高等数学视角下的弧微分公式推导及曲率公式适用条件  

Derivation of Arc Differential Formula from the Perspective of Advanced Mathematics and Conditions for the Application of the Curvature Formula

作　　者：强静 邵虎 张双圣 QIANG Jing;SHAO Hu;ZHANG Shuangsheng(School of Mathematics,China University of Mining and Technology,Xuzhou Jiangsu 221116,China;College of Environmental Engineering,Xuzhou University of Technology,Xuzhou Jiangsu 221018,China)

机构地区：[1]中国矿业大学数学学院,江苏徐州221116 [2]徐州工程学院环境工程学院,江苏徐州221018

出　　处：《大学数学》2023年第5期98-104,共7页College Mathematics

基　　金：江苏省高等教育教改研究重点项目(2021JSJG117);中国矿业大学教学研究重点项目(2021ZD02);中国矿业大学教学研究项目(2022KCSZ45)。

摘　　要：在利用弧微分公式推导曲率公式时,针对弧微分公式证明过程中“弧长和弦长的比值极限为1”假设不严谨的问题,本文在高等数学知识体系内,利用曲线直角坐标方程得到了弧长与有向弧段的值之间的关系,提出了一种证明弧微分公式的方法.之后,针对四种不同形式的曲线方程,推导并总结了不同方程下的曲率公式及其适用条件,并针对性地给出了两个典型例题.最后,给出了一个运用曲率公式求解实际工程问题的案例.In the derivation of the curvature formula using the arc differential formula,the assumption that the ratio limit of arc length to chord length is 1 is not rigorous in the proof of the arc differential formula.In this paper,in the body of knowledge of Advanced Mathematics,the relationship between the arc length and the value of a directed arc segment was obtained by using the curve rectangular coordinate equation,and a method for proving the arc differential formula was proposed.Then,according to 4 different forms of curve equations,the curvature formulas and their applicable conditions with different equations were deduced and summarized,and two typical examples were given.Finally,a case of using curvature formula to solve a practical engineering problem was given.

关 键 词：曲线方程 参数方程 极坐标 曲率 弧微分公式 

分 类 号：O13[理学—数学]