2024 Autumn - Mathematical Analysis B1(MATH1006.04)

Lecturer: Wang Huting

Time and Classroom: 1(3,4) 3(3,4) 5(1,2) at 5201

Textbook: Mathematical Analysis Vol.1 - Cheng Yi, Chen Qing and Li Ping

This page is for extended content of the course for the ones who are interested in deeper and broader knowledge.

For the basic information of this course, check the course website here.

Chapter 1 (Limit)

• 1.1 Real Numbers
  • The conception of countability and uncountability, cardinality of a set
  • Three ways to construct real numbers: the equivalence class of rational Cauchy sequence, infinite decimal, dedekind cut
  • Basic definitions about topology on the real line, the sixth axiom on completeness of the real number system: finite covering condition
• 1.2 Limit of a Sequence
  • Toeplitz’s Theorem (generalized Stolz’s theorem): try to use this to solve Example 1.2.20 from Page 25 and exercise 11 from page 53 in the textbook.
  • Wallis’ Formula
  • Stirling’s Formula: exercise 1(4) from Page 25 and exercise 10(2) from Page 53 are direct results from this.
• 1.3 Limit of a Function

1st Recitation Class: Note here

Chapter 2 (Continuity)

• 2.1 Basic Concepts
  • More general definition of continuity: topological language
  • Semi continuous function
• 2.2 Properties of Continuous Function on a Closed Interval
  • Some more properties of continuous function

Chapter 3 (Derivative)

• 3.1 Derivative
  • Everywhere continuous but nowhere differentiable function: Weierstrass function
• 3.2 Differential
• 3.3 Differential Mean Value Theorem
  • Darboux function: the function with intermediate property
• 3.4 Limit of Indeterminate Forms
• 3.5 Monotonicity and Convexity
  • Bounded convex function is constant
  • A convex function is differentiable at all but countably many points
• 3.6 Taylor Expansion
  • Analytic function: the function in exercise 3.6.10 is not analytic

5th Recitation Class: Note here

Chapter 4 (Indefinite Integral)

• 4.1 Basic Methods in Solving Indefinite Integral
  • Nonelementary indefinite integral
• 4.2 Indefinite Integral of Rational Function

Chapter 5 (Definite Integral)

• 5.1 Integration
  • Lebesgue integral
• 5.2 Integrability
• 5.3 Basic Applications
  • Calculating moment of inertia using integral
  • Minkowski’s inequality
• 5.4 Improper Integral

7th Recitation Class: Note here

Chapter 6 (Ordinary Differential Equation)

• 6.1 First Order Differential Equation
• 6.2 Second Order Linerar Differential Equation
  • More on (high order) constant coefficient linear equation

10th Recitation Class: Note here

Chapter 7 (Infinte Series)

• 7.1 Number Series
  • Advanced tests for convergence
• 7.2 Function Series
• 7.3 Power Series and Taylor Expansion
• 7.4 Applicatoin of Series