Differential Geometry and its Applications (Classroom Resource Materials) (Mathematical Association of America Textbooks)

496 Pages·2007·57.99 MB·English

by  John Oprea

This book was previously published
by Pearson Education, Inc.
© 2007 by
The Mathematical Association ofA merica (Icorporated)
Library of Congress Catalog Card Number 2007924394
ISBN 978-0-88385-748-9
Printed in the United States ofA merica
Current Printing (last digit):
10 9 8 7 6 5 4 3 2 1
Differential Geometry
and its
Applications
Second Edition
John Oprea
Cleveland State University
The Mathematical Association of America
Council on Publications
James Daniel, Chair
Classroom Resource Materials Editorial Board
Zaven A. Karian, Editor
Gerald M. Bryce
Douglas B. Meade
Wayne Roberts
Kay B. Somers
Stanley E. Seltzer
Susan G. Staples
George Exner
William C. Bauldry
Charles R. Hadlock
Shahriar Shahriari
Holly S. Zullo
CLASSROOM RESOURCE MATERIALS
Classroom Resource Materials is intended to provide supplementary classroom material for
students-laboratory exercises, projects, historical information, textbooks with unusual ap
proaches for presenting mathematical ideas, career information, etc.
101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett
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Calculus Mysteries and Thrillers, R. Grant Woods
Combinatorics: A Problem Oriented Approach, Daniel A. Marcus
Conjecture and Proof, Miklos Laczkovich
A Course in Mathematical Modeling, Douglas Mooney and Randall Swift
Cryptological Mathematics, Robert Edward Lewand
Differential Geometry and Its Applications, John Oprea
Elementary Mathematical Models, Dan Kalman
Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft
Essentials ofM athematics, Margie Hale
Exploratory Examplesfor Real Analysis, Joanne E. Snow and Kirk E. Weller
Fourier Series, Rajendra Bhatia
Geometry From Africa: Mathematical and Educational Explorations, Paulus Gerdes
Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz
and Karen Dee Michalowicz
Identification Numbers and Check Digit Schemes, Joseph Kirtland
Interdisciplinary Lively Application Projects, edited by Chris Arney
Inverse Problems: Activitiesfor Undergraduates, Charles W. Groetsch
Laboratory Experiences in Group Theory, Ellen Maycock Parker
Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor
Katz
Mathematical Connections: A Companionfor Teachers and Others, Al Cuoco
Mathematical Evolutions, edited by Abe Shenitzer and John Stillwell
Mathematical Modeling in the Environment, Charles Hadlock
Mathematics for Business Decisions Part 1: Probability and Simulation (electronic textbook),
Richard B. Thompson and Christopher G. Lamoureux
Mathematics for Business Decisions Part 2: Calculus and Optimization (electronic textbook),
Richard B. Thompson and Christopher G. Lamoureux
Math Made Visual: Creating Images for Understanding Mathematics, Claudi Alsina and Roger
B. Nelsen
Ordinary Differential Equations: A BriefE clectic Tour, David A. Sanchez
Oval Track and Other Permutation Puzzles, John O. Kiltinen
A Primer ofA bstract Mathematics, Robert B. Ash
Proofs Without Words, Roger B. Nelsen
Proofs Without Words II, Roger B. Nelsen
A Radical Approach to Real Analysis, 2nd edition, David M. Bressoud
Real Infinite Series, Daniel D. Bonar and Michael Khoury, Jr.
She Does Math!, edited by Marla Parker
Solve This: Math Activities for Students and Clubs, James S. Tanton
Student Manual for Mathematics for Business Decisions Part J: Probability and Simulation,
David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic
Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization,
David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic
Teaching Statistics Using Baseball, Jim Albert
Topology Now!, Robert Messer and Philip Straffin
Understanding our Quantitative World, Janet Andersen and Todd Swanson
Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa
Crannell, Gavin LaRose, Thomas Ratliff, Elyn Rykken
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Contents
Preface xiii
Note to Students xix
1 The Geometry of Curves 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Arclength Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Frenet Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Non-Unit Speed Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27
1.5 Some Implications of Curvature and Torsion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34
1.6 Green's Theorem and the Isoperimetric Inequality. . . . . . . . . . . . . . . . . . . . . . . . . .. 38
1.7 The Geometry of Curves and Maple ........................................ 42
2 Surfaces 67
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67
2.2 The Geometry of Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.3 The Linear Algebra of Surfaces ........................................... , 86
2.4 Normal Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91
2.5 Surfaces and Maple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3 Curvatures 107
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 107
3.2 Calculating Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. III
3.3 Surfaces of Revolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119
3.4 A Formula for Gauss Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 123
3.5 Some Effects of Curvature( s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 127
3.6 Surfaces of Delaunay ..................................................... 133
3.7 Elliptic Functions, Maple and Geometry .................................... 137
3.8 Calculating Curvature with Maple ......................................... 149
4 Constant Mean Curvature Surfaces 161
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 161
4.2 First Notions in Minimal Surfaces ......................................... 164
ix
4.3 Area Minimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 170
4.4 Constant Mean Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 173
4.5 Harmonic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 179
4.6 Complex Variables ....................................................... 182
4.7 Isothermal Coordinates ................................................... 184
4.8 The Weierstrass-Enneper Representations .................................. 187
4.9 Maple and Minimal Surfaces .............................................. 197
5 Geodesics, Metrics and Isometries 209
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 209
5.2 The Geodesic Equations and the Clairaut Relation ........................... 215
5.3 A Brief Digression on Completeness ....................................... 225
5.4 Surfaces not in]R3 ........................................................ 226
5.5 Isometries and Conformal Maps ........................................... 235
5.6 Geodesics and Maple ..................................................... 241
5.7 An Industrial Application ................................................. 262
6 Holonomy and the Gauss-Bonnet Theorem 275
6.1 Introduction ............................................................. 275
6.2 The Covariant Derivative Revisited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 277
6.3 Parallel Vector Fields and Holonomy ...................................... 280
6.4 Foucault's Pendulum ..................................................... 284
6.5 The Angle Excess Theorem ............................................... 286
6.6 The Gauss-Bonnet Theorem ............................................... 289
6.7 Applications of Gauss-Bonnet ............................................. 292
6.8 Geodesic Polar Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 297
6.9 Maple and Holonomy ..................................................... 305
7 The Calculus of Variations and Geometry 311
7.1 The Euler-Lagrange Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 311
7.2 Transversality and Natural Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . .. 318
7.3 The Basic Examples ...................................................... 322
7.4 Higher-Order Problems ................................................... 327
7.5 The Weierstrass E-Function ............................................... 334
7.6 Problems with Constraints ................................................ 346
7.7 Further Applications to Geometry and Mechanics ........................... 356
7.8 The Pontryagin Maximum Principle ........................................ 366
7.9 An Application to the Shape ofa Balloon ................................... 371
7.10 The Calculus of Variations and Maple ...................................... 380
8 A Glimpse at Higher Dimensions 397
8.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 397
8.2 Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 397
8.3 The Covariant Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 I
8.4 Christoffel Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 409
8.5 Curvatures .............................................................. 416
8.6 The Charming Doubleness ................................................ 430