Level 1: Pre-University to Year 1 1st Semester
Basic Mathematical Methods by Dr. Chan Man Ho
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Level 2: Year 1 1st Semester to Year 2 1st Semester, and Beyond
Extension Micro-modules in Mechanics by Prof. Kenneth Young
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Units and Dimensions
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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
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Coordinate Systems, Vectors and Matrices
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Coordinate Systems
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ENG )
Cartesian Coordinates in 3D, Cylindrical Coordinates, and Spherical Coordinates
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Solution )
Basics of Vector
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ENG )
Quantities with Direction and Magnitude, Basic Vector Algebra
Dot and Cross Products of Vectors
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ENG )
Dot and Cross Products, Properties and Applications
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Solution )
Basics of Matrix
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ENG )
Structure, Matrix Algebra, Inverse and Transpose, Applications
Inverse Matrix and Determinant
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ENG )
Finding Inverse Matrices and Determinants
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Solution )
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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
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Elementary Functions
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Exponential Functions and Logarithm
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普 |
ENG )
Euler Number e, Properties of Exponential Functions and Logarithm, Applications
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Solution )
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Some Elementary Functions
Exponential Function, Logarithm, Hyperbolic Functions, Trigonometric Functions
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Limit, Differentiation, Power Series, Approximations
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Limit
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ENG )
Properties and Calculation of Limits, L’ Hospital’s Rule
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Solution )
Differentiation
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普 |
ENG )
Definition, Slope of Graph, Higher Order Derivatives, Applications to Kinematics
Rules of Differentiation
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ENG )
Differentiation of Common Functions, Product Rule and Chain Rule
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Solution )
Approximation
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普 |
ENG )
Concepts and Applications, Taylor’s Series
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Solution )
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Power Series and Taylor's Series
Power Series, Exponential Function, Differentiation and Integration, Taylor Series, Newton's Method, Appendix: Convergence
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Integration and Applications
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Definition and Meaning of Integration
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ENG )
Definition, Area under Curve, Definite Integral, Applications to Kinematics
Differentiation and Integration, Common Integrals
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ENG )
Relationships between Differentiation and Integration, Integrals of Common Functions
Methods of Integration
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ENG )
Change of Variable and Integration by Parts
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Solution )
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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
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Line Integral, Curl, Gradient and Applications
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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
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Complex Number and Applications
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Basics of Complex Number
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ENG )
Real Part and Imaginary Part, Polar Representation, Basic Algebra
Relationship to Sinusoidal Functions
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ENG )
Exponential Form, Relationship with Sinusoidal Functions, Time Variation and Phase, Application to Oscillation
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Solution )
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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
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Ordinary Differential Equations in Oscillation Problems
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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
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Integrated Application: Rotations
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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
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