Curves and Singularities

A Geometrical Introduction to Singularity Theory

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By: J. Bruce, P. J. Giblin

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Published: 25th January 1993

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The 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