Calculus Volume 3

1023 Pages·2018·64.37 MB·english

by  Gilbert Strang

Volume 3

Calculus Volume 3

SENIOR CONTRIBUTING AUTHORS
EDWIN "JED" HERMAN, UNIVERSITY OF WISCONSIN-STEVENS POINT
GILBERT STRANG, MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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PRINT BOOK ISBN-10 1-938168-07-0
PRINT BOOK ISBN-13 978-1-938168-07-9
PDF VERSION ISBN-10 1-947172-16-6
PDF VERSION ISBN-13 978-1-947172-16-6
Revision Number C3-2016-003(03/18)-MJ
Original Publication Year 2016
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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter1: Parametric Equations and Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 Parametric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Calculus of Parametric Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.4 Area and Arc Length in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.5 Conic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Chapter2: Vectors in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.1 Vectors in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.2 Vectors in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.3 The Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
2.4 The Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
2.5 Equations of Lines and Planes in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
2.6 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
2.7 Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Chapter3: Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
3.1 Vector-Valued Functions and Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 260
3.2 Calculus of Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
3.3 Arc Length and Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
3.4 Motion in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Chapter4: Differentiation of Functions of Several Variables . . . . . . . . . . . . . . . . . . . . 333
4.1 Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
4.2 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
4.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
4.4 Tangent Planes and Linear Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 389
4.5 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
4.6 Directional Derivatives and the Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
4.7 Maxima/Minima Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
4.8 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
Chapter5: Multiple Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
5.1 Double Integrals over Rectangular Regions . . . . . . . . . . . . . . . . . . . . . . . . . . 478
5.2 Double Integrals over General Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
5.3 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
5.4 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
5.5 Triple Integrals in Cylindrical and Spherical Coordinates . . . . . . . . . . . . . . . . . . . 566
5.6 Calculating Centers of Mass and Moments of Inertia . . . . . . . . . . . . . . . . . . . . . 592
5.7 Change of Variables in Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
Chapter6: Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641
6.1 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
6.2 Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
6.3 Conservative Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
6.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
6.5 Divergence and Curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737
6.6 Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753
6.7 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789
6.8 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807
Chapter7: Second-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 831
7.1 Second-Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832
7.2 Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849
7.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863
7.4 Series Solutions of Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 884
Appendix A:Table of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897
Appendix B:Table of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903
Appendix C:Review of Pre-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013
This OpenStax book is available for free at http://cnx.org/content/col11966/1.2
Preface 1
PREFACE
WelcometoCalculusVolume3,anOpenStaxresource.Thistextbookwaswrittentoincreasestudentaccesstohigh-quality
learning materials, maintaining highest standards of academic rigor at little to no cost.
About OpenStax
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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
OurCalculusVolume3textbookadherestothescopeandsequenceofmostgeneralcalculuscoursesnationwide.Wehave
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
learnedandemphasizingconnectionsbetweentopicsandbetweentheoryandapplications.Thegoalofeachsectionisto
enablestudentsnotjusttorecognizeconcepts,butworkwiththeminwaysthatwillbeusefulinlatercoursesandfuture
careers.Theorganizationandpedagogicalfeaturesweredevelopedandvettedwithfeedbackfrommathematicseducators
dedicated to the project.
Volume 1
2 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 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
Chapter 7: Second-Order Differential Equations
Pedagogical foundation
ThroughoutCalculusVolume3youwillfindexamplesandexercisesthatpresentclassicalideasandtechniquesaswellas
modern applications and methods. Derivations and explanations are based on years of classroom experience on the part
oflong-timecalculusprofessors,strivingforabalanceofclarityandrigorthathasprovensuccessfulwiththeirstudents.
Motivationalapplicationscoverimportanttopicsinprobability,biology,ecology,business,andeconomics,aswellasareas
ofphysics,chemistry,engineering,andcomputerscience.StudentProjectsineachchaptergivestudentsopportunitiesto
exploreinterestingsidelightsinpureandappliedmathematics,fromnavigatingabankedturntoadaptingamoonlanding
vehicleforanewmissiontoMars.ChapterOpeningApplicationsposeproblemsthataresolvedlaterinthechapter,using
theideascoveredinthatchapter.ProblemsincludetheaveragedistanceofHalley'sCommentfromtheSun,andthevector
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-chapterExampleswalkstudentsthroughproblemsbyposingaquestion,steppingoutasolution,andthenaskingstudents
to practice the skill with a “Checkpoint” question. The book also includes assessments at the end of each chapter so
studentscanapplywhatthey’velearnedthroughpracticeproblems.Manyexercisesaremarkedwitha[T]toindicatethey
aresuitable forsolutionbytechnology,includingcalculators orComputerAlgebraSystems(CAS).Answersforselected
exercisesareavailableintheAnswerKeyatthebackofthebook.Thebookalsoincludesassessmentsattheendofeach
chapter so students can apply what they’ve learned through practice problems.
Early or late transcendentals
ThethreevolumesofCalculusaredesignedtoaccommodatebothEarlyandLateTranscendentalapproachestocalculus.
ExponentialandlogarithmicfunctionsareintroducedinformallyinChapter1ofVolume1andpresentedinmorerigorous
termsinChapter6inVolume1andChapter2inVolume2.Differentiationandintegrationofthesefunctionsiscoveredin
Chapters3–5inVolume1andChapter1inVolume2forinstructorswhowanttoincludethemwithothertypesoffunctions.
Thesediscussions,however,areinseparatesectionsthatcanbeskippedforinstructorswhoprefertowaituntiltheintegral
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