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Published: 25th January 1993
ISBN: 9780521429993
Number Of Pages: 340
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Published: 29th January 1993The differential geometry of curves and surfaces in Euclidean space has fascinated mathematicians since the time of Newton. Here the authors take a novel approach by casting the theory into a new light, that of singularity theory. The second edition of this successful textbook has been thoroughly revised throughout and includes a multitude of new exercises and examples. A new final chapter has been added that covers recently developed techniques in the classification of functions of several variables, a subject central to many applications of singularity theory. Also in this second edition are new sections on the Morse lemma and the classification of plane curve singularities. The only prerequisites for students to follow this textbook are a familiarity with linear algebra and advanced calculus. Thus it will be invaluable for anyone who would like an introduction to the modern theories of catastrophes and singularities.
"This delightfully written book overflows with beautiful mathematics, requiring only linear algebra, multi-variable calculus, and a little mathematical sophistication." The American Mathematical Monthly
| Preface to the second edition | p. xiii |
| Preface to the first edition | p. xv |
| Introductory example: a gravitational catastrophe machine | p. 1 |
| Curves, and functions on them | p. 10 |
| Parametrized curves | p. 13 |
| Tangent vectors | p. 15 |
| Contact | p. 18 |
| Reparametrization | p. 23 |
| Curvature | p. 28 |
| Functions on plane curves | p. 32 |
| Space curves | p. 41 |
| More about functions | p. 48 |
| Right equivalence | p. 48 |
| Flat functions | p. 54 |
| Jets | p. 55 |
| Regular values and smooth manifolds | p. 59 |
| Smooth maps and parametrized manifolds | p. 60 |
| Regular values | p. 65 |
| Tangent spaces | p. 74 |
| The Morse lemma | p. 85 |
| Morse lemma with parameters | p. 95 |
| Envelopes | p. 99 |
| Families and envelopes | p. 99 |
| Other definitions of envelope | p. 107 |
| Local structure of envelopes | p. 110 |
| Examples of envelopes | p. 118 |
| Envelopes and differential equations | p. 131 |
| Unfoldings | p. 133 |
| Unfoldings and (p)versal unfoldings | p. 134 |
| Versal unfoldings | p. 148 |
| Cusps | p. 153 |
| Unfoldings: applications | p. 160 |
| Distance-squared and height functions | p. 160 |
| Envelopes | p. 168 |
| Developable surfaces and the tangent developable of a space curve | p. 178 |
| Some different local pictures | p. 182 |
| Symmetry sets | p. 184 |
| Caustics by reflexion | p. 192 |
| Duals and differential equations | p. 200 |
| Transversality | p. 206 |
| Smooth manifolds | p. 207 |
| Transversality | p. 211 |
| Thom's transversality lemma | p. 216 |
| Algebraic surfaces and more on apparent contours | p. 220 |
| Open and dense properties | p. 222 |
| Generic properties of curves | p. 226 |
| The Monge-Taylor map | p. 227 |
| A transversality theorem for space curves | p. 237 |
| More on unfoldings | p. 240 |
| Determining the power series | p. 241 |
| Proof of convergence | p. 243 |
| Singular points, several variables, generic surfaces | p. 250 |
| The geometry of surfaces in Euclidean 3-space | p. 250 |
| Functions of one variable (again) | p. 253 |
| Families of differential equations | p. 257 |
| Differential equations | p. 262 |
| Functions of several variables | p. 263 |
| R-trivial families | p. 266 |
| A little algebra | p. 269 |
| The determinacy theorem | p. 273 |
| Classification of functions | p. 278 |
| Surfaces in Euclidean 3-space again | p. 284 |
| Moduli and simple singularities | p. 288 |
| Simple singularities | p. 292 |
| Null sets and Sard's theorem | p. 297 |
| Historical note | p. 301 |
| Further reading | p. 305 |
| References | p. 306 |
| Index of notation | p. 309 |
| Index | p. 311 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
ISBN: 9780521429993
ISBN-10: 0521429994
Audience:
Tertiary; University or College
Format:
Paperback
Language:
English
Number Of Pages: 340
Published: 25th January 1993
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.86 x 15.27
x 1.96
Weight (kg): 0.45
Edition Number: 2
Edition Type: Revised
