4 个星期速通微积分。总的学习内容被分成了 20 个模块，通过跳过部分内容，可以按照两种难度（入门/进阶），三种强度（每周 5 小时/ 10 小时/ 15 小时）定出 6 种学习方案。

隔壁生物医学工程系有一个成像科学的博士专业，招来的学生本科专业很杂，医学、生物、物理、数学、计算机……数学水平参差不齐。

为了让科研牛马们能尽快出活儿，他们在正规课程之外加开了一个数学速成班 (math crash course)。

计划用 2 个月的时间让没学过的人速通/学过的人复习线性代数和微积分，其中微积分 4 个星期。

总的学习内容被分成了 20 个模块，通过跳过部分内容，可以按照两种难度（入门/进阶），三种强度（每周 5 小时/ 10 小时/ 15 小时）定出 6 种学习方案。

本文把 calculus 翻译成了“微积分”，国内对这部分内容进行教学的课程，在偏数学的专业叫做“数学分析”，偏自然科学的专业叫做“高等数学”。

我觉得“高等数学”是比“微积分”更信达雅的翻译~~，数分是什么，真不熟~~。

翻译成“微积分”，这部分内容显然不止微分和积分，串联起两者的极限概念更基本，却没出现在名字里；

翻译成“高等数学”，”calcul-” 词根表示算术，”-us” 词尾是拉丁语中的名词后缀，两者一结合，显示了当时的英国精英面对百姓甚至同侪时，想要/需要/能够用拉丁语为自己的成果抬身价，“高等”二字精准挑明了这种自卑与自负一体两面的心态～

最终还是选择了前者，无非是不想在标题这种没有上下文的地方，无意中认领这种价值取向和心态。

模块

模块初学进阶
M1三角函数、简介简介、函数的导数
M2极限连续性、函数的导数求导法则 1
M3求导法则 1求导法则 2
M4线性近似、中值定理偏导数、切平面
M5求导法则 2积分思想、定积分
M6函数的极值微积分基本原理、不定积分
M7洛必达法则、最优化反常积分、概率密度
M8偏导数分部积分、概率分布的高阶矩
M9链式法则、多元函数的极值多元积分、联合概率分布
M10极坐标和复数变坐标系积分、随机变量变换
M11积分思想Delta 函数、复指数函数
M12定积分采样与 Delta 函数
M13微积分基本原理卷积
M14不定积分单点散布函数 (point spread)
M15三角积分、三角代换傅里叶变换 1
M16反常积分、概率密度傅里叶变换 2
M17分部积分、概率期望周期信号的傅里叶变换
M18分部积分、概率分布的高阶矩傅里叶变换的性质 1
M19多元积分、联合概率分布傅里叶变换的性质 2
M20变坐标系积分、随机变量变换奈奎斯特-香农采样定理

学习方案

难度精力第 1 周第 2 周第 3 周第 4 周
入门每周 5 小时M2, M3M5, M8M9, M11M12, M13
每周 10 小时M2, M3, M5M6, M8, M9M11, M12, M13M16, M17, M19, M20
每周 15 小时M1 — M5M6 — M10M11 — M15M16 — M20
选学进阶难度的各章
进阶每周 5 小时M2, M3M4, M5M6, M7M10, M11
每周 10 小时M3, M4, M5M7, M8, M9M10, M12, M13M14, M15, M17, M18
每周 15 小时M1 — M5M6 — M10M11 — M15M16 — M20

教材

课程使用的课本有三个：

Stewart J., Clegg D.K., Watson S. (2021). Calculus (9th ed.). Cengage.

Oppenheim A.V., Willsky A.S. (2016). Signals and Systems (2nd edition). Pearson

Prince J.L., Links J.M. (2022). Medical Imaging Signals and Systems (2nd edition). Pearson.

笔记

Introduction to Differentiation and Integration

  • [M1] Introduction and Derivative of a function
    • Tangent and Velocity:
      • Read Chapter 2.1: The Tangent and Velocity Problems (p. 78–81)
      • Complete the following exercises: 3, 7, 9 (p. 82)
    • Limit of a function:
      • Read Chapter 2.2: The Limit of a Function (p. 83-91)
      • Complete the following exercises: 7, 8, 23, 25,28 (p. 92-94) ****
      • Video: Limits
    • Continuity:
      • Read Chapter 2.5: Continuity (p.115-123).
      • Complete the following exercises: 12, 17, 20, 43, 47 (p. 124-126)
    • Derivative of a Function:
      • Read Chapter 2.7: Derivatives and Rates of Change (pg. 140-148).
      • Complete the following **exercises: 5, 6, 7, 8, 12 (**p.149-152)
      • Video: Derivatives
    • Powers and Polynomials:
      • Read Chapter 3.1: Derivatives of Polynomials and Exponential Functions (p. 174-181).
      • Complete the following **exercises: 6, 13, 16, 35 (**p. 181-184)
  • [M2] Derivative Rules: Power, Sine/Cosine, Product/Quotient, MVT, and L'Hopital Rule
    • Product Rule & Quotient Rule:
    • Derivatives of the Trigonometric Functions:
    • Linear Approximation using Derivative of a Function:
      • Read Chapter 3.10: Linear Approximations and Differentials (p.254-258).
      • Complete the following exercises: 4, 5, 17, 51 (p.258-260)
      • Video: Linear Approximation
    • Mean Value Theorem (MVT):
      • Read Chapter 4.2: The Mean Value Theorem (p.290-305).
      • Complete the following exercises: 5, 13, 25, 34 (p.305-309)
      • Video: Mean Value Theorem
    • L’Hôpital’s Rule:
      • Read Chapter 4.4: Indeterminate Forms and l’Hospital’s Rule (p. 309-319).
      • Complete the following exercises: 2, 5, 14, 45 (p.316-319)
      • Video: l’Hospital’s Rule
  • [M3] Derivative Rules: Chain and Inverse
    • The Chain Rule:
    • Inverse Functions and Their Derivatives:
    • Derivatives of Logarithmic and Inverse Trigonometric Functions:
      • Read the Chapter 3.6: Derivatives of Logarithmic and Inverse Trigonometric Functions (p. 217-223) and
      • complete the following exercises: 11, 18, 28, 62, 71 (p. 224-225)
      • Video: Derivative of Logarithmic Functions
  • [M4] Partial Derivatives and Tangent Planes
  • [M5] Idea of the Integral and Definite Integrals
    • Idea of the Integral
      • Read Chapter 5.1 (p.371-p.381), which introduces the integral and relates it to the problems of finding the area under a smooth curve and computing total distance traveled from velocity.
      • Complete the following exercises: 5, 11, 15 (p.381-p.383)
      • Challenge problems: 17, 25, 34 (p.383-p.384)
    • The Definite Integral
      • Read Chapter 5.2 (p. 384-394), which formally introduces the definite integral and describes it properties and techniques on how to compute it.
      • Complete the following exercises: 3, 7, 11, 35, 41, 43, 65, 67 (p.394-397)
      • Challenge problems: 23, 29, 47 (p.395-p.396)

Application of Integration

  • [M6] The Fundamental Theorem of Calculus and Indefinite Integrals
    • The Fundamental Theorem of Calculus
    • Antiderivative
    • Indefinite Integrals
      • Read Chapter 5.4 (p. 409-415), which introduces the indefinite integral and its properties.
      • Complete the following exercises: 5, 9, 15, 27, 31 (p.415)
      • Challenge problems: 49, 53, 57 (p.415-p.416)
  • [M7] Improper Integrals & Probability Densities
  • [M8] Integration by Parts & Expectation Values
  • [M9] Integration of Multiple Variables & Joint Probability Distributions
    • Integrals of Multiple Variables
      • Double Integrals
        • Read Chapter 14.1 (p.521-p.526)
        • Complete the following exercises: 4, 29, 31 (pg. 526-527)
      • Triple Integrals
        • Read Chapter 14.3 (p.536-p.539).
        • Complete the following exercises: 3, 15 (pg. 540)
    • Joint Probability Distributions
  • [M10] Integral Change of Coordinates and Transformations of Random Variables

Introduction to Signals and Systems

Properties of Fourier Transform

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