重积分是高等数学的重要组成部分,不仅在几何、物理上都有着重要的应用,而且在数学、物理学、工程学、统计学等多个学科中都有广泛应用。本文从微积分的思想起源为切入点介绍了重积分的定义和性质,接下来介绍了重积分在几何以及在物理...重积分是高等数学的重要组成部分,不仅在几何、物理上都有着重要的应用,而且在数学、物理学、工程学、统计学等多个学科中都有广泛应用。本文从微积分的思想起源为切入点介绍了重积分的定义和性质,接下来介绍了重积分在几何以及在物理上的应用,最后结合向量给出两类曲线积分的定义和联系。两类曲线积分的知识点在竞赛和考研中频频出现。用向量的观点来看两类曲线积分的联系,可以加深学生的理解,有助于培养学生的数学逻辑思维。Multiple integrals are an important component of advanced mathematics, with significant applications not only in geometry and physics, but also in various disciplines such as mathematics, physics, engineering, and statistics. This article introduces the definition and properties of multiple integrals from the perspective of the origin of calculus. Then, we introduce the applications of multiple integration in geometry and physics. Finally, the definitions and connections between two types of curve integrals are given by combining vectors. The knowledge of two types of curve integrals frequently appears in competitions and postgraduate entrance exams. It can deepen students’ understanding and help cultivate their mathematical logical thinking by the perspective of vectors to find the connection between two types of curve integrals.展开更多
文摘重积分是高等数学的重要组成部分,不仅在几何、物理上都有着重要的应用,而且在数学、物理学、工程学、统计学等多个学科中都有广泛应用。本文从微积分的思想起源为切入点介绍了重积分的定义和性质,接下来介绍了重积分在几何以及在物理上的应用,最后结合向量给出两类曲线积分的定义和联系。两类曲线积分的知识点在竞赛和考研中频频出现。用向量的观点来看两类曲线积分的联系,可以加深学生的理解,有助于培养学生的数学逻辑思维。Multiple integrals are an important component of advanced mathematics, with significant applications not only in geometry and physics, but also in various disciplines such as mathematics, physics, engineering, and statistics. This article introduces the definition and properties of multiple integrals from the perspective of the origin of calculus. Then, we introduce the applications of multiple integration in geometry and physics. Finally, the definitions and connections between two types of curve integrals are given by combining vectors. The knowledge of two types of curve integrals frequently appears in competitions and postgraduate entrance exams. It can deepen students’ understanding and help cultivate their mathematical logical thinking by the perspective of vectors to find the connection between two types of curve integrals.
