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Glossary
Chapter 1: What mathematical modelling is
- Overview
- 1.1 The problem-solving cycle
- 1.2 Formulating a problem and decomposing it
- 1.3 Assumptions and what they cost you
- 1.4 Translating a situation into mathematics
- 1.5 Computing and interpreting a solution
- 1.6 Evaluating, refining and reporting
- 1.7 How to write a modelling report
- Review
Chapter 2: Networks and graphs
- Overview
- 2.1 Networks and their terminology
- 2.2 Connected, disconnected, directed and undirected graphs
- 2.3 Representing a network: adjacency matrices
- 2.4 Matrix algebra and the product of adjacency matrices
- 2.5 Trees and spanning trees
- 2.6 Minimum spanning trees: Kruskal's algorithm
- 2.7 Minimum spanning trees: Prim's algorithm
- 2.8 Shortest paths and Dijkstra's algorithm
- 2.9 Multi-stage problems and dynamic programming
- 2.10 Bellman's principle of optimality
- Review
Chapter 3: Describing motion
- Overview
- 3.1 Displacement, velocity and acceleration
- 3.2 Motion with constant acceleration
- 3.3 Motion graphs
- 3.4 Differentiation and the chain rule
- 3.5 The fundamental theorem of calculus
- 3.6 Deriving the equations of motion by calculus
- 3.7 Projectile motion
- 3.8 Time of flight and range
- Review
Chapter 4: Forces and Newton's laws
- Overview
- 4.1 Forces and Newton's laws
- 4.2 Resultant force and equilibrium
- 4.3 Motion on a smooth horizontal plane
- 4.4 Motion on an inclined plane
- 4.5 Tension and connected particles
- 4.6 Pulleys
- Review
Chapter 5: Friction and resistance
- Overview
- 5.1 Friction and the normal reaction
- 5.2 The coefficient of friction
- 5.3 Motion on rough planes
- 5.5 Drag proportional to a power of speed
- Review
Chapter 6: Momentum, impulse and collisions
- Overview
- 6.1 Momentum and impulse
- 6.2 Conservation of momentum
- 6.3 Collisions in one dimension
- 6.4 Newton's experimental law of restitution
- 6.5 Elastic and inelastic collisions
- Review
Chapter 7: Circular motion and oscillation
- Overview
- 7.1 Angular velocity and period
- 7.2 Centripetal acceleration and force
- 7.3 Circular motion in context
- 7.5 Hooke's law and elastic strings
- Review
Chapter 8: Modelling a changing world
- Overview
- 8.1 Rates of change and differential equations
- 8.2 Exponential growth and decay
- 8.3 Recurrence relations as discrete models
- 8.4 First-order difference equations
- 8.5 Feedback and limited growth
- Review
Chapter 9: The modelling project
- Overview
- 9.1 Reading the brief and choosing a problem
- 9.2 Building the model and stating assumptions
- 9.3 Testing the model against data
- 9.4 Iterating and improving
- 9.5 Communicating the result
- 9.6 Critical evaluation of models
- Review
About this book

Chapter 2: Networks and graphs

Networks as models of connection — vertices, edges, weights — and the graphs, matrices and algorithms used to reason about them.

The chapter problem

Wiring the new building. You will connect a set of rooms with the least total length of cable, then find the shortest route between two of them. By the end of the chapter you can say which algorithm answers which question.

Where it is used: Transport, delivery routes, cabling, broadband and utility networks.

Your turn to connect it: Which journey you take, or which network reaches your area, is decided by a problem like this one.

Sections

2.1 Networks and their terminology — tier O · refs S2
2.2 Connected, disconnected, directed and undirected graphs — tier O · refs S2
2.3 Representing a network: adjacency matrices — tier O · refs S2
2.4 Matrix algebra and the product of adjacency matrices — tier O · refs S2
2.5 Trees and spanning trees — tier O · refs S2
2.6 Minimum spanning trees: Kruskal's algorithm — tier O · refs S2
2.7 Minimum spanning trees: Prim's algorithm — tier O · refs S2
2.8 Shortest paths and Dijkstra's algorithm — tier O · refs S2
2.9 Multi-stage problems and dynamic programming — tier O · refs S2
2.10 Bellman's principle of optimality — tier O · refs S2
2.11 Comparing the algorithms and choosing one — tier H · refs S2