Introduction to Advanced Calculus

所在平台: Coursera

课程主页: https://www.coursera.org/learn/introduction-to-advanced-calculus

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课程简介

课程名称：高级微积分简介 课程概述：本课程“高级微积分简介”是“微积分简介”课程的自然延续，虽然具备一定微积分基础的学生可以直接入读。课程重点关注数学在科学、工程和商业中的应用基础，特别侧重于函数的级数表示以及微分方程理论的引入。 课程大纲： 第一周 - 微分： 本模块回顾了导数的极限定义，深入探讨中值定理和介值定理等基本原理，导致方程近似解法的形成。引入了如洛必达法则和牛顿法等新技术，帮助快速求解方程根。模块最后引入双曲函数，补充了对圆函数的理解。 第二周 - 积分： 本模块回顾曲线下的面积、黎曼和的概念，引入定积分和微积分基本定理，及不定积分。探讨替换积分法，包括一些难题，并重温对数和指数的性质，介绍分部积分法和部分分数积分法，涉及线性代数的相关原理。此外，还介绍了旋转体体积的盘体法和壳体法以及与弧长相关的表面积公式。模块最后探讨了不定积分、多种变体和对比技术，包括涉及托里切利喇叭的画家悖论，提出有限体积但无限表面积的概念。 第三周 - 函数的级数表示： 本模块回顾了与序列相关的概念，包括单调收敛定理，以及级数的定义（作为序列和的极限）。引入几何级数、调和级数和交替调和级数，阐述收敛的比值测试和交替测试。介绍幂级数表示，包括泰勒级数和麦克劳林级数的显式公式，分析重要函数如指数、对数、圆函数和双曲函数的级数表示。通过截断无限级数，研究使用泰勒和麦克劳林多项式的函数近似，这引出了泰勒定理并证明欧拉数e是无理数，交替调和级数收敛到自然对数2。 第四周 - 微分方程简介： 本模块作为微分方程理论的入门，从可分离方程开始，推广到导数与函数值成正比的简单情况，用于建模指数增长和衰减。引入抑制或死亡因素，形成逻辑斯蒂方程及其解，广泛应用于科学和人口动态中。讨论平衡解及其稳定性，并介绍可使用积分因子法求解的一类一阶线性微分方程。最后引入具有常数系数的二阶方程及其解空间的二维性，涉及与线性代数和矩阵指数的连接，概述相互作用的人口模型。 通过本课程，学生将建立起高级微积分的理论框架，为进一步研究数学在各领域的应用打下坚实基础。

课程大纲

Name:Week 1 - Differentiation

Description:This module begins by reviewing limit definitions of the derivative, looking in depth at underlying results and principles such as the Mean Value Theorem and the Intermediate Value Theorem, leading to methods for finding approximate solutions of equations. New techniques are introduced, such as L'Hopital's Rule for finding difficult limits and the lightning fast Newton's Method for homing in on roots of equations. The module finishes by adding hyperbolic functions to the toolkit, complementing existing knowledge of circular functions.

Name:Week 2 - Integration

Description:This module begins by reviewing areas under curves, the method of Riemann sums, leading to definite integrals, and the Fundamental Theorem of Calculus, leading to indefinite integrals. It then reviews integration by substitution, including difficult examples, and revisits logarithms and exponentials and their properties, using the constructive late transcendental method (compared with the existential early transcendental method). The module then introduces the method of integration by parts and the method of partial fractions, including a sketch of underlying related principles from linear algebra. The module then introduces the disc and shell methods for finding volumes of revolution, formulae for finding surface areas of revolutions, related to arc length, and the concept of work from physics. The module finishes with an introduction to improper integrals, their many variations and contrasting techniques, including a discussion of the painter's paradox, involving Torricelli's trumpet, which has a finite volume but infinite surface area.

Name:Week 3 - Series Representations of Functions

Description:This third module begins by reviewing concepts related to sequences, including the Monotone Convergence Theorem, which is used frequently to guarantee convergence of limits and series under certain conditions. The module then introduces series, which are sums of sequences, which go on forever, and defined formally as limits of partial sums, which may or may not converge. Geometric, harmonic and alternating harmonic series are introduced, leading to the Ratio Test and the Alternating Test for convergence. Power series representations are introduced, including explicit formulae for Taylor and Maclaurin series, in terms of iterated derivatives and factorials. Important functions, such as exponential, logarithmic, circular and hyperbolic functions, are analysed, compared and contrasted, from the point of view of series representations. Approximations of functions are studied using Taylor and Maclaurin polynomials, which result by truncating the respective infinite series. This leads to Taylor's Theorem, which enables one to control the quality of the approximation and make predictions using a remainder term. The method is also used to prove Euler's number e is irrational and that the alternating harmonic series converges to the natural logarithm of 2.

Name:Week 4 - Introduction to Differential Equations

Description:This fourth and final module serves as an introduction to the vast theory of differential equations. It begins with the class of separable equations, generalising the simplest cases where the derivative of a function is proportional to the value of the function, used to model exponential growth and decay. Introducing an inhibition or death factor, leads to the logistic equation and its solution, the logistic function, used to model wide ranging phenomena in science and population dynamics. A discussion of equilibrium solutions and their stability ensues. The module then considers a class of first order linear differential equations, which may be solved using an integrating factor method, an instance of the Conjugation Principle, used widely in mathematics to solve difficult problems or avoid obstacles. The module then considers second order equations with constant coefficients, which have solution spaces that are two-dimensional, analogous to planes in space. The module finishes with an introduction to solutions of systems of equations, which model interacting populations, in a symbiotic or predator-prey relationship, including a brief overview of connections with concepts in linear algebra and the matrix exponential.

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