Table of Contents
1. Introduction and Organisation
1.1 Individual Projects
1.2 Group Projects
1.3 Case Studies
2. Assessment
2.1 Introduction
2.2 Interim Reports
2.3 Verbal Presentations
2.4 Final Report
2.5 Moderating
2.6 Assessment of Case Studies
3. Individual Projects
3.1 Introduction
3.2 Selecting a Project
3.3 Report Writing
3.4 Non-Euclidean Geometry
3.4.1 Scope
3.4.2 Project Details
3.5 Boomerangs
3.5.1 Scope
3.5.2 Project Details
3.6 Hurricane Dynamics
3.6.1 Scope
3.6.2 Project Details
3.7 Hypergeometric Functions
3.7.1 Scope
3.7.2 Project Details
3.8 Summary
3.9 Project Examples
4. Group Projects
4.1 Introduction
4.2 Setting up Group Projects
4.2.1 Peer Assessment
4.2.2 Dividing into Groups
4.3 Estuarial Diffusion
4.4 Graphs and Networks
4.5 Fourier Transforms
4.6 Orbital Motion
4.7 Conclusion
4.8 Further Suggestions
5. Case Studies
5.1 Introduction
5.2 Ocean Surface Dynamics
5.3 Non-linear Oscillations
5.4 Traffic Flow
5.5 Contour Integral Solutions to ODEs
5.6 Optimisation
5.7 Euler and Series
5.8 Summary
5.9 Exercises
A. Project Example 1: Topics in Galois Theory
A.1 Galois' Approach
A.1.1 Preparation
A.1.2 The Galois Resolvent
A.1.3 The Galois Group
A.1.4 Soluble Equations and Soluble Groups
A.2 The Modern Approach
A.2.1 Field Extension
A.2.2 The Galois Group
A.2.3 Applying Galois Theory
A.3 Soluble Groups
A.3.1 Normal Subgroup Series
A.3.2 Normal Subgroups
A.3.3 Simple Groups
A.3.4 p-Groups
A.4 Geometrical Constructions
A.4.1 Constructible Points
A.4.2 Impossibility Proofs
A.4.3 Performing Algebraic Operations by Construction
A.4.4 Regular n-gons
B. Project Example 2: Algebraic Curves
B.1 Basic De.nitions and Properties
B.1.1 Complex Algebraic Curves and Real Algebraic CurvesB.1.2 Projective Spaces
B.1.3 A.ne and Projective Curves
B.1.4 Singular Points
B.2 Intersection of Two Curves and Points of Inflection
B.2.1 Bezout's Theorem
B.2.2 Points of Inflection on a Curve
B.3 Conics and Cubics
B.3.1 Conics
B.3.2 Cubics
B.3.3 Additive Group Structure on a Cubic
B.4 Complex Analysis
B.4.1 Holomorphic Functions and Entire Functions
B.4.2 Closed Curve Theorem and Line Integrals
B.4.3 Liouville's Theorem and Fundamental Theorem of Algebra
B.4.4 Properties of Holomorphic Functions
B.4.5 General Cauchy Closed Curve Theorem
B.4.6 Isolated Singularities and Removable Singularities
B.4.7 Laurent Expansions
B.4.8 Residue Theorem
B.4.9 Conformal Mapping
B.5 Topology and Riemann Surfaces
B.5.1 Topology of Complex Algebraic Curves
B.5.2 Riemann Surfaces
B.5.3 Degeneration of a Cubic
B.5.4 Singularities and Riemann Surfaces
B.6 Further Topics
B.6.1 The Weierstrass Function
B.6.2 Differential Forms on a Riemann Surface
B.6.3 Abel's Theorem
C. Project Example 3: Water Waves on a Sloping Beach
C.1 Abstract
C.2 Introduction
C.3 Surface Waves
C.3.1 The Current
C.3.2 The Boundary Conditions
C.3.3 A Separable Solution of Laplace's Equation
C.4 [No Title]
C.4.1 The Velocity of the Waves
C.4.2 The Group Velocity of the Waves
C.4.3 The Motion of the Particles
C.4.4 Breaking Waves in Shallow Water
C.5 [No Title]
C.5.1 Plane Waves
C.5.2 Wave Rays
C.5.3 The Waves Approaching a Beach
C.5.4 Wave rays in shallow water
C.6 [No Title]
C.6.1 Conclusion and Discussion
Index
