实分析

学分：6

• 微积分是人类最有意义的科学成就之一，它把之前不可解决的物理问题变成常规计算。虽然微积分源自古代，至 17 世纪晚期才由牛顿在推导他的运动和重力法则时发展， 而莱布尼茨发明了我们今天仍在使用的微积分符号。

• 分析是数学中研究微积分背后理论的分支，通过引入极限把微积分置于牢固的逻辑基础上。本模块课程从这一严谨的角度介绍微分和积分。

• 介绍实变量函数及其导数的概念形式，复习和扩展微分和积分的常用技能和应用，学习简单的一阶与二阶微分方程。

课程结束时，学生应能：

• 陈述函数以及相关符号的定义；会描画一元函数的图像。

• 解基本的不等式，包括涉及二次方程式和系数的不等式。

• 运用标准方法，计算实变量函数的导数和积分。

• 懂得在适当的环境下应用微分和积分。

• 陈述导数的定义，用基本原理计算导数；

• 陈述微积分基本定理，并了解其定理的证明。

• 解简单的一阶与二阶常微分方程。

Real Analysis 实分析

Credits: 6

• Calculus is one of mankind’s most significant scientific achievements,

transforming previously intractable physical problems into often routine calculations. Although its roots trace back into antiquity, it was developed in the late 17th century by Newton, when developing his laws of motion and gravitation, and Leibniz, who developed the notation we still use today.

• Analysis is the branch of mathematics that underpins the theory behind the calculus, placing it on a firm logical foundation through the introduction of the notion of a limit. This module introduces differentiation and integration from this rigorous point of view.

• The notion of a function of a real variable and its derivative are formalised. The familiar techniques and applications of differentiation and integration are reviewed and extended. Simple first and second order ordinary differential equations are studied.

By the end of this module, students should be able to:

• State the definition of a function and related notions and be able to sketch graphs of functions of a real variable.

• Solve basic inequalities, including those involving quadratic terms and moduli.

• Calculate derivatives and integrals of functions of a real variable using standard techniques.

• Apply differentiation and integration in appropriate situations.

• State the definition of the derivative and calculate derivatives from first principles.

• State the Fundamental Theorem of Calculus and have an appreciation of its proof.

• Solve simple examples of first and second order ordinary differential equations.