. $，体积微元：$\text dv=r\text dr\text d\theta\text dz$
$$
\iiint_\varOmega f(x,y,z)\text dz=\iiint_\varOmega f(r\cos\theta,r\sin\theta,z)r\ \text dr\text d\theta\text dz
$$
球坐标：$\left\{ \begin{array}{l}
x=r\sin\varphi\cos \theta \ \ \ \ \ \ \ \ \ \ \ 0\le r < +\infty\\
y=r\sin\varphi\sin \theta \ \ \ \ \ \ \ \ \ \ \ \ 0\le \theta \le 2\pi\\
z=r\cos\varphi\ \ \ \ \ \ \ \ \ \ \ \ \ -\infty < z<+\infty\\
\end{array} \right. $，体积微元：$\text dv=r^2\sin\varphi\ \text dr\text d\varphi\text d\theta$
$$
\iiint_\varOmega f(x,y,z)\text dz=\iiint_\varOmega f(r\sin\varphi\cos \theta,r\sin\varphi\sin \theta,r\cos\varphi)r^2\ \text dr\text d\theta\text dz
$$
对称性：若$\varOmega$关于$xOy$对称，则$\iiint_{\varOmega}{f}\left( x,y,z \right) \text{d}z=\left\{ \begin{array}{l}
2\iiint_{\varOmega _1}{f}\left( x,y,z \right) \text{d}z\ \ \ \ \ \ f\text{为偶函数}\\
0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f\text{为奇函数}\\
\end{array} \right. $
线面积分
弧长线积分(第一类线积分)
$$\int_Lf\text ds$$
性质
与积分路径无关：$\int_{L\left( AB \right)}{f\left( x,y \right)}\text{d}s=\int_{L\left( BA \right)}{f\left( x,y \right)}\text{d}s$
轴对称性：
- $L$关于$y$轴对称$\Rightarrow$$\int_L{f\left( x,y \right)}\text{d}s=\left\{ \begin{array}{l} 2\int_{L:x\ge 0}{f\left( x,y \right)}\text{d}s\ \ \ \ \ f\left( x,y \right) \text{关于}x\text{为偶函数}\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f\left( x,y \right) \text{关于}x\text{为奇函数}\\ \end{array} \right. $
- $L$关于$x$轴对称$\Rightarrow$$\int_L{f\left( x,y \right)}\text{d}s=\left\{ \begin{array}{l} 2\int_{L:y\ge 0}{f\left( x,y \right)}\text{d}s\ \ \ \ \ f\left( x,y \right) \text{关于}y\text{为偶函数}\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f\left( x,y \right) \text{关于}y\text{为奇函数}\\ \end{array} \right. $
轮换对称性：$L$关于$y=x$对称，则$\int_Lf(x,y)\text ds=\int_Lf(y,x)\text ds$
计算
- $L:\left\{ \begin{array}{l} x=x\left( t \right)\\ y=y\left( t \right)\\ \end{array}\left( \alpha \le t\le \beta \right) \right. \\\int_L{f\left( x,y \right)}\text{d}s=\int_{\alpha}^{\beta}{f\left( x\left( t \right) ,y\left( t \right) \right) \sqrt{x'^2\left( t \right) +y'^2\left( t \right)}}\text{d}t$
- $L:y=y(x)\ (a\le x\le b)\\\int_L{f\left( x,y \right)}\text{d}s=\int_a^b f(x,y(x))\sqrt{1+y'^2(t)}\text dx$
- $L:r=r(\theta)(\alpha\le\theta \le\beta)\\\int_Lf(x,y)\text ds=\int_{\alpha}^{\beta}f(r\cos\theta,r\sin\theta)\sqrt{r^2+r'^2}\text d\theta$
坐标线积分(第二类线积分)
$$\int_L P\text dx+Q\text dy$$
性质
与路径方向有关：$\int_{AB} P\text dx+Q\text dy=-\int_{BA} P\text dx+Q\text dy$
线积分与路径无关判定：以下等价
- 线积分$\int_LP\text dx+Q\text dy$与路径无关
- $\oint_LP\text dx+Q\text dy=0$，其中$L$为$D$光滑曲线
- $\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$
- $P(x,y)\text dx+Q(x,y)\text dy=\text dF(x,y)$
两类线积分联系：$\oint_LP\text dx+Q\text dy=\oint_L(P\cos\alpha+Q\cos\beta)\text ds\\(\cos \alpha =\pm \dfrac{\varphi '}{\sqrt{\varphi '^2+\varPsi '^2}}\ \ \ \ \cos \beta =\pm \dfrac{\varPsi '}{\sqrt{\varphi '^2+\varPsi '^2}})\left\{ \begin{array}{l} \alpha <\beta \ \ \ \ +\\ \alpha >\beta \ \ \ \ -\\ \end{array} \right. $
计算
直接法：$L:\left\{ \begin{array}{l} x=x\left( t \right)\\ y=y\left( t \right)\\ \end{array} \right. t\in \left[ \alpha ,\beta \right] $，则$\int_LP\text dx+Q\text dy=\int_{\alpha}^{\beta}(P(x,y)x'+Qy'(t))\text dt$
格林公式：区域$D$由光滑曲线$L$围成，则$\oint_L{P\text{d}x+Q\text{d}y}=\iint_D{\left( \dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y} \right)}\text{d}\sigma $
改变路径：（平行于坐标轴的直线积分）
$\int_{(x_1,y_1)}^{(x_2,y_2)}P\text dx+Q\text dy$$=\int_{x_1}^{x_2}P(x,y_1)\text dx+\int_{y_1}^{y_2}Q(x_2,y)\text dy\\=\int_{x_1}^{x_2}P(x,y_2)\text dx+\int_{y_1}^{y_2}Q(x_1,y)\text dy$特例：$L:y=y(x)\ \ a\le x \le b$ $\int_LP\text dx+Q\text dy=\int_a^b(P+Qy’(x))\text dx$
特例：$L:x=x(y)\ \ c\le y \le d$ $\int_LP\text dx+Q\text dy=\int_a^b(Px’(y)+Q)\text dx$
斯托克斯公式
$\oint_LP\text dx+Q\text dy+R\text dz$$=\iint_{\sum}\left| \begin{matrix} \cos \alpha& \cos \beta& \cos \gamma\\ \dfrac{\partial}{\partial x}& \dfrac{\partial}{\partial y}& \dfrac{\partial}{\partial z}\\ P& Q& R\\ \end{matrix} \right|\text{d}S\\=\iint_{\sum}\left( \dfrac{\partial R}{\partial y}-\dfrac{\partial Q}{\partial z} \right) \text{d}y\text{d}z+\left( \dfrac{\partial P}{\partial z}-\dfrac{\partial R}{\partial x} \right) \text{d}z\text{d}x+\left( \dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y} \right) \text{d}x\text{d}y$对面积面积分(第一类面积分)
$$\iint_\sum f(x,y,z)\text dS$$
性质
与积分曲面方向无关：$\iint_\sum\iint_{-\sum}$
奇偶性：$\sum$关于$xOy$对称，则$\iint_\sum f(x,y,z)\text dS=\left\{ \begin{array}{l} 2\iint_{\sum{}}{f\left( x,y,z \right)}\text{d}S\ \ \ \ f\text{是关于}z\text{的偶函数}\\ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f\text{是关于}z\text{的奇函数}\\ \end{array} \right. $
计算
$\iint_\sum f(x,y,z)\text dS=\iint_{D_{xy}}f(x,y,z(x,y))\sqrt{1+z’^2_x+z’^2_y}$
对坐标面积分(第二类面积分)
$$\iint_{\sum{}}{P\text{d}y\text{d}z+Q\text{d}z\text{d}x+R\text{d}x\text{d}y}$$
性质
与积分曲面方向无关：$-\iint_\sum\iint_{-\sum}$
两类面积分联系：$\iint_{\sum{}}{\left( P\cos \alpha +Q\cos \beta +R\cos \gamma \right)}\text{d}S=\iint_{\sum{}}{\left( P\text{d}x\text{d}y+Q\text{d}z\text{d}x+R\text{d}x\text{d}y \right)}$
计算
直接法：有向曲面$\sum:z=z(x,y)$，则$\iint_\sum R(x,y,z)\text dx\text dy=±\iint_\sum R(x,y,z(x,y))\text dx\text dy$
高斯公式：闭区域$\varOmega$，则$\iint_{\mathbb{}}\kern{-12pt}\subset\kern{0pt}_\sum P\text dy\text dz+Q\text dz\text dx+R\text d\text y=\iiint_\varOmega \left( \dfrac{\partial P}{\partial x}+\dfrac{\partial Q}{\partial y}+\dfrac{\partial R}{\partial z} \right) \text{d}v$
| $\cos\alpha$ | $\cos\beta$ | $\cos\gamma$ | 封闭曲面 |
|---|---|---|---|
| $>0$前侧 | $>0$右侧 | $>0$上侧 | $>0$外侧 |
| $<0$后侧 | $<0$左侧 | $<0$下侧 | $<0$内侧 |
