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所在平台: Udemy |
课程主页: https://www.udemy.com/course/advanced-calculus-part-one-the-fundamentals/
课程评论：没有评论
**课程名称：研究生微积分** **课程概述：** 本课程旨在为你提供严格的数学基础，深入探讨微积分背后的数学原理。你将接触到本科微积分中已知的定理的证明，并学习更深奥的概念，例如紧集、一致连续、Lipschitz 连续、极限上确界（limit supremum）等，所有这些都将在度量空间的通用框架下进行。通过完成选自实际考试的困难的家庭作业，你将为研究生资格考试（又称 Prelim 考试）做好充分准备。本课程对 GRE 数学专业科目考试也大有裨益。 课程将教授如何撰写准确严谨的证明。讲师将展示如何着手解决问题，如何将零散的观察整合成一个完整的证明。通过讲师亲自示范的例子，你将学会如何以连贯且严谨的方式呈现解决方案，以满足研究生（在资格考试中）的标准。 **第一部分内容涵盖：** * 度量空间中的序列及其极限 * 极限上确界（limit supremum）和极限下确界（limit infimum） * 连续性与半连续性 * 级数与收敛判别法 * 拓扑学中连续性的定义 * 紧集上的连续函数 * 连通集上的连续函数 * 局部性质 * 连续性模（modulus of continuity） * 一致连续函数、Lipschitz 映射和 Hölder 映射、绝对连续函数 **未来可能开设的课程：** * 单变量微分积分学（包括逐点收敛的函数序列和级数、等度收敛、幂级数、解析函数和泰勒级数） * 多变量微积分（需要一个完整的独立课程，甚至可能需要两个课程） **课程材料组织方式：** * **视频讲座：** 每部分（共7部分）都包含讲座，涵盖定义、关键示例和反例，以及最重要的定理及其证明（如果证明具有启发性）。 * **包含Prelim试题的家庭作业：** 每部分的第一个讲座包含一份可下载的 PDF 文件，其中包含过去 Prelim 考试的试题（或难度相当的试题）。这些就是你的家庭作业！鼓励你投入尽可能多的时间尝试独立寻找解决方案。 * **部分练习题的解答：** 在每部分的最后一个视频讲座中，讲师将演示家庭作业中标有“*”号的练习题的解法。讲师不会直接给出最终的简洁答案，而是会引导你经历（最初杂乱无章的）发现解决方案的过程。讲师将教授如何做出小的观察，然后将它们组合起来形成一个解决方案。最后，讲师将展示如何以严谨的方式书写数学内容，以满足考试和数学惯例的标准。这种写作技巧不容忽视。 讲师对本课程上线感到非常兴奋，并期待你的反馈。
This course provides you with the rigorous mathematics that underlies calculus. You will see proofs of results you have seen in undergraduate calculus and be introduced to much deeper notions, such as compact sets, uniform continuity, Lipschitz continuity, limit supremum, etc., all in the generality of metric spaces as well. By working on difficult homework questions, chosen from actual exams, you will be ready to take Qualifying, a.k.a. Prelim, exams in graduate schools. This course can also help with GRE in math subject.You will learn how to write accurate and rigorous proofs. I will show you how to approach a problem and how to bring together scattered observation to formulate a proof. Then, via examples that I do myself, I show you how to present your solutions in a coherent and rigorous way that will meet the standard expected of graduate students (in qualifying exams).In this part 1, we coversequences, and their limits (in metric spaces)limit supremum and limit infimumcontinuity and semi-continuityseries, convergence teststopological definition of continuitycontinuous functions on compact setscontinuous functions on connected setslocal propertiesmodulus of continuityuniformly continuous functions, Lipschitz and Hölder maps, absolutely continuous functionsA possible future course will cover single-variable differential and integral calculus: uniform convergence of sequences and series of functions, equi-continuity, power series, analytic functions and Taylor series. A whole separate course (or even two) is needed to cover multivariable calculus - but that is far into future.Here is how material is organized:Video Lectures. Each section (7 total) begins with lectures that cover definitions, provide key examples and counter-examples, and present the most important theorems, accompanied with proofs if the proofs are instructive.Homework sets with prelim questions. The first lecture of each section contains a downloadable PDF containing question from past prelim exam (or at a comparable difficulty level to them). This is your homework! You are encouraged to spend as much time as you can to try to find solutions on your own.Solutions to select exercises from homework sets. In the last video lectures of each section, I show solutions to the *-ed exercises from the homework. I do not just give you a final clean solution. Instead, I walk you through the (initially messy) process that leads one to discover a solution. I teach you how to make small observations and then bring them together to form a solution. Finally, I show you how to write math in a rigorous way that will meet standards of exams and of mathematical conventions. This latter skill cannot be overlooked.I am so excited to have this course out and cannot wait for your feedback.
