Volume Calculus Volume SENIOR CONTRIBUTING AUTHORS EDWIN "JED" HERMAN, UNIVERSITY OF WISCONSIN-STEVENS POINT GILBERT STRANG, MASSACHUSETTS INSTITUTE OF TECHNOLOGY OpenStax Rice University 6100 Main Street MS-375 Houston, Texas 77005 To learn more about OpenStax, visit https://openstax.org Individual print copies and bulk orders can be purchased through our website ©2018 Rice University Textbook content produced by OpenStax is licensed under a Creative Commons Attribution NonCommercial ShareAlike 4.0 International License (CC BY-NC-SA 4.0) Under this license, any user of this textbook or the textbook contents herein can share, remix, and build upon the content for noncommercial purposes only Any adaptations must be shared under the same type of license In any case of sharing the original or adapted material, whether in whole or in part, the user must provide proper attribution as follows: - - - - If you noncommercially redistribute this textbook in a digital format (including but not 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Access The future of education OpenStax.org I like free textbooks and I cannot lie Table of Contents Preface Chapter 1: Parametric Equations and Polar Coordinates 1.1 Parametric Equations 1.2 Calculus of Parametric Curves 1.3 Polar Coordinates 1.4 Area and Arc Length in Polar Coordinates 1.5 Conic Sections Chapter 2: Vectors in Space 2.1 Vectors in the Plane 2.2 Vectors in Three Dimensions 2.3 The Dot Product 2.4 The Cross Product 2.5 Equations of Lines and Planes in Space 2.6 Quadric Surfaces 2.7 Cylindrical and Spherical Coordinates Chapter 3: Vector-Valued Functions 3.1 Vector-Valued Functions and Space Curves 3.2 Calculus of Vector-Valued Functions 3.3 Arc Length and Curvature 3.4 Motion in Space Chapter 4: Differentiation of Functions of Several Variables 4.1 Functions of Several Variables 4.2 Limits and Continuity 4.3 Partial Derivatives 4.4 Tangent Planes and Linear Approximations 4.5 The Chain Rule 4.6 Directional Derivatives and the Gradient 4.7 Maxima/Minima Problems 4.8 Lagrange Multipliers Chapter 5: Multiple Integration 5.1 Double Integrals over Rectangular Regions 5.2 Double Integrals over General Regions 5.3 Double Integrals in Polar Coordinates 5.4 Triple Integrals 5.5 Triple Integrals in Cylindrical and Spherical Coordinates 5.6 Calculating Centers of Mass and Moments of Inertia 5.7 Change of Variables in Multiple Integrals Chapter 6: Vector Calculus 6.1 Vector Fields 6.2 Line Integrals 6.3 Conservative Vector Fields 6.4 Green’s Theorem 6.5 Divergence and Curl 6.6 Surface Integrals 6.7 Stokes’ Theorem 6.8 The Divergence Theorem Chapter 7: Second-Order Differential Equations 7.1 Second-Order Linear Equations 7.2 Nonhomogeneous Linear Equations 7.3 Applications 7.4 Series Solutions of Differential Equations Appendix A: Table of Integrals Appendix B: Table of Derivatives Appendix C: Review of Pre-Calculus Index 27 44 64 73 101 102 123 146 165 186 211 228 259 260 270 283 305 333 334 352 369 389 406 422 438 458 477 478 501 526 546 566 592 610 641 642 663 689 711 737 753 789 807 831 832 849 863 884 897 903 905 1013 This OpenStax book is available for free at http://cnx.org/content/col11966/1.2 Preface PREFACE Welcome to Calculus Volume 3, an OpenStax resource This textbook was written to increase student access to high-quality learning materials, maintaining highest standards of academic rigor at little to no cost About OpenStax OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 25 books for college and AP® courses used by hundreds of thousands of students OpenStax Tutor, our low-cost personalized learning tool, is being used in college courses throughout the country Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed About OpenStax's resources Customization Calculus Volume is licensed under a Creative Commons Attribution 4.0 International (CC BY) license, which means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most relevant to the needs of your course Feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer You can even provide a direct link in your syllabus to the sections in the web view of your book Instructors also have the option of creating a customized version of their OpenStax book The custom version can be made available to students in low-cost print or digital form through their campus bookstore Visit your book page on OpenStax.org for more information Errata All OpenStax textbooks undergo a rigorous review process However, like any professional-grade textbook, errors sometimes occur Since our books are web based, we can make updates periodically when deemed pedagogically necessary If you have a correction to suggest, submit it through the link on your book page on OpenStax.org Subject matter experts review all errata suggestions OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on OpenStax.org Format You can access this textbook for free in web view or PDF through OpenStax.org, and for a low cost in print About Calculus Volume Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency Volume covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations Coverage and scope Our Calculus Volume textbook adheres to the scope and sequence of most general calculus courses nationwide We have worked to make calculus interesting and accessible to students while maintaining the mathematical rigor inherent in the subject With this objective in mind, the content of the three volumes of Calculus have been developed and arranged to provide a logical progression from fundamental to more advanced concepts, building upon what students have already learned and emphasizing connections between topics and between theory and applications The goal of each section is to enable students not just to recognize concepts, but work with them in ways that will be useful in later courses and future careers The organization and pedagogical features were developed and vetted with feedback from mathematics educators dedicated to the project Volume Preface Chapter 1: Functions and Graphs Chapter 2: Limits Chapter 3: Derivatives Chapter 4: Applications of Derivatives Chapter 5: Integration Chapter 6: Applications of Integration Volume Chapter 1: Integration Chapter 2: Applications of Integration Chapter 3: Techniques of Integration Chapter 4: Introduction to Differential Equations Chapter 5: Sequences and Series Chapter 6: Power Series Chapter 7: Parametric Equations and Polar Coordinates Volume Chapter 1: Parametric Equations and Polar Coordinates Chapter 2: Vectors in Space Chapter 3: Vector-Valued Functions Chapter 4: Differentiation of Functions of Several Variables Chapter 5: Multiple Integration Chapter 6: Vector Calculus Chapter 7: Second-Order Differential Equations Pedagogical foundation Throughout Calculus Volume you will find examples and exercises that present classical ideas and techniques as well as modern applications and methods Derivations and explanations are based on years of classroom experience on the part of long-time calculus professors, striving for a balance of clarity and rigor that has proven successful with their students Motivational applications cover important topics in probability, biology, ecology, business, and economics, as well as areas of physics, chemistry, engineering, and computer science Student Projects in each chapter give students opportunities to explore interesting sidelights in pure and applied mathematics, from navigating a banked turn to adapting a moon landing vehicle for a new mission to Mars Chapter Opening Applications pose problems that are solved later in the chapter, using the ideas covered in that chapter Problems include the average distance of Halley's Comment from the Sun, and the vector field of a hurricane Definitions, Rules, and Theorems are highlighted throughout the text, including over 60 Proofs of theorems Assessments that reinforce key concepts In-chapter Examples walk students through problems by posing a question, stepping out a solution, and then asking students to practice the skill with a “Checkpoint” question The book also includes assessments at the end of each chapter so students can apply what they’ve learned through practice problems Many exercises are marked with a [T] to indicate they are suitable for solution by technology, including calculators or Computer Algebra Systems (CAS) Answers for selected exercises are available in the Answer Key at the back of the book The book also includes assessments at the end of each chapter so students can apply what they’ve learned through practice problems Early or late transcendentals The three volumes of Calculus are designed to accommodate both Early and Late Transcendental approaches to calculus Exponential and logarithmic functions are introduced informally in Chapter of Volume and presented in more rigorous terms in Chapter in Volume and Chapter in Volume Differentiation and integration of these functions is covered in Chapters 3–5 in Volume and Chapter in Volume for instructors who want to include them with other types of functions These discussions, however, are in separate sections that can be skipped for instructors who prefer to wait until the integral definitions are given before teaching the calculus derivations of exponentials and logarithms This OpenStax book is available for free at http://cnx.org/content/col11966/1.2 Answer Key 1001 x 111 Conservative, f (x, y) = ye + x sin(y) 113 ∮ C (2ydx + 2xdy) = 32 115 F(x, y) = (10x + 3y)i + (3x + 10y)j 117 F is not conservative z 119 F is conservative and a potential function is f (x, y, z) = xye 121 F is conservative and a potential function is f (x, y, z) = z 123 F is conservative and a potential function is f (x, y, z) = x y + y z 125 F is conservative and a potential function is f (x, y) = e x y 127 ∫ C 129 ∫ C 131 ∮ 133 ∮ 135 ∫ C 137 ∫ C F · dr = e + F · dr = 41 G · dr = −8π C1 F · dr = C2 F · dr = 150 F · dr = −1 139 × 10 31 erg 141 ∫ C F · ds = 0.4687 143 circulation = πa and flu = 147 ∫ C 2xydx + (x + y)dy = 32 149 ∫ C sin x cos ydx + (xy + cos x sin y)dy = 12 151 ∮ C 153 ∫ C 155 ∮ C 157 ∮ C 159 ∮ C (−ydx + xdy) = π xe −2x dx + ⎛⎝x + 2x y 2⎞⎠dy = y dx − x dy = −24π −x ydx + xy dy = 8π ⎛ ⎝x + y 2⎞⎠dx + 2xydy = 161 A = 19π 163 A = 165 ∫ 8π C+ ⎛ ⎝y + x 3⎞⎠dx + x dy = 9π 167 A = 169 A = 171 ∫ C ⎛ ⎝x y − 2xy + y 2⎞⎠ds = 1002 173 Answer Key ∫ xdx + ydy = 2π x2 + y2 C 225 175 W = 177 W = 12π 179 W = 2π 181 ∮ 183 ∫ C 185 ∫ C 187 ∮ C 189 ∮ C 191 ∮ C 193 ∮ C 195 ∫ C 197 ∫ C 199 ∮ C 201 ∮ C 203 ∫ C 205 ∮ C y dx + x dy = + x 3dx + 2xydy = ⎛ ⎝3y ⎞ ⎛ − e sin x⎞⎠dx + ⎝7x + y + 1⎠dy = 36π F · dr = (y + x)dx + (x + sin y)dy = xydx + x y dy = 22 21 F · dr = 15π sin(x + y)dx + cos(x + y)dy = F · dr = π F· ^ n ds = F · nds = 32π ⎡ ⎣−y C ⎛ ⎝y + sin(xy) + xy cos(xy)⎤⎦dx + ⎡⎣x + x cos(xy)⎤⎦dy = 4.7124 x⎞ +e ⎠dx + ⎛⎝2x + cos⎛⎝y 2⎞⎠⎞⎠dy = 207 False 209 True 211 True 213 curl F = i + x j + y k ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 2 2 2 215 curl F = ⎝xz − xy ⎠i + ⎝x y − yz ⎠j + ⎝y z − x z⎠k 217 curl F = i + j + k 219 curl F = −yi − zj − xk 221 curl F = 223 div F = 3yz + 2y sinz + 2xe 2z 225 div F = 2(x + y + z) 227 div F = x + y2 229 div F = a + b 231 div F = x + y + z 233 Harmonic 235 div (F × G) = 2z + 3x 237 div F = 2r 239 curl r = This OpenStax book is available for free at http://cnx.org/content/col11966/1.2 Answer Key 1003 r 241 curl = r 2x k 243 curl F = x + y2 245 div F = 247 div F = − 2e −6 249 div F = 251 curl F = j − 3k 253 curl F = 2j − k 255 a = 257 F is conservative 259 div F = cosh x + sinh y − xy 261 (bz − cy)i(cx − az)j + (ay − bx)k 263 curl F = 2ω 265 F × G does not have zero divergence 2 267 ∇ · F = −200k⎡⎣1 + 2⎛⎝x + y + z 2⎞⎠⎤⎦e −x + y + z 269 True 271 True 273 r(u, v) = 〈 u, v, − 3u + 2v 〉 for −∞ ≤ u < ∞ and −∞ ≤ v < ∞ 275 r(u, v) = 〈 u, v, (16 − 2u + 4v) 〉 for |u| < ∞ and |v| < ∞ π 277 r(u, v) = 〈 cos u, sin u, v 〉 for ≤ u ≤ , ≤ v ≤ 279 A = 87.9646 281 ∬ SzdS = 8π ⎛ ⎞ 2 283 ∬ S⎝x + y ⎠zdS = 16π 4π 285 ∬ F · NdS = S 287 m ≈ 13.0639 289 m ≈ 228.5313 291 ∬ SgdS = ⎛ ⎞ 293 ∬ S⎝x + y − z⎠dS ≈ 0.9617 ⎛ 2⎞ 4π 295 ∬ ⎝x + y ⎠dS = 297 ∬ x zdS = 1023 2π S 299 ∬ (z + y)dS ≈ 10.1 S S 301 m = πa 13 303 ∬ F · NdS = 24 305 ∬ F · NdS = S S 307 y + z⎞⎛1 ∫ ∫ ⎛⎝4 − 3y + 16 ⎠⎝4 0 309 ∫ ∫ ⎡⎣x − 2(8 − 4x) + z⎤⎦ 0 ⎞ 17⎠dzdy 17dzdx 1004 Answer Key 311 ∬ ⎛⎝x z + y z⎞⎠dS = πa S 2 313 ∬ S x yzdS = 171 14 315 ∬ yzdS = 2π 317 ∬ S(xi + yj) · dS = 16π S 319 m = πa 192 321 F ≈ 4.57 lb 323 8πa 325 The net flux is zero 327 ∬ S(curl F · N)dS = πa 329 ∬ S(curl F · N)dS = 18π 331 ∬ S(curl F · N)dS = −8π 333 ∬ S(curl F · N)dS = 335 ∫ C 337 ∫ C F · dS = F · dS = −9.4248 339 ∬ curl F · dS = S 341 ∬ Scurl F · dS = 2.6667 343 ∬ (curl F · N)dS = − S 345 ∫ C ⎛1 ⎞ π ⎝2 y dx + zdy + xdz⎠ = − 347 ∬ S(curl F · N)dS = −3π 349 ∫ C (ck × R) · dS = 2πc 351 ∬ Scurl F · dS = 353 ∮ F · dS = −4 355 ∬ Scurl F · dS = 357 ∬ Scurl F · dS = −36π 359 ∬ Scurl F · N = 361 ∮ C F · dr = 363 ∬ Scurl(F) · dS = 84.8230 365 A = ∬ S(∇ × F) · ndS = 367 ∬ S(∇ × F) · ndS = 2π ⎛ ⎞ 369 C = π ⎝cos φ − sin φ⎠ 371 ∮ C F · dr = 48π This OpenStax book is available for free at http://cnx.org/content/col11966/1.2 Answer Key 1005 373 ∬ S(∇ × F) · n = 375 377 ∫ S 379 ∫ S 381 ∫ S F · nds = 75.3982 F · nds = 127.2345 F · nds = 37.6991 F · nds = 9πa S π 385 ∬ SF · dS = 387 ∬ F · dS = 383 ∫ S 389 ∬ SF · dS = 241.2743 391 ∬ DF · dS = −π 2π 393 ∬ F · dS = S 395 16 6π 128 π 397 − 399 −703.7168 401 20 403 ∬ SF · dS = 405 ∬ F · NdS = S 407 ∬ S ‖ R ‖ R · nds = 4πa 4π 409 ∭ z dV = R 15 ∬ F · dS = 6.5759 411 S 413 ∬ SF · dS = 21 415 ∬ SF · dS = 72 417 ∬ SF · dS = −33.5103 419 ∬ SF · dS = πa b 421 ∬ F · dS = π 21π 423 ∬ F · dS = S S ⎛ ⎞ −1 425 −⎝1 − e ⎠ Review Exercises 427 False 429 False 431 1006 Answer Key y 433 Conservative, f (x, y) = xy − 2e 435 Conservative, f (x, y, z) = x y + y z + z x 16 437 − 32 ⎛3 − 1⎞ 439 ⎠ ⎝ 441 Divergence: e x + xe xy + xye xyz , curl: xze xyz i − yze xyz 443 −2π 445 −π 447 31π/2 449 2(2 + π) 451 2π/3 Chapter Checkpoint 7.1 a Nonlinear b Linear, nonhomogeneous 7.4 Linearly independent 7.5 y(x) = c e 3x + c xe 3x 7.6 ⎞ x⎛ a y(x) = e ⎝c cos3x + c sin3x⎠ b xy j + ye k y(x) = c e −7x + c xe −7x 7.7 y(x) = −e −2x + e 5x x 7.8 y(x) = e (2cos3x − sin3x) This OpenStax book is available for free at http://cnx.org/content/col11966/1.2 Answer Key 1007 7.9 y(t) = te −7t At time t = 0.3, y(0.3) = 0.3e (−7 * 0.3) = 0.3e −2.1 ≈ 0.0367 The mass is 0.0367 ft below equilibrium At time t = 0.1, y′(0.1) = 0.3e −0.7 ≈ 0.1490 The mass is moving downward at a speed of 0.1490 ft/sec 7.10 y(x) = c e −x + c e 4x − 1008 Answer Key 7.11 y(t) = c e 2t + c te 2t + sint + cost 7.12 a y(x) = c e 4x + c e x − xe x b y(t) = c e −3t + c e 2t − 5cos2t + sin2t 3x + , 7.13 z = 11x 7.14 a b z = 2x + 11x y(x) = c cos x + c sin x + cos x ln|cos x| + x sin x x(t) = c e t + c te t + te t ln|t| 14 Hz 7.15 x(t) = 0.1cos(14t) (in meters); frequency is 2π ≈ 0.637, 7.16 x(t) = 17sin(4t + 0.245), frequency = 2π A = 17 7.17 x(t) = 0.6e −2t − 0.2e −6t −8t + 4te −8t 7.18 x(t) = e 7.19 x(t) = −0.24e −2t cos(4t) − 0.12e −2t sin(4t) −2t cos(4t) − 2e −2t sin(4t) 7.20 x(t) = − cos(4t) + sin(4t) + e −2t −2t Transient solution: e cos(4t) − 2e sin(4t) Steady-state solution: − cos(4t) + sin(4t) −t −t 7.21 q(t) = −25e cos(3t) − 7e sin(3t) + 25 7.22 ∞ ∑ (−1) n 2n x = a e −x n! a y(x) = a b y(x) = a (x + 1) n=0 Section Exercises linear, homogenous nonlinear linear, homogeneous 11 y = c e 5x + c e −2x 13 y = c e −2x + c xe −2x 15 y = c e 5x/2 + c e −x −x/2 ⎛ 3x 3x ⎞ 17 y = e ⎝c cos + c sin ⎠ 19 y = c e −11x + c e 11x 21 y = c cos9x + c sin9x 23 y = c + c x ⎛⎛ + 22⎞⎠/3⎞⎠x 25 y = c e ⎝⎝ + c2 e ⎛⎛ ⎝⎝1 − 22⎞⎠/3⎞⎠x 27 y = c e −x/6 + c xe −x/6 29 y = c + c e 9x 31 y = −2e −2x + 2e −3x 33 y = 3cos(2x) + 5sin(2x) This OpenStax book is available for free at http://cnx.org/content/col11966/1.2 Answer Key 1009 35 y = −e 6x + 2e −5x −x/5 −x/5 + xe 37 y = 2e 39 y = ⎛ ⎞ 6x ⎛ ⎞ −7x ⎝e − e −7 ⎠e − ⎝e − e −7 ⎠e 41 No solutions exist 2x 2e + 2x xe 43 y = 2e − e2 45 y = 4cos3x + c sin3x, infinite y many solutions 47 5y″ + 19y′ − 4y = 49 a y = 3cos(8x) + 2sin(8x) b 51 a y = e (−5/2)x ⎡ ⎛ 35 ⎞ 35 ⎛ 35 ⎞⎤ ⎣−2cos ⎝ x⎠ + 35 sin ⎝ x⎠⎦ b 55 y = c e −4x/3 + c e x − e −2x 57 y = c cos4x + c sin4x + 20 59 y = c e 2x + c xe 2x + 2x + 5x −x −x 61 y = c e + c xe + sin x − cos x 2 1010 Answer Key 63 y = c cos x + c sin x − x cos2x − sin2x −5x + c xe −5x + x e −5x + 65 y = c e 25 67 a y p(x) = Ax + Bx + C 35 b y p(x) = − x + x − 3 ⎛ ⎞ −x 69 a y p(x) = ⎝Ax + Bx + C⎠e ⎛ ⎞ −x b y p(x) = ⎝1 x − x − 33 ⎠e 32 ⎛ ⎞ ⎛ ⎞ x x 71 a y p(x) = ⎝Ax + Bx + C⎠e cos x + ⎝Dx + Ex + F ⎠e sin x ⎛ ⎞ x ⎛ ⎞ x 2 b y p(x) = ⎝− x − 11 x − 27 ⎠e cos x + ⎝− x + x + 39 ⎠e sin x 10 10 25 250 25 250 −2x + e 3x 73 y = c + c e 15 75 y = c e 2x + c e −4x + xe 2x 3x −3x 8x − 77 y = c e + c e 3 79 y = c cos2x + c sin2x − x cos2x + sin2x ln(sin2x) 347 + e 7x + x e 7x − xe 7x 81 y = − 343 343 49 5x 5x −5x 57 + e + xe + e 83 y = − 25 25 25 10 x ln x 85 y p = + π Hz 87 x″ + 16x = 0, x(t) = cos(4t) − 2sin(4t), period = sec, frequency = π π Hz 89 x″ + 196x = 0, x(t) = 0.15cos(14t), period = sec, frequency = π 91 a x(t) = 5sin(2t) Hz b period = π sec, frequency = π c π d t = sec 93 a x(t) = e −t/5 ⎛⎝20cos(3t) + 15sin(3t)⎞⎠ b underdamped 95 a x(t) = 5e −4t + 10te −4t This OpenStax book is available for free at http://cnx.org/content/col11966/1.2 Answer Key 1011 b critically damped −π/4 ft below 97 x(π) = 7e ⎛ ⎞ 32 16 sin ⎛ 128t⎞ ⎝ ⎠ 99 x(t) = sin(4t) + cos ⎝ 128t⎠ − −6t ⎛ ⎞ cos(10t) ⎝0.051cos(8t) + 0.03825sin(8t)⎠ − 101 q(t) = e 103 q(t) = e −10t (−32t − 5) + 5, I(t) = 2e 105 y = c + 5c 107 y = c 109 y = c ∞ ∞ ∑ n=1 (160t + 9) (−x/5) = c + 5c e −x/5 n! ∞ (x) 2n (x) 2n + + c1 ∑ (2n) ! (2n + 1) ! n=0 ∑ x 2n = c e x n! n=0 ∞ 20 n ∑ n=0 ∞ −10t ∞ x 2n + c ∑ x 2n + n 1 ⋅ ⋅ ⋅ ⋯ (2n + 1) n = n! n=0 c 113 y = c x + x2 111 y = c ∑ 115 y = − 3x + 2x − 12x + 16x − 120x + ⋯ 3! 4! 6! 7! Review Exercises 117 True 119 False 121 second order, linear, homogeneous, λ − = 123 first order, nonlinear, nonhomogeneous 125 y = c sin(3x) + c cos(3x) x x 2 127 y = c e sin(3x) + c e cos(3x) + x + 25 2x 129 y = c e −x + c e −4x + x + e − 18 16 131 y = c e (−3/2)x + c xe 133 y = e −2x sin ⎛⎝ 2x⎞⎠ 1−x ⎛ ⎞ 4x e 135 y = ⎝e − 1⎠ e −1 ⎛ g⎞ 137 θ(t) = θ cos ⎝ t⎠ l 141 b = a (−3/2)x + x + x − 16 27 27 1012 This OpenStax book is available for free at http://cnx.org/content/col11966/1.2 Answer Key Index 1013 INDEX Symbols δ ball, 364, 472 δ disk, 352, 472 A acceleration vector, 305, 327 angular coordinate, 44, 96 angular frequency, 864 arc-length function, 285, 327 arc-length parameterization, 286, 327 Archimedean spiral, 56 B Bessel functions, 888 binormal vector, 292, 327 boundary conditions, 843, 892 boundary point, 358, 471 boundary-value problem, 843, 892 Brahe, 315 C cardioid, 54, 96 chain rule, 272 chambered nautilus, 8, 56 characteristic equation, 838, 892 circulation, 683, 823 cissoid of Diocles, 72 Clairaut’s theorem, 658 closed curve, 670, 689, 823, 823 closed set, 359, 471 Cobb-Douglas function, 464 Cobb-Douglas production function, 388 complementary equation, 849, 892 complex conjugates, 839 complex number, 839 component, 249 component functions, 260, 327 components, 109 conic section, 73, 96 connected region, 690, 823 connected set, 359, 471 conservative field, 654, 823 constant multiple rule, 272 constraint, 471 constraints, 458 contour map, 342, 373, 471 coordinate plane, 249 coordinate planes, 125 critical point of a function of two variables, 438, 471 cross product, 165, 249 cross-partial property, 657 curl, 743, 823 curtate cycloid, 22 curvature, 288, 327 cusp, 96 cusps, 18 cycloid, 17, 96 cylinder, 211, 249 cylindrical coordinate system, 228, 249 D definite integral of a vectorvalued function, 276, 327 derivative, 270 derivative of a vector-valued function, 270, 327 determinant, 171, 249 differentiable, 396, 471 direction angles, 153, 249 direction cosines, 153, 249 direction vector, 186, 249 directional cosines, 434 directional derivative, 423, 471 directrix, 74, 96 discriminant, 90, 96, 443, 471 divergence, 737, 823 divergence theorem, 808, 823 domain, 643 dot product, 146 dot product or scalar product, 249 double integral, 480, 633 double Riemann sum, 480, 633 E Earth’s orbit, eccentricity, 87, 96 electrical potential, 437 Electrical power, 405 electrical resistance, 404 electrostatic fields, 815 ellipsoid, 215, 249 Elliptic Cone, 222 elliptic cone, 249 elliptic paraboloid, 217, 249 Elliptic Paraboloid, 222 epitrochoid, 26 equivalent vectors, 102, 249 Ernest Rutherford, 385 error term, 396 Euler’s formula, 840 expected values, 518 F Faraday’s law, 799 flow line, 661 flux, 681, 823 flux integral, 778, 823 focal parameter, 88, 96 focus, 74, 96 force, 116 Fourier’s law of heat transfer, 822 Frenet frame of reference, 297, 327 Fubini’s theorem, 484, 633 Fubini’s thereom, 548 function of two variables, 334, 471 Fundamental Theorem for Line Integrals, 807, 823 Fundamental Theorem for Line Integrals., 693 Fundamental Theorem of Calculus, 807 G Gauss’ law, 815, 823 Gauss’s law for magnetism, 739 general bounded region, 550 general form, 76, 96 general form of the equation of a plane, 195, 249 general solution to a differential equation, 837 generalized chain rule, 412, 471 gradient, 427, 471 gradient field, 654, 823 graph of a function of two variables, 344, 471 gravitational force, 702 Green’s theorem, 711, 807, 823 grid curves, 762, 823 H harmonic function, 724 heat equation, 379 heat flow, 781, 823 helix, 264, 265, 327 higher-order partial derivatives, 377, 471 homogeneous functions, 420 homogeneous linear equation, 832, 892 Hooke’s law, 863 hurricanes, 648 1014 Hyperboloid of One Sheet, 221 hyperboloid of one sheet, 249 Hyperboloid of Two Sheets, 221 hyperboloid of two sheets, 249 hypocycloid, 18 I implicit differentiation, 415 improper double integral, 514, 633 indefinite integral of a vectorvalued function, 276, 327 independence of path, 823 independent of path, 696 independent random variables, 517 independent variables, 410 initial point, 102, 249, 261 initial-value problems, 842 interior point, 358, 471 intermediate variable, 471 intermediate variables, 407 inverse-square law, 816, 823 iterated integral, 484, 633 J Jacobian, 614, 633 joint density function, 517 K Kepler, 315 Kepler’s laws of planetary motion, 315, 327 L Lagrange multiplier, 459, 471 Laplace operator, 748 Laplace’s equation, 379, 724 level curve of a function of two variables, 341, 471 level surface of a function of three variables, 348, 472 limaỗon, 54, 96 limit of a function of two variables, 353 limit of a vector-valued function, 265, 327 line integral, 663, 823 linear approximation, 394, 472 linearly dependent, 836, 892 linearly independent, 836, 892 local extremum, 441 lunes of Alhazen, 525 M magnitude, 102, 110, 249 major axis, 79, 96 Index mass flux, 777, 824 mass of a wire, 678 method of Lagrange multipliers, 459, 472 method of undetermined coefficients, 850, 892 method of variation of parameters, 857, 892 minor axis, 79, 96 mixed partial derivatives, 378, 472 N nappe, 96 nappes, 73 nonhomogeneous linear equation, 832, 892 normal, 151 normal component of acceleration, 308, 327 normal form of Green’s theorem, 719 normal plane, 297, 327 normal vector, 195, 249 normalization, 114, 249 O objective function, 458, 472 octants, 125, 249 one-to-one transformation, 611, 633 open set, 359, 472 optimization problem, 458, 472 orientation, 10, 96 orientation of a curve, 670, 824 orientation of a surface, 775, 824 orthogonal, 151 orthogonal vectors, 151, 250 osculating circle, 298, 327 osculating plane, 327 overdamped, 869 P parallelepiped, 177, 250 parallelogram method, 104, 250 parameter, 9, 96 parameter domain, 753 parameter domain (parameter space), 824 parameter space, 753 parameterization of a curve, 16, 96 parameterized surface, 753 parameterized surface (parametric surface), 824 parametric curve, 10, 96 This OpenStax book is available for free at http://cnx.org/content/col11966/1.2 parametric equations, 9, 96 parametric equations of a line, 187, 250 parametric surface, 753 partial derivative, 369, 472 partial differential equation, 379, 472 particular solution, 849, 892 path independent, 696 perpendicular, 151 piecewise smooth curve, 675, 824 planar transformation, 610, 633 plane curve, 262, 327 polar axis, 47, 96 polar coordinate system, 44, 96 polar equation, 96 polar equations, 52 polar rectangle, 526, 633 pole, 47, 96 potential function, 655, 824 power series, 884 principal unit normal vector, 292, 327 principal unit tangent vector, 275, 327 product rule, 272 projectile motion, 311, 327 prolate cycloid, 23 Q Quadric surfaces, 215 quadric surfaces, 250 R radial coordinate, 44, 97 radial field, 644, 824 radius of curvature, 298, 327 radius of gyration, 600, 633 region, 359, 472 regular parameterization, 760, 824 reparameterization, 265, 327 resolution of a vector into components, 157 resonance, 879 Reuleaux triangle, 525 right-hand rule, 123, 250 RLC series circuit, 879, 892 rose, 54, 97 rotational field, 647, 824 rulings, 211, 250 S saddle point, 442, 472 scalar, 104, 250 scalar equation of a plane, 195, Index 250 scalar line integral, 664, 824 scalar multiplication, 104, 250 scalar projection, 156, 250 simple curve, 689, 824 simple harmonic motion, 864, 892 simple pendulum, 404 simply connected region, 690, 824 skew lines, 192, 250 smooth, 292, 327 space curve, 262, 328 space-filling curve, 55, 97 space-filling curves, 18 speed, 121 sphere, 130, 250 spherical coordinate system, 235, 250 spring-mass system, 863 standard equation of a sphere, 131, 250 standard form, 75, 97 standard position, 262 standard unit vectors, 115, 250 standard-position vector, 109, 250 steady-state solution, 877, 892 Stokes’ theorem, 789, 807, 824 stream function, 723, 824 sum and difference rules, 272 superposition principle, 835 surface, 337, 472 surface area, 763, 824 surface independent, 796, 824 surface integral, 778, 824 surface integral of a scalarvalued function, 768, 824 surface integral of a vector field, 777, 824 symmetric equations of a line, 250 symmetric equations of a line, 187 symmetry, 57, 537 T Tacoma Narrows Bridge, 879 tangent plane, 389, 472 tangent vector, 275, 328 tangential component of acceleration, 308, 328 tangential form of Green’s theorem, 712 terminal point, 102, 250, 261 three-dimensional rectangular 1015 coordinate system, 123, 250 topographical map, 340 Torque, 180 torque, 250 total differential, 400, 472 trace, 250 traces, 214 transformation, 610, 634 transient solution, 877 tree diagram, 410, 472 triangle inequality, 105, 250 triangle method, 104, 250 triple integral, 547, 634 triple integral in cylindrical coordinates, 568, 634 triple integral in spherical coordinates, 575, 634 triple scalar product, 175, 251 Type I, 502, 634 Type II, 502, 634 U unit vector, 114, 251 unit vector field, 651, 824 V vector, 102, 251 vector addition, 105, 251 vector difference, 105, 251 vector equation of a line, 187, 251 vector equation of a plane, 195, 251 vector field, 643, 824 vector line integral, 671, 824 vector parameterization, 262, 328 vector product, 169, 251 vector projection, 156, 251 vector sum, 104, 251 vector-valued function, 260, 328 vector-valued functions, 305 velocity vector, 305, 328 vertex, 74, 97 vertical trace, 344, 472 W wave equation, 379 William Thomson (Lord Kelvin), 383 witch of Agnesi, 20 work done by a force, 251 work done by a vector field, 679 work done by the force, 160 Z zero vector, 102, 251 ... College Park Peter Olszewski, Penn State Erie, The Behrend College Steven Purtee, Valencia College Alice Ramos, Bethel College This OpenStax book is available for free at http://cnx.org/content/col11966/1.2... assumed the ant climbed onto the tire at the very edge, where the tire touches the ground As the wheel rolls, the ant moves with the edge of the tire (Figure 1. 13) As we have discussed, we have a lot... in the figure, we let b denote the distance along the spoke from the center of the wheel to the ant As before, we let t represent the angle the tire has rotated through Additionally, we let C