Fields, Flows and Waves

This book introduces the undergraduate to the diversity and versatility of fundamental mathematics in the real world, without the need for multiple preparatory courses first. It illustrates, through examples, how core undergraduate topics that students have already encountered - such as linear algebra, matrices and differential equations - are natural tools for constructing, analysing and interpreting phenomena in the real world.

THE ONLY BOOK ON CONTINUA AT A BROAD, INTRODUCTORY LEVEL FOSTERS THE STUDENT'S DESIRE TO APPLY MATHEMATICS, WITHOUT ASSUMING A STRONG PHYSICS BACKGROUND SHOWS THE VERSATILITY OF STANDARD TECHNIQUES IN DESCRIBING MANY NATURALLY ARISING PHENOMENA

Contenu

1. The Continuum Description.- 1.1 Densities and Fluxes.- 1.2 Conservation and Balance Laws in One Dimension.- 1.3 Heat Flow.- 1.4 Steady Radial Flow in Two Dimensions.- 1.5 Steady Radial Flow in Three Dimensions.- 2. Unsteady Heat Flow.- 2.1 Thermal Energy.- 2.2 Effects of Heat Supply.- 2.3 Unsteady, Spherically Symmetric Heat Flow.- 3. Fields and Potentials.- 3.1 Gradient of a Scalar.- 3.2 Gravitational Potential.- 3.3 Continuous Distributions of Mass.- 3.4 Electrostatics.- 4. Laplace's Equation and Poisson's Equation.- 4.1 The Ubiquitous Laplacian.- 4.2 Separable Solutions.- 4.3 Poisson's Equation.- 4.4 Dipole Solutions.- 5. Motion of an Elastic String.- 5.1 Tension and Extension; Kinematics and Dynamics.- 5.2 Planar Motions.- 5.3 Properties of the Wave Equation.- 5.4 D'Alembert's Solution, Travelling Waves and Wave Reflections.- 5.5 Other One-dimensional Waves.- 6. Fluid Flow.- 6.1 Kinematics and Streamlines.- 6.2 Volume Flux and Mass Flux.- 6.3 Two-dimensional Flows of Incompressible Fluids.- 6.4 Pressure in a Fluid.- 6.5 Bernoulli's Equation.- 6.6 Three-dimensional, Incompressible Flows.- 7. Elastic Deformations.- 7.1 The Kinematics of Deformation.- 7.2 Polar Decomposition.- 7.3 Stress.- 7.4 Isotropic Linear Elasticity.- 8. Vibrations and Waves.- 8.2 Guided Waves.- 8.3 Love Waves in Elasticity.- 8.4 Elastic Plane Waves.- 9. Electromagnetic VVaves and Light.- 9.1 Physical Background.- 9.2 Waveguides.- 10. Chemical and Biological Models.- 10.1 Diffusion of Chemical Species.- 10.2 Population Biology.- 10.3 Biological Waves.- Solutions.