微積分二 - Calculus (II)

This course, Calculus (II), is a continuation of Calculus (I) from the previous semester. It delves into the exploration of numbers and shapes, which are fundamental to various scientific and quantitative fields. The course traces the historical development of calculus from ancient methods of calculating areas to its formalization by Newton and Leibniz in the 17th century. It emphasizes the interrelationship between differentiation and integration through the Fundamental Theorem of Calculus, providing tools to solve complex problems. The curriculum covers advancements in calculus made by mathematicians like the Bernoulli family, Euler, and Lagrange, highlighting their applications in physics and engineering. It also addresses the rigor introduced in the 18th and 19th centuries by mathematicians such as Cauchy, Dirichlet, Riemann, and Weierstrass, who sought to solidify calculus's logical foundations. This period saw the development of modern set theory by Cantor, which provides a framework for rigorously defining concepts like limits, continuity, and real numbers. Unlike introductory calculus courses (Calculus I & II), this course (often referred to as Math Calculus) focuses on clarifying key concepts that are often treated intuitively in other courses. It bridges the gap between rigorous logical treatment and intuitive understanding, while still including practical examples, though potentially fewer than in more introductory courses. The aim is to provide a glimpse into the foundational logic underpinning modern mathematical analysis.

Curriculum

The course covers a wide range of topics in Calculus II, organized into 53 lectures. Key areas include functions, limits, continuity, differentiation, integration, ordinary differential equations, multivariable calculus, vector analysis, and curve/surface theory.

1. Full Course (53 Lectures)
   • Introduction (I): Review of Differentiation, Integration, and Series; Reconstructing Logarithmic and Exponential Functions via Integration
   • Introduction (II): Review of Power Series; Exponential Function; Sine and Cosine Functions and their Periodicity; What is π?
   • Introduction (III): Arc Length and Rectifiable Curves; Schwarz's Example of a Surface
   • Introduction (IV): Fundamental Theorem of Algebra
   • Abel's Lemma for Series Rewriting and its Applications
   • Abel's Theorem on Power Series
   • About ODEs
   • Review: Concepts and Geometric Illustrations of ODEs; First Integrals of ODEs
   • Existence and Uniqueness Theorem for ODE Solutions
   • Review: Picard Iteration Method; Existence and Uniqueness Theorem for ODE Solutions
   • Example of an ODE Initial Value Problem where Uniqueness Fails when the Lipschitz Condition is Not Met; Maximal Extension of ODE Solutions
   • Examples of ODEs in Physics: Law of Universal Gravitation, Simple Pendulum
   • First Integrals, Conservative Fields, and Potential Energy
   • Angular Functions: Review of Polar Coordinates; Angular Functions of Continuously Differentiable Planar Motion
   • Autonomous ODEs; Solutions Remaining Bounded in Compact Sets for Infinite Time; Phase Diagrams (Example: Simple Pendulum)
   • Revisiting Solutions of Linear ODEs with Constant Coefficients: Picard Iteration Method vs. Matrix Exponential
   • Solution to Angular Function Problems (Cont. 3/17 (B)); Existence and Uniqueness of Linear ODE Solutions
   • All Continuously Differentiable Planar Motions Have Continuous Angular Functions
   • Discussion on Pendulums: Review of the Simple Pendulum
   • Discussion on Pendulums 2: Huygens' Pendulum
   • Review of Integration Concepts: Upper and Lower Sums and Integrals; Integrable Functions
   • Some Point Set Topology Concepts: Interior, Exterior, and Boundary Points of a Set in a Metric Space
   • Iterated Integrals 1: Fubini's Theorem (Basic Version)
   • Integration Concept: Figures and their Integrals
   • Integration Concept: Fubini's Theorem (Review), Fubini's Theorem (Advanced Version)
   • Multivariable Differentiation Theory 1: Review of Differentiability; Derivative Matrix/Jacobian Matrix; Chain Rule
   • Multivariable Differentiation Theory 2: Gradient Vector; Mean Value Theorem (Mixed Version of Integration and Partial Derivatives)
   • Multivariable Differentiation Theory 3: Local Optimization (Maxima and Minima) and Critical Points; Convexity of Functions; Second Derivative Test for Convexity and Extrema
   • How to Describe Objects in Space 1: Examples of Parameterizing Objects Described by Equations (Implicit Functions and Implicit Differentiation)
   • Multivariable Differentiation Theory 4: Implicit Functions and Implicit Differentiation: Implicit Function Theorem (Single Equation Case)
   • Multivariable Differentiation Theory 5: Lagrange Multiplier Method
   • Multivariable Differentiation Theory 6: Implicit Function Theorem (Multiple Equations Case)
   • Multivariable Differentiation Theory 7: Informal Discussion on the Concept of Implicit Functions and Implicit Differentiation
   • How to Describe Objects in Space 2: Parametric View (Image View) vs. Equation View (Preimage View)
   • Inverse Function Theorem 1: (Uninspired Explanation)
   • Inverse Function Theorem 2
   • Inverse Function Theorem 3: Proof of the Inverse Function Theorem via Iteration; Relationship between Inverse Function Theorem and Implicit Function Theorem
   • Change of Variables in Integration 1: Elementary Maps; Decomposing Change of Variables into a Composition of Elementary Maps
   • Change of Variables in Integration 2: Basic Properties of Maps and Figures
   • Change of Variables in Integration 3: Volume of Spheres in Arbitrary Dimensions; High-Dimensional Spherical Coordinates; Orthogonal Coordinate Systems
   • Improper Integrals in Multiple Variables 1: Absolutely Convergent Improper Integrals
   • Improper Integrals in Multiple Variables 2: Examples of Absolutely Convergent Improper Integrals
   • Basic Concepts of Curves 1: Regular Curves (in Plane or Space); Signed Curvature of Planar Curves; Curvature and Torsion of Space Curves; Frenet Frame
   • Definition of Centroid and Center of Mass (To be used in 'Basic Concepts of Curves 2')
   • Basic Concepts of Curves 2: Application of the Change of Variables Formula with Frenet Frame: Volume Problems
   • Surface Area 1: Definition and Rationale of Surface Area for Parametric Surfaces in 3D Space
   • Surface Area 2: Generalization: How to Define the Volume of an m-dimensional Manifold in n-dimensional Space
   • Classical Vector Analysis 1: Work and Line Integrals of Vector Fields along Given Paths
   • Classical Vector Analysis 2: Concepts of Gradient Fields (Conservative Fields); Necessary Condition for a Vector Field to be a Gradient Field
   • Classical Vector Analysis 3: Informal Discussion on the Concept of Homotopy; Simply Connected Spaces
   • Classical Vector Analysis 4: Detailed Discussion on 'A Vector Field is a Gradient Field if and Only If its Integral along a Continuously Differentiable Curve Depends Only on the Endpoints'
   • Classical Vector Analysis 5: Vector Fields Satisfying the Necessary Condition for a Gradient Field Have Zero Integral along the Boundary Curves of a Square Map
   • Classical Vector Analysis 6: Green's Theorem for Vector Fields in Planar Regions

Careers

Graduates from mathematics programs at National Taiwan University are well-prepared for a variety of careers. The rigorous analytical and problem-solving skills developed through courses like Calculus (II) are highly valued in numerous sectors.

• Data Scientist
• Actuary
• Financial Analyst
• Operations Research Analyst
• Cryptographer
• Software Developer
• Researcher
• Educator

Frequently asked questions

What are the main topics covered in Calculus (II)?

Calculus (II) covers functions, limits, continuity, differentiation, integration, ordinary differential equations, multivariable calculus, vector analysis, and curve/surface theory over 53 lectures.

What is the duration of the Calculus (II) course?

The course is described as 'Full Course (53 Lectures)'.

What are the estimated living costs for students at National Taiwan University?

The estimated cost of living per semester in Taipei is TWD 84,000 – 113,000 (USD 2,800 – 3,767), which includes books, housing, and general expenses.

Are tuition fees included in the living cost estimate?

No, tuition fees are separate from the living cost estimate and can be found on the 'Finances' page.

What are the potential career paths for graduates of mathematics programs?

Graduates are well-prepared for roles such as Data Scientist, Actuary, Financial Analyst, Software Developer, and Researcher, or can pursue graduate studies.