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Catastrophe Theory and Its Applications

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Catastrophe Theory and Its Applications Cover

 

Synopses & Reviews

Publisher Comments:

First integrated treatment of main ideas behind René Thom's theory of catastrophes stresses detailed applications in the physical sciences. Mathematics of theory explained with a minimum of technicalities. Over 200 illustrations clarify text designed for researchers and postgraduate students in engineering, mathematics, physics and biology. 1978 edition. Bibliography.

Synopsis:

First integrated treatment of main ideas behind Rene Thom's theory of catastrophes. Focus on detailed physical applications. 200 illus. 1978 edition. Bibliography.


Synopsis:

First integrated treatment of main ideas behind René Thom's theory of catastrophes stresses detailed applications in the physical sciences. Mathematics of theory explained with a minimum of technicalities. Over 200 illustrations. 1978 edition.


Table of Contents

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 Kármán 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 rôle of fluctuations

    11. Spatial variation

    12. Partition functions

    13. Renormalization group

    14. Structural stability of renormalization

    The rôle 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

  &

Product Details

ISBN:
9780486692715
With:
Stewart, Ian
Author:
Poston, T.
Author:
Mathematics
Author:
Poston, Tim
Author:
Stewart, Ian
Publisher:
Dover Publications
Location:
Mineola, N.Y. :
Subject:
General
Subject:
Mathematics
Subject:
Advanced
Subject:
Catastrophes (mathematics)
Subject:
General Mathematics
Subject:
Catastrophe Theory
Subject:
Applications of Catastrophe Theory
Subject:
Ian Stewart
Subject:
Mathematics - General
Subject:
Statistics
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Publication Date:
20120231
Binding:
TRADE PAPER
Grade Level:
General/trade
Language:
English
Illustrations:
Yes
Pages:
512
Dimensions:
9.25 x 6.5 in 1.56 lb

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Product details 512 pages Dover Publications - English 9780486692715 Reviews:
"Synopsis" by , First integrated treatment of main ideas behind Rene Thom's theory of catastrophes. Focus on detailed physical applications. 200 illus. 1978 edition. Bibliography.


"Synopsis" by ,
First integrated treatment of main ideas behind René Thom's theory of catastrophes stresses detailed applications in the physical sciences. Mathematics of theory explained with a minimum of technicalities. Over 200 illustrations. 1978 edition.


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