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Copy path线积分定义的一个注.tex
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线积分定义的一个注.tex
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\documentclass[a4paper, 12pt]{article} % Font size (can be 10pt, 11pt or 12pt) and paper size (remove a4paper for US letter paper)
\usepackage{amsmath,amsfonts,bm}
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% ----------------------------------------------------------------------------------------
% TITLE
% ----------------------------------------------------------------------------------------
\begin{document}
\title{\textbf{线积分定义的一个注}}
% \setlength\epigraphwidth{0.7\linewidth}
\author{\small{叶卢庆}\\{\small{杭州师范大学理学院,学
号:1002011005}}\\{\small{Email:h5411167@gmail.com}}} % Institution
\renewcommand{\today}{\number\year. \number\month. \number\day}
\date{\today} % Date
% ----------------------------------------------------------------------------------------
\maketitle % Print the title section
% ----------------------------------------------------------------------------------------
% ABSTRACT AND KEYWORDS
% ----------------------------------------------------------------------------------------
% \renewcommand{\abstractname}{摘要} % Uncomment to change the name of the abstract to something else
% \begin{abstract}
% \end{abstract}
% \hspace*{3,6mm}\textit{关键词:} % Keywords
% \vspace{30pt} % Some vertical space between the abstract and first section
% ----------------------------------------------------------------------------------------
% ESSAY BODY
% ----------------------------------------------------------------------------------------
设函数 $f(x,y,z)$ 在光滑曲线(导函数连续) $l$ 上有定义且连续.$l$ 的方程
为
$$
\begin{cases}
x=x(t),\\
y=y(t),\\
z=z(t)\\
\end{cases}(t_0\leq t\leq T)
$$
则
$$
\int_lf(x,y,z)ds=\int_{t_0}^Tf[(x(t),y(t),z(t)]\sqrt{x_t'^2+y_t'^2+z_t'^2}dt.
$$
\begin{remark}
该定理有明显的物理意义.我们发现 $\sqrt{x_t'^2+y_t'^2+z_t'^2}$ 就是质
点在 $t$ 时刻的运动速度.我认为该定理本质上就是变量替换公式在发挥作用.
\end{remark}
\begin{proof}[证明]
对时间区间 $[t_0,T]$ 进行 $n$ 等分,成为
$$[t_0,t_0+\frac{T-t_0}{n}],\cdots,[t_0+(n-1)\frac{T-t_0}{n},t_0+n
\frac{T-t_0}{n}]$$
在第 $i$ 个时间区间上,任选一个值,这个值是$f(x(t_{i-1}),y(t_{i-1}),z(t_{i-1}))$.根据定义易得
$$
\int_lf(x,y,z)ds=\lim_{n\to\infty}\sum_{i=1}^nf(x(t_{i-1}),y(t_{i-1}),z(t_{i-1}))\Delta S_{i-1}.
$$
其中 $\Delta S_{i-1}$ 是第 $i$ 段时间段走过的弧长,根据微分中值定理,易得
$$
\frac{\Delta S_{i-1}}{\frac{T-t_0}{n}}=\sqrt{(x_{t_0+i\lambda\frac{T-t_0}{n}}'^2+y_{t_0+i\lambda\frac{T-t_0}{n}}'^2+z_{t_0+i\lambda\frac{T-t_0}{n}}'^2)},0<\lambda<1.
$$
结合导函数的连续性,可得
$$
\Delta S_{i-1}=\sqrt{(x_{t_0+i\frac{T-t_0}{n}}'^2+y_{t_0+i\frac{T-t_0}{n}}'^2+z_{t_0+i\frac{T-t_0}{n}}'^2)}\frac{T-t_0}{n}+o(\frac{T-t_0}{n}).
$$
将其代入上上式,得到
\begin{align*}
\int_lf(x,y,z)ds&=\lim_{n\to\infty}\sum_{i=1}^nf(x(t_{i-q}),y(t_{i-1}),z(t_{i-1}))\sqrt{(x_{t_0+i\frac{T-t_0}{n}}'^2+y_{t_0+i\frac{T-t_0}{n}}'^2+z_{t_0+i\frac{T-t_0}{n}}'^2)}\frac{T-t_0}{n}\\&+\lim_{n\to\infty}no(\frac{1}{n})\\&=\lim_{n\to\infty}\sum_{i=1}^nf(x(t_{i-q}),y(t_{i-1}),z(t_{i-1}))\sqrt{(x_{t_0+i\frac{T-t_0}{n}}'^2+y_{t_0+i\frac{T-t_0}{n}}'^2+z_{t_0+i\frac{T-t_0}{n}}'^2)}\frac{T-t_0}{n}\\&=\int_{t_0}^Tf[(x(t),y(t),z(t)]\sqrt{x_t'^2+y_t'^2+z_t'^2}dt.
\end{align*}
\end{proof}
% ----------------------------------------------------------------------------------------
% BIBLIOGRAPHY
% ----------------------------------------------------------------------------------------
\bibliographystyle{unsrt}
\bibliography{sample}
% ----------------------------------------------------------------------------------------
\end{document}