Integral Special Notes - Definition of Integral

一、定积分

$f(x)$​ 定义在 $[a,b]$​,any points $[a,b]$​ for the small area, 分点$a=x_0<x_1<x_2<\cdots <x_n=b$​,称为 $[a,b]$​​​ 的一个分划.

若 $\exists I\in \mathbf{R}$​​​, 对 $[a,b]$​​​ of any division and $\forall {\xi }_i\in \left[x_{i-1},x_i\right]$​​​ made and $\sum _{i=1}^{n}f\left({\xi }_i\right)\Delta x_i$​​​,均有

$\underset{\lambda \to 0}{\lim }\sum _{i=1}^{n}f\left({\xi }_i\right)\Delta x_i=I\left(\lambda =\underset{1\le i\le n}{\max }\Delta x_i\right)$,

则称 $f(x)$​​​ 在 $[a,b]$​​​ 可积,记为 $f\in R[a,b]$,$I$​ 称为 $f(x)$​ 在 $[a,b]$​ 的定积分,记为 $I={\int }_a^{b}f(x)dx$​​​ .

对于定积分 $I={\int }_a^{b}f(x)dx$​,$b$​​ Called the upper limit of points,$a$​ called the lower limit of integration,$x$​ called the integral variable,$dx$​​ called integral calculus.

二、二重积分

设 $D$ 是 $xOy$ A bounded closed region of a plane,函数 $z=f(x,y)$ 在 $D$ 定义,$I$ 为实数,

If several curves will be used $D$ Arbitrarily divided into small areas $\Delta D_1,\Delta D_2,\cdots ,\Delta D_n$,

任取 $\left({\xi }_i,{\eta }_i\right)\in \Delta D_i(i=1,2,\cdots ,n)$,$\Delta {\sigma }_i$ 表示 $\Delta D_i$ 的面积,

$f\left({\xi }_i,{\eta }_i\right)\Delta {\sigma }_i$ called the integral element,Sum the integral elements to get the following integral sum formula： $\sum _{i=1}^{n}f\left({\xi }_i,{\eta }_i\right)\Delta {\sigma }_i$

记 $\lambda =\underset{1\le i\le n}{\max }\left\{d_i\right\}$,$d_i$ is a small area $\Delta D_i$ 的直径,若总有： $\underset{\lambda \to 0}{\lim }\sum _{i=1}^{n}f\left({\xi }_i,{\eta }_i\right)\Delta {\sigma }_i=I$

则称函数 $z=f(x,y)$ 在有界闭区域 $D$ 上可积,$I$ 称为$z=f(x,y)$ 在 $D$ 的二重积分,记为 $\underset{D}{\iint }f(x,y)d\sigma$ .

其中,$\iint -$ Double integral sign,$D-$ 积分区域,$f(x,y)-$被积函数,

$x,y-$积分变量,$f(x,y)d\sigma -$被积表达式,$d\sigma -$area element (面积微元) .

若函数 $f(x,y)$ 在 $D$ 上可积,则 $\underset{D}{\iint }f(x,y)d\sigma =\underset{\lambda \to 0}{\lim }\sum _{i=1}^{n}f\left({\xi }_i,{\eta }_i\right)\Delta {\sigma }_i=I$.

About the definition“总有”的含义：

The function value of the points taken for all small regions,Both working and taking the limit result in a unique number that exists and is deterministic $I$,

且极限 $I$ The value of is related to the region segmentation method and the points in the region $\left({\xi }_i,{\eta }_i\right)$ method is irrelevant.

三、三重积分

设 $\Omega$ 是 $R^3$ A bounded closed region in ,函数 $f(x,y,z)$ 在 $\Omega$ 上定义,$I$ 为实数,If the area $\Delta {\Omega }_1,\Delta {\Omega }_2,\cdots ,\Delta {\Omega }_n$,任取 $\left({\xi }_i,{\eta }_i,{\varsigma }_i\right)\in \Delta {\Omega }_i$,

作和 $\sum _{i=1}^{n}f\left({\xi }_i,{\eta }_i,{\varsigma }_i\right)\Delta V_i(\Delta V_i是\Delta {\Omega }_i的体积)$,The following limits always exist and are unique（It has nothing to do with the three-dimensional division method and the point selection method）:

$\underset{i\to 0}{\lim }\sum _{i=1}^{n}f\left({\xi }_i,{\eta }_i,{\varsigma }_i\right)\Delta V_i=I$ ( 其中 $\lambda =\underset{1\le i\le n}{\max }\left\{d_i\right\},d_i$ is a small area $\Delta {\Omega }_i$ 的直径 ),

则称函数 $f(x,y,z)$ 在 $\Omega$ 可积,$I$ 称为 $f$ 在 $\Omega$ 的三重积分,记为： $\underset{\Omega }{\iiint }f(x,y,z)dV(dV-体积元素)$

若 $\underset{\Omega }{\iiint }f(x,y,z)dV$ 存在,则 $\underset{\Omega }{\iiint }f(x,y,z)dxdydz$

四、第一类曲线积分

设 $f(x,y,z)$ on a bounded surface $\Sigma$ is defined and bounded,若 $\underset{\lambda \to 0}{\lim }\sum _{i=1}^n\left({\xi }_i,{\eta }_i,{\varsigma }_i\right)\Delta S_i$ Limits exist and are unique,

This limit value is called $f(x,y,z)$ on a bounded surface $\Sigma$ 上的第一类曲面积分,又称Quantitative function surface integral.

称 $f(x,y,z)$ 在 $\Sigma$ 上可积,记作 ${\iint }_{\Sigma }f(x,y,z)dS$ .即：
$$\underset{\Sigma }{\iint }f(x,y,z)dS=\underset{\lambda \to 0}{\lim }\sum _{i=1}^n\left({\xi }_i,{\eta }_i,{\varsigma }_i\right)\Delta S_i$$
否则称 $f(x,y,z)$ 在 $\Sigma$ Not integrable.

The limit exists and the only meaning：The sum limit value has nothing to do with the division method of the surface and the way of picking points on the surface.

五、第一类曲面积分

设 $f(x,y,z)$ on a bounded surface $\Sigma$ is defined and bounded,若 $\underset{\lambda \to 0}{\lim }\sum _{i=1}^n\left({\xi }_i,{\eta }_i,{\varsigma }_i\right)\Delta S_i$ Limits exist and are unique,

This limit value is called $f(x,y,z)$ on a bounded surface $\Sigma$ 上的第一类曲面积分,又称Quantitative function surface integral.

称 $f(x,y,z)$ 在 $\Sigma$ 上可积,记作 ${\iint }_{\Sigma }f(x,y,z)dS$ .即：
$$\underset{\Sigma }{\iint }f(x,y,z)dS=\underset{\lambda \to 0}{\lim }\sum _{i=1}^n\left({\xi }_i,{\eta }_i,{\varsigma }_i\right)\Delta S_i$$
否则称 $f(x,y,z)$ 在 $\Sigma$ Not integrable.

The limit exists and the only meaning：The sum limit value has nothing to do with the division method of the surface and the way of picking points on the surface.

六、第二类曲线积分

设向量 $\vec{A}(P)$ in bounded smooth curves ${\Gamma }_{AB}$ 上有定义,且有界（ $\vec{A}(P)$ The components of are bounded functions）,

$\overrightarrow{T^0}(P)$ 表示曲线 ${\Gamma }_{AB}$ 上点 $P$ The unit vector of the tangent at and with the specified direction（由 $A$ 到 $B$ ）一致.

If the first kind of curve integral ${\int }_{{\Gamma }_{AB}}(\vec{A}(P)\cdot \overrightarrow{T^0}(P))ds$ 存在,The value of this integral is called a vector $\vec{A}(P)$ 沿曲线 ${\Gamma }_{AB}$ 由 $A$ 到 $B$ 的第二类曲线积分,又称Vector-valued function curve integral.

七、第二类曲面积分

设 $\Sigma$ 是有界分片光滑曲面,$\vec{A}(x,y,z)=\{P(x,y,z),Q(x,y,z),R(x,y,z)\}$,

定义在 $\Sigma$ 上的向量且有界 ( $P,Q,R$ 有界),$(x,y,z)\in \Sigma$ 处的单位法向量,

$\overrightarrow{n^0}(x,y,z)=\{\cos \alpha ,\cos \beta ,\cos \gamma \}$ 与指定的侧一致,若 $\underset{\Sigma }{\iint }(\vec{A}\cdot \overrightarrow{n^0})dS$ 存在,

该积分值称为 $\vec{A}$ 沿曲面 $\Sigma$ 指定侧的第二类曲面积分或向量值曲面积分.

八、Nicknames for various points

The first kind of curve integral is also called the curve integral of the quantitative function、对弧长的曲线积分.

Surface integrals of the first kind are also called surface integrals of quantitative functions、对面积的曲面积分.

The second kind of curve integral is also called the curve integral of vector-valued functions、对坐标的曲线积分.

The second type of surface integral is also called the surface integral of vector-valued functions、对坐标的曲面积分.