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Catastrophe Theory and Its Applications


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Preface 1 Smooth and sudden changes 1. Catastrophes 2. The Zeeman catastrophe machine 3. Gravitational catastrophe machines 4. Catastrophe theory 2 Multidimensional geometry 1. Set-theoretic notation 2. Euclidean space 3. Linear transformations 4. Matrices 5. Quadratic forms 6. Two-variable cubic forms 7. Polynomial geometry 3 Multidimensional calculus 1. Distance in Euclidean space 2. The derivative as tangent 3. Contours 4. Partial derivatives 5. Higher derivatives 6. Taylor series 7. Truncated algebra 8. The Inverse Function Theorem 9. The Implicit Function Theorem 4 Critical points and transversality 1. Critical points 2. The Morse Lemma 3. Functions of a single variable 4. Functions of several variables 5. The Splitting Lemma 6. Structural stability 7. Manifolds 8. Transversality 9. Transversality and stability 10. Transversality for mappings 11. Codimension 5 Machines revisited 1. The Zeeman machine 2. The canonical cusp catastrophe 3. Dynamics of the Zeeman machine 4. The gravitational machines 5. Formulation of a general problem 6 Structural stability 1. Equivalence of families 2. Structural stabillty of families 3. Physical interpretations of structural stability 4. The Morse and Splitting Lemmas for families 5. Catastrophe geometry 7 Thom's classification theorem 1. Functions and families of functions 2. One-parameter families 3. Non-transversaliity and symmetry 4. Two-parameter families 5. "Three-, four- and five-parameter families" 6. Higher catastrophes 7. Thom's theorem 8 Determinacy and unfoldings 1. Determine and strong determinacy 2. One-variable jet spaces 3. Infinitesimal changes of variable 4. Weaker determinacy conditions 5. Transformations that move the origin 6. Tangency and transversality 7. Codimension and unfoldings 8. Transversality and universality 9. Strong equivalence of unfoldings 10. Numbers associated with singularities 11. Inequalities 12. Summary of results and calculation methods 13. Examples and calculations 14. Compulsory remarks on terminology 9 The first seven catastrophe geometries 1. The objects of study 2. The fold catastrophe 3. The cusp catastrophe 4. The swallowtail catastrophe 5. The butterfly catastrophe 6. The elliptic umbilic 7. The hyperbolic umbilic 8. The parabolic umbilic 9. Ruled surfaces 10 Stability of ships Static equilibrium 1. Buoyancy 2. Equilibrium 3. Stability 4. The vertical-sided ship 5. Geometry of the buoyancy locus 6. Metacentres Ship shapes 7. The elliptical ship 8. The rectangular ship 9. Three dimensions 10. Oil-rigs 11. Comparison with current methods 11. The geometry of fluids Background on fluid mechanics 1. What we are describing 2. Stream functions 3. Examples of flows 4. Rotation 5. Complex variable methods Stability and experiment 6. Changes of variable 7. Heuristic programme 8. Experimental realization Combining polymer molecules 9. Non-Newtonian behaviour 10. Extensional flows Degenerate flows 11. The six-roll mill 12. The non-local bifurcation set of the elliptic umbilic 13. The six-roll mill with polymer solution 14. The 2n-roll mill 12 Optics and scattering theory Ray optics 1. Caustics 2. The rainbow 3. Variational principles 4. Scattering Wave optics 5. Asymptotic solutions of wave equations 6. Oscillatory integrals 7. Universal unfoldings 8. Orders of caustics Applications 9. Scattering from a crystal lattice 10. Other caustics 11. Mirages 12. Sonic booms 13. Giant ocean waves 13 Elastic structures General theory 1. Objects under stress 2. Elastic equilibria 3. Infinite-dimensional peculiarities Euler struts 4. Finite element vision 5. Classical (1744) variational version 6. Perturbation analysis 7. Modern functional analysis 8. The buckling of a spring 9. The pinned strut The geometry of collapse 10. Imperfection sensitivity 11. "(r, s)-Stability" 12. Optimization 13. Symmetry: rods and shells Buckling plates 14. The von Karman equations 15. Unfolding a double eigenvalue Dynamics 16. Soft modes 17. Stiffness 14 Thermodynamics and phase transitions Equations of state 1. van der Waals' equation 2. Ferromagnetism Thermodynamic potentials 3. Entropy 4. Transforming the maximum entropy principle 5. Legendre transformations 6. Explicit potentials 7. The Landau theory Fluctuations and critical exponents 8. Classical exponents 9. Topological tinkering 10. The role of fluctuations 11. Spatial variation 12. Partition functions 13. Renormalization group 14. Structural stability of renormalization The role of symmetry 15. Even functions 16. The shapes of rotating stars 17. Symmetry breaking 18. Tricritical points 19. Crystal symmetries 20. Spectrum singularities 15 Laser physics Preliminaries 1. Atoms 2. Field 3. Interaction 4. Measurement The laser catastrophe 5. Unfolded Hamiltonian 6. Equations of motion 7. Mean field approximation 8. Boundary conditions 9. Non-equilibrium stationary manifold Experiments 10. Laser transition 11. Optical bistability 12. Photocount distributions Analytic correspondence 13. Equilibrium boundary conditions 14. Equilibrium manifold 15. Thermodynamic phase transition 16. Critical behaviour 17. Analytic correspondence of experiments &

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