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Differential Geometry and Its Applications This Book Was Previously Published by Pearson Education, Inc AMS / MAA CLASSROOM RESOURCE MATERIALS VOL 59 John Oprea Differential Geometry and Its Applications | Second Edition | Differential Geometry and its Applications This book was previously published by Pearson Education, Inc. Originally published by The Mathematical Association of America, 2007. ISBN: 978-1-4704-5050-2 LCCN: 2007924394 Copyright © 2007, held by the Amercan Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2019 The American Mathematical Society retains all rights except those granted to the United States Government. ⃝1 The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 24 23 22 21 20 19 10.1090/clrm/059 AMS/MAA CLASSROOM RESOURCE MATERIALS VOL 59 Differential Geometry and its Applications Second Edition John Oprea P1: RTJ dgmain maab004 May 25, 2007 17:5 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 P1: RTJ dgmain maab004 May 25, 2007 17:5 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 Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein 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 of Mathematics, Margie Hale Exploratory Examples for 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: Activities for 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 Companion for 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 Brief Eclectic Tour, David A. Sanchez´ Oval Track and Other Permutation Puzzles, John O. Kiltinen A Primer of Abstract Mathematics, Robert B. Ash P1: RTJ dgmain maab004 May 25, 2007 17:5 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 1: 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 P1: RTJ dgmain maab004 May 25, 2007 17:5 To my mother and father, Jeanne and John Oprea. P1: RTJ dgmain maab004 May 25, 2007 17:5 P1: RTJ dgmain maab004 May 25, 2007 17:5 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 SomeImplicationsofCurvatureandTorsion................................ 34 1.6 Green’s Theorem and the Isoperimetric Inequality . .......................... 38 1.7 TheGeometryofCurvesandMaple........................................ 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 NormalCurvature........................................................ 91 2.5 Surfaces and Maple ...................................................... 96 3 Curvatures 107 3.1 Introduction . ............................................................ 107 3.2 CalculatingCurvature.................................................... 111 3.3 Surfaces of Revolution ................................................... 119 3.4 AFormulaforGaussCurvature............................................ 123 3.5 SomeEffectsofCurvature(s).............................................. 127 3.6 Surfaces of Delaunay . .................................................... 133 3.7 Elliptic Functions, Maple and Geometry . ................................... 137 3.8 CalculatingCurvaturewithMaple......................................... 149 4 Constant Mean Curvature Surfaces 161 4.1 Introduction . ............................................................ 161 4.2 First Notions in Minimal Surfaces ......................................... 164 ix P1: RTJ dgmain maab004 May 25, 2007 17:5 x Contents 4.3 AreaMinimization....................................................... 170 4.4 ConstantMeanCurvature................................................. 173 4.5 Harmonic Functions . ..................................................... 179 4.6 ComplexVariables....................................................... 182 4.7 IsothermalCoordinates................................................... 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 IsometriesandConformalMaps........................................... 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 TheBasicExamples...................................................... 322 7.4 Higher-OrderProblems................................................... 327 7.5 The Weierstrass E-Function. .............................................. 334 7.6 ProblemswithConstraints................................................ 346 7.7 Further Applications to Geometry and Mechanics ........................... 356 7.8 The Pontryagin Maximum Principle. ....................................... 366 7.9 An Application to the Shape of a Balloon . .................................. 371 7.10 TheCalculusofVariationsandMaple...................................... 380 8
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