多变量与向量分析
学分: 6
现实世界中的大多数模型有多个变量；微积分方法可以解答现实世界中发生的问题。通常，这些问题需通过坐标函数（如说明热量分布或速度势位的坐标函数）、同时运用多变量函数理论中的偏微分法或重积分法解答。建立了两个或多个变量函数平稳点的理论和分类方法，可以识别最大值和最小值，包括受约束的函数。介绍了微分算子梯度、散度、旋度及拉普拉斯。这些定理尤其用于有关线积分、面积分和体积积分的积分定理（散度定理和格林和斯托克斯定理），并用于物理守恒定律的数学公式。本课程将详述应用数学和分析学使用的基本概念。
在本课程结课时，学生应能够：
• 运用多实元函数微积分的概念和基本计算方法
• 运用多种分析工具和数值法解答多实元函数微积分问题，即发现并分析多元函数的稳定点
• 计算线性和多重（面、体）积分计算笛卡儿曲线坐标和正交曲线坐标中的梯度、散度、旋度及拉普拉斯算
• 说明并运用向量分析中的积分定理，即格林定理、斯托克斯定理和散度定理
• 认识保守向量场及其属性
Multivariable and Vector Analysis 多变量与向量分析
Credits: 6
Most models of real-world situations depend on more than one variable and the techniques of calculus can be extended to solve problems arising in such situations. Typically these are problems whose solutions are functions of position, describing, for example, heat distribution or velocity potential, and involve the partial differentiation or multiple integration of functions of more than one variable. The theory and classification of stationary points of functions of two or more variables is developed allowing maxima and minima, including those subjects to constraints, to be identified.
The differential operators div, grad, curl and the Laplacian are introduced. These are used in particular in the integral theorems (the Divergence theorem and the theorems of Green and Stokes) that relate line, surface and volume integrals and are used in the mathematical formulation of physical conservation laws.
This module develops fundamental ideas that are used both in applied mathematics and in the development of analysis.
By the end of this module, students should be able to:
• use the notation and basic manipulative techniques of the calculus of functions of several real variables
• apply a variety of analytic and numerical techniques to solve problems in the calculus of several real variables, e.g. to find and analyse the stationary points of functions of more than one variable
• evaluate line and multiple (surface and volume) integrals
• evaluate grad, div, curl and the Laplacian in Cartesian and orthogonal curvilinear coordinates
• State and apply the integral theorems of vector analysis, namely Stokes' and Green’s theorems and the divergence theorem
• recognise conservative vector fields and their properties
