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Calculus in 3D Geometry, Vectors, and Multivariate Calculus AMS / MAA TEXTBOOKS VOL 40 Calculus in 3D Geometry, Vectors, and Multivariate Calculus Zbigniew Nitecki 10.1090/text/040 AMS/MAA TEXTBOOKS VOL 40 Calculus in 3D Geometry, Vectors, and Multivariate Calculus Zbigniew Nitecki Committee on Books Jennifer J. Quinn, Chair MAA Textbooks Editorial Board Stanley E. Seltzer, Editor William Robert Green, Co-Editor Bela Bajnok Suzanne Lynne Larson Jeffrey L. Stuart Matthias Beck John Lorch Ron D. Taylor, Jr. Heather Ann Dye Michael J. McAsey Elizabeth Thoren Charles R. Hampton Virginia Noonburg Ruth Vanderpool 2010 Mathematics Subject Classification. Primary 26-01. For additional information and updates on this book, visit www.ams.org/bookpages/text-40 Library of Congress Cataloging-in-Publication Data Names: Nitecki, Zbigniew, author. Title: Calculus in 3D: Geometry, vectors, and multivariate calculus / Zbigniew Nitecki. Description: Providence, Rhode Island: MAA Press, an imprint of the American Mathematical Society, [2018] | Series: AMS/MAA textbooks; volume 40 | Includes bibliographical references and index. Identifiers: LCCN 2018020561 | ISBN 9781470443603 (alk. paper) Subjects: LCSH: Calculus–Textbooks. | AMS: Real functions – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA303.2.N5825 2018 | DDC 515/.8–dc23 LC record available at https://lccn.loc.gov/2018020561. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary ac- knowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permit- ted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. © 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ⃝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 1 23 22 21 20 19 18 Contents Preface v Idiosyncracies v Format viii Acknowledgments ix 1 Coordinates and Vectors 1 1.1 Locating Points in Space 1 1.2 Vectors and Their Arithmetic 11 1.3 Lines in Space 18 1.4 Projection of Vectors; Dot Products 25 1.5 Planes 31 1.6 Cross Products 39 1.7 Applications of Cross Products 56 2 Curves and Vector-Valued Functions of One Variable 67 2.1 Conic Sections 67 2.2 Parametrized Curves 80 2.3 Calculus of Vector-Valued Functions 92 2.4 Regular Curves 102 2.5 Integration along Curves 113 3 Differential Calculus for Real-Valued Functions of Several Variables 123 3.1 Continuity and Limits 123 3.2 Linear and Affine Functions 127 3.3 Derivatives 132 3.4 Level Curves 144 3.5 Surfaces and Tangent Planes I: Graphs and Level Surfaces 158 3.6 Surfaces and Tangent Planes II: Parametrized Surfaces 167 3.7 Extrema 176 3.8 Higher Derivatives 190 3.9 Local Extrema 197 4 Integral Calculus for Real-Valued Functions of Several Variables 205 4.1 Integration over Rectangles 205 4.2 Integration over General Planar Regions 217 4.3 Changing Coordinates 228 4.4 Integration Over Surfaces 236 4.5 Integration in Three Variables 248 5 Integral Calculus for Vector Fields and Differential Forms 263 5.1 Line Integrals of Vector Fields and 1-Forms 263 5.2 The Fundamental Theorem for Line Integrals 272 5.3 Green’s Theorem 278 5.4 Green’s Theorem and 2-forms in ℝ2 289 iii iv Contents 5.5 Oriented Surfaces and Flux Integrals 293 5.6 Stokes’ Theorem 299 5.7 2-forms in ℝ3 306 5.8 The Divergence Theorem 317 5.9 3-forms and the Generalized Stokes Theorem (Optional) 329 A Appendix 335 A.1 Differentiability in the Implicit Function Theorem 335 A.2 Equality of Mixed Partials 336 A.3 The Principal Axis Theorem 339 A.4 Discontinuities and Integration 344 A.5 Linear Transformations, Matrices, and Determinants 347 A.6 The Inverse Mapping Theorem 353 A.7 Change of Coordinates: Technical Details 356 A.8 Surface Area: The Counterexample of Schwarz and Peano 363 A.9 The Poincare Lemma 367 A.10 Proof of Green’s Theorem 374 A.11 Non-Orientable Surfaces: The Möbius Band 376 A.12 Proof of Divergence Theorem 377 A.13 Answers to Selected Exercises 379 Bibliography 393 Index 397 Preface The present volume is a sequel to my earlier book, Calculus Deconstructed: A Second Course in First-Year Calculus, published by the Mathematical Association of America in 2009. I have used versions of this pair of books for several years in the Honors Calcu- lus course at Tufts, a two-semester “boot camp” intended for mathematically inclined freshmen who have been exposed to calculus in high school. The first semester of this course, using the earlier book, covers single-variable calculus, while the second semes- ter, using the present text, covers multivariate calculus. However, the present book is designed to be able to stand alone as a text in multivariate calculus. The treatment here continues the basic stance of its predecessor, combining hands- on drill in techniques of calculation with rigorous mathematical arguments. Nonethe- less, there are some differences in emphasis. On one hand, the present text assumes a higher level of mathematical sophistication on the part of the reader: there is no ex- plicit guidance in the rhetorical practices of mathematicians, and the theorem-proof format is followed a little more brusquely than before. On the other hand, the material being developed here is unfamiliar territory for the intended audience to a far greater degree than in the previous text, so more effort is expended on motivating various ap- proaches and procedures, and a substantial number of technical arguments have been separated from the central text, as exercises or appendices. Where possible, I have followed my own predilection for geometric arguments over formal ones, although the two perspectives are naturally intertwined. At times, this may feel like an analysis text, but I have studiously avoided the temptation to give the general, 푛-dimensional versions of arguments and results that would seem natural to a mature mathematician: the book is, after all, aimed at the mathematical novice, and I have taken seriously the limitation implied by the “3D” in my title. This has the ad- vantage, however, that many ideas can be motivated by natural geometric arguments. I hope that this approach lays a good intuitive foundation for further generalization that the reader will see in later courses. Perhaps the fundamental subtext of my treatment is the way that the theory de- veloped earlier for functions of one variable interacts with geometry to handle higher- dimension situations. The progression here, after an initial chapter developing the tools of vector algebra in the plane and in space (including dot products and cross products), is to first view vector-valued functions of a single real variable in termsof parametrized curves—here, much of the theory translates very simply in a coordinate- wise way—then to consider real-valued functions of several variables both as functions with a vector input and in terms of surfaces in space (and level curves in the plane), and finally to vector fields as vector-valued functions of vector variables. Idiosyncracies There are a number of ways, some apparent, some perhaps more subtle, in which this treatment differs from the standard ones: Conic Sections: I have included in § 2.1 a treatment of conic sections, starting with a version of Apollonius’s formulation in terms of sections of a double cone (and ex- plaining the origin of the names parabola, hyperbola, and ellipse), then discussing v vi Preface the focus-directrix formulation following Pappus, and finally sketching how this leads to the basic equations for such curves. I have taken a quasi-historical ap- proach here, trying to give an idea of the classical Greek approach to curves which contrasts so much with our contemporary calculus-based approach. This is an ex- ample of a place where I think some historical context enriches our understanding of the subject. This can be treated as optional in class, but I personally insist on spending at least one class on it. Parametrization: I have stressed the parametric representation of curves and sur- faces far more, and beginning somewhat earlier, than many multivariate texts. This approach is essential for applying calculus to geometric objects, and it is also a beautiful and satisfying interplay between the geometric and analytic points of view. While Chapter 2 begins with a treatment of the conic sections from a clas- sical point of view, this is followed by a catalogue of parametrizations of these curves and, in § 2.4, by a more careful study of regular curves in the plane and their relation to graphs of functions. This leads naturally to the formulation of path integrals in § 2.5. Similarly, quadric surfaces are introduced in § 3.4 as level sets of quadratic polynomials in three variables, and the (three-dimensional) Im- plicit Function Theorem is introduced to show that any such surface is locally the graph of a function of two variables. The notion of parametrization of a surface is then introduced and exploited in § 3.6 to obtain the tangent planes of surfaces.
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