# 曲线积分

${\displaystyle W=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {s} }$

## 向量分析

### 标量场的曲线积分

#### 定义

${\displaystyle \int _{C}f\,\mathrm {d} s=\int _{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,\mathrm {d} t.}$

f称为积分函数C是积分路径。不严格地说，ds可以被看作积分路径上的一段很小的“弧长”。曲线积分的结果不依赖于参量化函数r

### 向量场的曲线积分

${\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot \,\mathrm {d} \mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,\mathrm {d} t.}$

### 与路径无关的条件

${\displaystyle \nabla G=\mathbf {F} ,}$

${\displaystyle {\operatorname {d} \over \operatorname {d} \!t}G{\bigl (}\mathbf {r} (t){\bigr )}=\nabla G{\bigl (}\mathbf {r} (t){\bigr )}\cdot \mathbf {r} '(t)=\mathbf {F} {\bigl (}\mathbf {r} (t){\bigr )}\cdot \mathbf {r} '(t)}$

${\displaystyle \int _{C}\mathbf {F} (\mathbf {r} )\cdot \,\mathrm {d} \mathbf {r} =\int _{a}^{b}\mathbf {F} {\bigl (}\mathbf {r} (t){\bigr )}\cdot \mathbf {r} '(t)\,\mathrm {d} t=\int _{a}^{b}{\frac {\mathrm {d} G{\bigl (}\mathbf {r} (t){\bigr )}}{\mathrm {d} t}}\,\mathrm {d} t=G{\bigl (}\mathbf {r} (b){\bigr )}-G{\bigl (}\mathbf {r} (a){\bigr )}}$

## 複曲线积分

${\displaystyle \int _{L}f(z)\,\mathrm {d} z}$

${\displaystyle \sum _{k=1}^{n}f(\gamma (t_{k}))[\gamma (t_{k})-\gamma (t_{k-1})]=\sum _{k=1}^{n}f(\gamma _{k})\Delta \gamma _{k}.}$

${\displaystyle \gamma }$连续可微时，曲线积分可以用一个实变函数的积分表示：

${\displaystyle \int _{L}f(z)\,\mathrm {d} z=\int _{a}^{b}f{\bigl (}\gamma (t){\bigr )}\,\gamma \,'(t)\,\mathrm {d} t.}$

${\displaystyle L}$为闭合曲线时，积分的起点和终点重合，这时${\displaystyle f}$沿${\displaystyle L}$的曲线积分通常记作

${\displaystyle \oint _{L}f(z)\,dz}$

${\displaystyle \int _{L}f{\overline {\mathrm {d} z}}={\overline {\int _{L}{\overline {f}}\mathrm {d} z}}=\int _{a}^{b}f{\bigl (}\gamma (t){\bigr )}\,{\overline {\gamma '(t)}}\,\mathrm {d} t.}$

### 例子

{\displaystyle {\begin{aligned}\oint _{L}f(z)\,\mathrm {d} z&=\int _{0}^{2\pi }{1 \over e^{it}}ie^{it}\,\mathrm {d} t=i\int _{0}^{2\pi }e^{-it}e^{it}\,\mathrm {d} t\\&=i\int _{0}^{2\pi }\,\mathrm {d} t=i(2\pi -0)=2\pi i.\end{aligned}}}

## 注释

1. ^ 分为曲线积分curve integralcurvilinear integral）或路徑積分path integral或contour integral，参考留数定理
2. ^ 路径积分中当积分路径为闭合曲线时，又称为环路积分围道积分

## 参考文献

1. ^ Ahlfors, Lars. Complex Analysis 2nd edition. : 103.