数学与应用数学-格林公式的应用.pdf

格林公式的应用 摘 要 格林公式是英国著名的数学家、物理学家乔治格林在1928年提出的。格林公式 及其应用是高等数学的重要内容之一，在多元积分学教学内容体系中处于承上启下、 承前启后的地位。格林公式与牛顿- 莱布尼茨公式、高斯公式、斯托克斯公式，都体现了整体运算与边界运算之间的联系 ，为二重积分的进一步理论研究和实际应用提供了新途径。在使用“ 曲线积分与路线 的无关性”时，要求积分区域是单连通的，从而利用格林公式计算得到任意封闭曲线 的积分为零，但如何计算复连通区域内的曲线积分的问题却很少。因此本文对格林公 式的各种应用做出归纳，并研究在具有有限多个孤立奇点的区域上，如何运用格林公 式简化较为复杂的第二类曲线积分的计算问题。 关键词：格林公式；曲线积分；封闭曲线；复连通 The Application of the Green’s Formula ABSTRACT The Green's formula is a famous British mathematician and physicist George green put forward in 1928. Green's formula and its application is one of the important contents of higher mathematics. Green's formula, Newton-leibniz formula, Gauss formula and Stoke formula all embody the connection between the integral operation and boundary operation, which provides a new way for the further theoretical research and practical application of double integral. When "independence between curve integral and route" is used, it is required that the integral region is simply connected, so the integral of any closed curve is obtained by using green's formula to be zero. However, how to calculate the curve integral in the complex connected region is rarely discussed. Therefore, this paper summarizes the various applications of green's formula, and studies how to use green's formula to simplify the calculation of the second type of complicated curvilinear integral in the region with a finite number of isolated singularities. Keywords ：Green formula; Curve integral; closed curve; reconnected 目 录 一、引言 1 二、利用格林公式计算积分 1 （一）积分区域为闭合区域1 1．计算曲线积分 1 2 ．计算二重积分 2 （二）积分区域为非闭合区域3 （三）积分区域为复连通区域5 1．挖洞法 5