Micro-modules in Mathematics for Introductory Undergraduate Physics

Department of Physics
The Chinese University of Hong Kong

Updated: 19 Sept, 2023

Level 1: Pre-University to Year 1 1st Semester

Basic Mathematical Methods by Dr. Chan Man Ho
Level 2: Year 1 1st Semester to Year 2 1st Semester, and Beyond

Extension Micro-modules in Mechanics by Prof. Kenneth Young
Units and Dimensions
Conversion of Units and Dimensional Analysis
  • Units and Dimensions, Consistency and Conversion of Units, Dimensional Analysis

  • Physical Laws and Natural Units
  • Using Dimensionless Variables, Law of Physics in Dimensionless Variables, Fundamental Constants and Characteristic Scales, Using Natural Units with Examples, Deeper Understanding
  • Coordinate Systems, Vectors and Matrices
    Coordinate Systems ( | | ENG )
  • Cartesian Coordinates in 3D, Cylindrical Coordinates, and Spherical Coordinates
  • ( Supplement | Solution )

    Basics of Vector ( | | ENG )
  • Quantities with Direction and Magnitude, Basic Vector Algebra
  • Dot and Cross Products of Vectors ( | | ENG )
  • Dot and Cross Products, Properties and Applications
  • ( Supplement | Solution )

    Basics of Matrix ( | | ENG )
  • Structure, Matrix Algebra, Inverse and Transpose, Applications
  • Inverse Matrix and Determinant ( | | ENG )
  • Finding Inverse Matrices and Determinants
  • ( Supplement | Solution )
    Common Coordinates Systems and Beyond
  • Introduction, Cartesian Coordinates, Polar Coordinates, Cylindrical Coordinates, Spherical Coordinates, Formulas for Distance, Formulas for Area & Volume; Appendix – Cartesian Coordinates in Higher Dimensions, Spherical Coordinates in Higher Dimensions, Minkowski Space, General Coordinates

  • Vector Algebra
  • Introduction, Algebraic Formulation, Dot Product, Cross Product – Physical Motivation and Main Properties, Expression in Terms of Components, Parity, Some Identities, Appendix – Vectors in N-dimensional Space, Abstract Linear Space and Wedge Product

  • Ellipse
  • Introduction, Sum of Distances from Two Foci, Canonical Formula, Conic Section, Measured from One Focus
  • Elementary Functions
    Exponential Functions and Logarithm ( | | ENG )
  • Euler Number e, Properties of Exponential Functions and Logarithm, Applications
  • ( Supplement | Solution )
    Some Elementary Functions
  • Exponential Function, Logarithm, Hyperbolic Functions, Trigonometric Functions
  • Limit, Differentiation, Power Series, Approximations
    Limit ( | | ENG )
  • Properties and Calculation of Limits, L’ Hospital’s Rule
  • ( Supplement | Solution )

    Differentiation ( | | ENG )
  • Definition, Slope of Graph, Higher Order Derivatives, Applications to Kinematics
  • Rules of Differentiation ( | | ENG )
  • Differentiation of Common Functions, Product Rule and Chain Rule
  • ( Supplement | Solution )

    Approximation ( | | ENG )
  • Concepts and Applications, Taylor’s Series
  • ( Supplement | Solution )
    Power Series and Taylor's Series
  • Power Series, Exponential Function, Differentiation and Integration, Taylor Series, Newton's Method, Appendix: Convergence
  • Integration and Applications
    Definition and Meaning of Integration ( | | ENG )
  • Definition, Area under Curve, Definite Integral, Applications to Kinematics
  • Differentiation and Integration, Common Integrals ( | | ENG )
  • Relationships between Differentiation and Integration, Integrals of Common Functions
  • Methods of Integration ( | | ENG )
  • Change of Variable and Integration by Parts
  • ( Supplement | Solution )
    Definition, Properties and Numerical Methods of Integration
  • Definition and Properties, Numerical Methods – General Formalism, Different Methods of Evaluation, Fundamental Theorem of Calculus, Appendix: Derivation of Numerical Algorithms

  • Methods of Integration
  • Change of Variables, Parameters - Dependence on Parameters, Differentiating with respect to Parameters, Integration by Parts, Integrating Term by Term, Resources

  • Application of Integration to Work and Energy
  • Definition of Work, Potential Energy, Force from Potential Energy, Conservation of Energy – Work and KE, Work and PE, and Significance
  • Line Integral, Curl, Gradient and Applications
      Line Integral
  • Definition of Line Integral and Numerical Evaluation, Reduction to Ordinary Integrals, Other Integrals along a Line

  • Curl and Gradient
  • Dependence on Path?–Reduction to Closed Loops, Condition to be Independent, Stokes' Theorem; Introducing a Scalar Function; The Related Scalar Field and Some Applications

  • Work: Higher Dimensions
  • Definition of Work – Constant Force, Linearity, Variable Force, Line Integral; Reducing to Ordinary Integral; Dependence on Path – Reduction to Closed Loops and Examples, Condition for Conservative Force

  • Energy: Higher Dimensions
  • Potential Energy, Force From PE – Force components in Terms of Partial Derivatives, Gradient Operator, Curl of Gradient is Zero; Equipotential Surfaces, Conservation of Energy
  • Complex Number and Applications
    Basics of Complex Number ( | | ENG )
  • Real Part and Imaginary Part, Polar Representation, Basic Algebra
  • Relationship to Sinusoidal Functions ( | | ENG )
  • Exponential Form, Relationship with Sinusoidal Functions, Time Variation and Phase, Application to Oscillation
  • ( Supplement | Solution )
    Basics of Complex Number
  • Motivation, Arithmetic, Polar Representation; Algebra - Quadratic Equations, Counting Roots and General Polynomials, The Square Root Function and Logarithm Function

  • Differentiation and Integration of Complex Variables
  • Differentiation – Rules, Cauchy-Riemann Conditions, Harmonic Functions; Integration – Rules, Cauchy Integral Theorem, Residue & Meromorphic Functions, Cauchy Integral Formula, Roots; Vector Representation
  • Ordinary Differential Equations in Oscillation Problems
      Simple Harmonic Motion
  • Introduction and Kinematics, Graphical and Complex Representation, Equation of Motion and Solution in Polar Form, Solution in Alternate Form and Matching Initial Condition, Adding a Constant Force, Generality of SHM

  • Ordinary Differential Equations
  • Introduction, Numerical Method; Linearity and Superposition – Linear and Superposition, Real Equation, Complex Solution, and Linear with Constant Coefficients; ODEs with Constant Coefficients; Inhomogeneous – Case Defining the Problem and Splitting into Two Parts, Constant Coefficient, Constant, Inhomogeneity and Harmonic Inhomogeneity, Impulsive Inhomogeneity

  • Damped Harmonic Motion
  • Introduction and Kinematics – the Model, Numerical Solution, Guessing a Solution, Checking the Solution; Complex Method; Different Cases of Damping – Under-damped and Over-damped, Critically Damped; Energy

  • Forced Oscillations
  • Introduction, Case without Damping, Case with Damping – Complex solution, Physical Solution, Comparison with Undamped Case and Work done by External Force

  • Forced Oscillations: Beyond Steady-state Response
  • Introduction, Transients, Secular Solution – Zero Damping, Small Damping; Impulsive Force, Need for Another Method and Impulse, General Solution

  • Application of Oscillations
  • Introduction, IR Spectroscopy, Electrical Circuits, Selective Response

  • Integrated Application: Rotations
      Newton’s Second Law and Angular Momentum
  • Kinematics, KE and Moment of Inertia, Torque and Analog of Newton’s Second Law, Angular Momentum; Rotational Equilibrium; Appendix–Verifying the Energy, Euler-Lagrange Equation

  • Center of Mass
  • Introduction, Position of Center of Mass, Continuous Distribution, Equilibrium and Center of Gravity, Moments about Center of Mass, Motion of Center of Mass, Appendix: Moments

  • Moment of Inertia
  • Introduction and Simple Cases, Two Theorems – Parallel Axes Theorem, Perpendicular Axes Theorem; Continuous Distribution

  • Rolling
  • Introduction, Friction – Phenomenological Law, Limiting Friction and Coefficient of Friction, Examples; No-slip Condition; Rolling down an Incline – Three Methods to Solve the Problem, Other Examples; Examples with Slipping

  • Vector Formulations
  • Introduction – Right Hand Rule and Specifying a Rotation, Non-commutativity and Reducing to Special Case; Angular Displacement and Velocity, Torque; Equation of Motion – Derivation and Applying to Non-rigid Body, Precession of a Top; Angular Momentum; Moment of Inertia – An Example, Derivation of Key Formula, Properties of Key Formula; Appendix: Vector Identities