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volume 3 calculus volume 3 senior contributing authors gilbert strang, massachusetts institute of technology edwin “ jed ” herman, university of wisconsin at stevens point 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. ©2016 rice university. textbook content produced by openstax is licensed under a creative commons attribution non - commercial 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 limited to pdf and html ), then you must retain on every page the following attribution : “ access for free at openstax. org. ” - if you noncommercially redistribute this textbook in a print format, then you must include on every physical page the following attribution : “ access for free at openstax. org. ” - if you noncommercially redistribute part of this textbook, then you must retain in every digital format page view ( including but not limited to pdf and html ) and on every physical printed page the following attribution : “ access for free at openstax. org. ” - if you use this textbook as a bibliographic reference, please include https : / / openstax. org / details / books / calculus - volume - 3 in your citation. for questions regarding this licensing, please contact support @ openstax. org. trademarks the openstax name, openstax logo, openstax book covers, openstax cnx name, openstax cnx logo, opensta | openstax_calculus_volume_3_-_web | [
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calculus - volume - 3 in your citation. for questions regarding this licensing, please contact support @ openstax. org. trademarks the openstax name, openstax logo, openstax book covers, openstax cnx name, openstax cnx logo, openstax tutor name, openstax tutor logo, connexions name, connexions logo, rice university name, and rice university logo are not subject to the license and may not be reproduced without the prior and express written consent of rice university. hardcover book isbn - 13 978 - 1 - 938168 - 07 - 9 b & w paperback book isbn - 13 978 - 1 - 50669 - 805 - 2 digital version isbn - 13 978 - 1 - 947172 - 16 - 6 original publication year 2016 2 3 4 5 6 7 8 9 10 cjp 24 16 openstax openstax provides free, peer - reviewed, openly licensed textbooks for introductory college and advanced placement® courses and low - cost, personalized courseware that helps students learn. a nonprofit ed tech initiative based at rice university, we ’ re committed to helping students access the tools they need to complete their courses and meet their educational goals. rice university openstax is an initiative of rice university. as a leading research university with a distinctive commitment to undergraduate education, rice university aspires to path - breaking research, unsurpassed teaching, and contributions to the betterment of our world. it seeks to fulfill this mission by cultivating a diverse community of learning and discovery that produces leaders across the spectrum of human endeavor. philanthropic support openstax is grateful for the generous philanthropic partners who advance our mission to improve educational access and learning for everyone. to see the impact of our supporter community and our most updated list of partners, please visit openstax. org / foundation. arnold ventures chan zuckerberg initiative chegg, inc. arthur and carlyse ciocca charitable foundation digital promise ann and john doerr bill & melinda gates foundation girard foundation google inc. the william and flora hewlett foundation the hewlett - packard company intel inc. rusty and john jaggers the calvin k. kazan | openstax_calculus_volume_3_-_web | [
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chegg, inc. arthur and carlyse ciocca charitable foundation digital promise ann and john doerr bill & melinda gates foundation girard foundation google inc. the william and flora hewlett foundation the hewlett - packard company intel inc. rusty and john jaggers the calvin k. kazanjian economics foundation charles koch foundation leon lowenstein foundation, inc. the maxfield foundation burt and deedee mcmurtry michelson 20mm foundation national science foundation the open society foundations jumee yhu and david e. park iii brian d. patterson usa - international foundation the bill and stephanie sick fund steven l. smith & diana t. go stand together robin and sandy stuart foundation the stuart family foundation tammy and guillermo trevino valhalla charitable foundation white star education foundation schmidt futures william marsh rice university study where you want, what you want, when you want. access. the future of education. openstax. org when you access your book in our web view, you can use our new online highlighting and note - taking features to create your own study guides. our books are free and flexible, forever. get started at openstax. org / details / books / calculus - volume - 3 contents preface 1 parametric equations and polar coordinates 7 1 introduction 7 1. 1 parametric equations 8 1. 2 calculus of parametric curves 22 1. 3 polar coordinates 38 1. 4 area and arc length in polar coordinates 55 1. 5 conic sections 63 chapter review 84 vectors in space 89 2 introduction 89 2. 1 vectors in the plane 90 2. 2 vectors in three dimensions 109 2. 3 the dot product 132 2. 4 the cross product 149 2. 5 equations of lines and planes in space 169 2. 6 quadric surfaces 192 2. 7 cylindrical and spherical coordinates 209 chapter review 228 vector - valued functions 235 3 introduction 235 3. 1 vector - valued functions and space curves 235 3. 2 calculus of vector - valued functions 243 3. 3 arc length and curvature 256 3. 4 motion in space 274 chapter review 296 differentiation of functions of several variables 301 4 introduction 301 4. 1 functions of several variables 302 4. 2 limits and continuity 316 4. 3 partial derivatives 330 4. 4 tangent planes and linear approximation | openstax_calculus_volume_3_-_web | [
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valued functions 243 3. 3 arc length and curvature 256 3. 4 motion in space 274 chapter review 296 differentiation of functions of several variables 301 4 introduction 301 4. 1 functions of several variables 302 4. 2 limits and continuity 316 4. 3 partial derivatives 330 4. 4 tangent planes and linear approximations 347 4. 5 the chain rule 361 4. 6 directional derivatives and the gradient 374 4. 7 maxima / minima problems 386 4. 8 lagrange multipliers 402 chapter review 413 multiple integration 419 5 introduction 419 5. 1 double integrals over rectangular regions 420 5. 2 double integrals over general regions 440 5. 3 double integrals in polar coordinates 464 5. 4 triple integrals 484 5. 5 triple integrals in cylindrical and spherical coordinates 503 5. 6 calculating centers of mass and moments of inertia 528 5. 7 change of variables in multiple integrals 545 chapter review 567 vector calculus 575 6 introduction 575 6. 1 vector fields 576 6. 2 line integrals 593 6. 3 conservative vector fields 616 6. 4 green ’ s theorem 636 6. 5 divergence and curl 660 6. 6 surface integrals 675 6. 7 stokes ’ theorem 706 6. 8 the divergence theorem 722 chapter review 738 second - order differential equations 745 7 introduction 745 7. 1 second - order linear equations 746 7. 2 nonhomogeneous linear equations 758 7. 3 applications 769 | openstax_calculus_volume_3_-_web | [
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7. 4 series solutions of differential equations 786 chapter review 794 table of integrals 799 a table of derivatives 805 b review of pre - calculus 807 c answer key 811 index 905 access for free at openstax. org 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 3 is licensed under a creative commons attribution - noncommercial - sharealike 4. 0 international ( cc - by - nc - sa ) 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, do not use the content for commercial purposes, and distribute the content under the same cc - by - nc - sa license. 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 | openstax_calculus_volume_3_-_web | [
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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 3 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 3 covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second - order differential equations. coverage and scope our calculus volume 3 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 | openstax_calculus_volume_3_-_web | [
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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 1 • chapter 1 : functions and graphs • chapter 2 : limits • chapter 3 : derivatives • chapter 4 : applications of derivatives preface 1 • chapter 5 : integration • chapter 6 : applications of integration volume 2 • 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 3 • 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 • | openstax_calculus_volume_3_-_web | [
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chapter 7 : second - order differential equations pedagogical foundation throughout calculus volume 3 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. 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 1 of volume 1 and presented in more rigorous terms in chapter 6 in volume 1 and chapter 2 in volume 2. differentiation and integration of these functions is covered in chapters 3 – 5 in volume 1 and chapter 1 in volume 2 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 | openstax_calculus_volume_3_-_web | [
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