+612 9045 4394
 
CHECKOUT
Curves and Singularities : A Geometrical Introduction to Singularity Theory - J. Bruce

Curves and Singularities

A Geometrical Introduction to Singularity Theory

Hardcover Published: 29th January 1993
ISBN: 9780521419857
Number Of Pages: 340

Share This Book:

Hardcover

$473.25
or 4 easy payments of $118.31 with Learn more
Ships in 7 to 10 business days

Other Available Editions (Hide)

  • Paperback View Product Published: 25th January 1993
    $95.40

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 editionp. xiii
Preface to the first editionp. xv
Introductory example: a gravitational catastrophe machinep. 1
Curves, and functions on themp. 10
Parametrized curvesp. 13
Tangent vectorsp. 15
Contactp. 18
Reparametrizationp. 23
Curvaturep. 28
Functions on plane curvesp. 32
Space curvesp. 41
More about functionsp. 48
Right equivalencep. 48
Flat functionsp. 54
Jetsp. 55
Regular values and smooth manifoldsp. 59
Smooth maps and parametrized manifoldsp. 60
Regular valuesp. 65
Tangent spacesp. 74
The Morse lemmap. 85
Morse lemma with parametersp. 95
Envelopesp. 99
Families and envelopesp. 99
Other definitions of envelopep. 107
Local structure of envelopesp. 110
Examples of envelopesp. 118
Envelopes and differential equationsp. 131
Unfoldingsp. 133
Unfoldings and (p)versal unfoldingsp. 134
Versal unfoldingsp. 148
Cuspsp. 153
Unfoldings: applicationsp. 160
Distance-squared and height functionsp. 160
Envelopesp. 168
Developable surfaces and the tangent developable of a space curvep. 178
Some different local picturesp. 182
Symmetry setsp. 184
Caustics by reflexionp. 192
Duals and differential equationsp. 200
Transversalityp. 206
Smooth manifoldsp. 207
Transversalityp. 211
Thom's transversality lemmap. 216
Algebraic surfaces and more on apparent contoursp. 220
Open and dense propertiesp. 222
Generic properties of curvesp. 226
The Monge-Taylor mapp. 227
A transversality theorem for space curvesp. 237
More on unfoldingsp. 240
Determining the power seriesp. 241
Proof of convergencep. 243
Singular points, several variables, generic surfacesp. 250
The geometry of surfaces in Euclidean 3-spacep. 250
Functions of one variable (again)p. 253
Families of differential equationsp. 257
Differential equationsp. 262
Functions of several variablesp. 263
R-trivial familiesp. 266
A little algebrap. 269
The determinacy theoremp. 273
Classification of functionsp. 278
Surfaces in Euclidean 3-space againp. 284
Moduli and simple singularitiesp. 288
Simple singularitiesp. 292
Null sets and Sard's theoremp. 297
Historical notep. 301
Further readingp. 305
Referencesp. 306
Index of notationp. 309
Indexp. 311
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521419857
ISBN-10: 0521419859
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 340
Published: 29th January 1993
Publisher: CAMBRIDGE UNIV PR
Country of Publication: GB
Dimensions (cm): 22.86 x 15.24  x 2.24
Weight (kg): 0.67
Edition Number: 2
Edition Type: Revised