Cengage has dramatically enhanced the online experience Interactive activities, animations, exercises, and topic-based lecture videos have been added to Stewart’s Calculus: Concepts & Contexts, e, WebAssign component Multivariable Calculus Concepts and Contexts | 4e Calculus and the Architecture of Curves and A s s o c ia te s T h e interio r atrium is dom inated by a cu rva ce o u s foursto ry sta in le ss steel sculp tural shell that su g g ests a p re h isto ric creature and ho u se s a central co n ference sp ace Th e high ly co m p le x stru ctu re s that Frank G eh ry d e sig n s w o u ld be im p o ssib le to build w ith o u t the com puter T h e C A TIA so ftw are that h is a rch i tects and engin eers use to produce the co m p u ter m odels is based on princip les of c a lc u lu s —fitting cu rves by m atching tangent lin e s, m aking sure the curvatu re isn 't too larg e, and controlling param etric su rface s "C o nsequently," sa y s G ehry, "w e have a lot of freed o m I can play w ith shapes." Th e pro cess starts w ith G e h ry's initial sketches, w hich are translated into a su c ce s sion of p hysical m o dels (H und reds of different p hysical m o dels w ere constructed during the design of the building, first with basic w ooden blocks and then e vo lvin g into m ore sculptural fo rm s.)T h e n an engineer u se s a digitizer to record the coo rdinates of a se rie s of points on a physical m o d e l.Th e digitized points are fed into a com puter and the CA TIA so ftw are is used to link these points w ith sm ooth cu rve s (It jo ins curves so that th e ir tangent lines coincide; you can use the sam e idea to design the shap es of letters in the Laboratory Project on page 208 of this bo ok.)The architect has considerable free dom in creating these cu rves, guided by d isp lays of the cu rve , its d e rivative , and its cu rva tu re Th e n the curves are Courtesy of Gehry Partners, LLP T h e co ver photograph sh o w s the DZ B ank in B e rlin , designed and built 1995-2001 by Frank G e h ry Courtesy of Gehry Partners, LLP co n n ected to each other by a p aram e tric su rfa ce , and again the architect can so in m any p o ssib le w a y s w ith the g u id an ce of d isp la y s of the g eo m etric c h a cte ristic s of the su rfa ce T h e C A T IA m odel is then used to produce an o th e r p h ysica l m o del, w h ic h , in turn, su g g e sts m o d ifica tio n s and leads to ad ditional co m p u ter and p h ysica l m o d els T h e C A T IA p ro g ram w a s develo p ed in France by D a ssa u it S y s tè m e s , o rig in a lly fo r d esign ing a irp la n e s , and w a s su b se q u e n tly e m p lo yed in the a u to m o tive in d u stry Frank G e h ry, b ecause of his co m p le x sc u lp tu l s h a p e s, is the firs t to use it in arch ite ctu re It h elp s him a n sw e r his q u e s tio n , "H o w w ig g ly can you get and still m ake a b u ild in g ?" Multivariable Calculus Concepts and Contexts I 4e James Stewart M c M a s te r University and University of Toronto TRƯỞNG ĐẠt Mọe o y v WHGN THƯ ỵ i /Ạ CENGAGE Australia • Brazil • Mexico • Singapore • United Kingdom • United States Ĩ Í ' CENGAGE Multivariable Calculus: Concepts and Contexts, Fourth Edition Enhanced Edition James Stewart © 2019, 2010 C engage Learning, Inc U nless o th e rw is e no te d , all c o n te n t is © Cengage A LL RIGHTS RESERVED N o p a rt o f th is w o rk covered by th e c o p y P ro d u c t D ire c to r: M a rk S antee P ro d u c t M a n a g e r: G a ry W h a le n C o n te n t D e v e lo p e r: Lynh Pham rig h t he rein m ay be re prod uced o r d is trib u te d in any fo rm o r by any m eans, e xce p t as p e rm itte d by U.S c o p y rig h t law, w ith o u t th e p rio r w r itte n p e rm issio n o f th e c o p y rig h t ow ner P ro d u c t A s s is ta n t: A b b y D eV euve M a rk e tin g M a n a g e r: Ryan A h e rn C o n te n t P ro je c t M a n a g e r: J e n n ife r R isden P ro d u c tio n S ervice: G p h ic W o rld P h o to R esearcher: Lu m ina D a ta m a tic s T e xt R esearcher: Lu m ina D a ta m a tic s For product information and technology assistance, contact us at Cengage Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be e-mailed to permissionrequest@cengage.com C o p y E d ito r: K athi Tow nes Illustrator: Brian Betsill L ib ry o f C ongress C o n tro l N um b er: 20 08940620 A r t D ire c to r: V e rn o n Boes ISBN: -1-337 -68 781-2 Text D e s ig n e r: Jeanne C albrese C o ve r D e s ig n e r: Ire n e M o rris ; D e n ise D a v id so n C o ve r Im age: th o m a s m a y e rc h iv e c o m In te rio r Im age, page iv: C h ris tia n M u e lle r /S h u tte rs to c k c o m Cengage 20 Channel Center Street Boston, MA 02210 USA C o m p o s ito r: G p h ic W o rld Cengage is a lea d in g p ro v id e r o f custo m ized learnin g s o lu tio n s w ith e m ployees re sid in g in nearly d iffe re n t co u n trie s and sales in m o re th a n 125 c o u n trie s a round th e w o rld Find y o u r local represen ta tiv e at www.cengage.com C engage Learning p ro d u cts are re prese nted in Canada by N elson E ducation, Ltd To learn m o re a b o u t Cengage p la tfo rm s and services, v is it www.cengage.com To re g iste r o r access y o u r o n lin e learning s o lu tio n o r purchase m a terials fo r y o u r course, v is it www.cengagebrain.com Printed in the United States of America Print Number: 01 Print Year: 2017 K10TI5 Trademarks Derive is a registered trademark of Soft Warehouse, Inc Maple is a registered trademark of Waterloo Maple, Inc Mathematica is a registered trademark of Wolfram Research, Inc Contents Preface xi To the Student xx 553 Infinite Sequences and S erie s 8.1 Sequences 554 8.2 Series 565 The Integral and Comparison Tests: Estimating Sums Laboratory Project ■ Logistic Sequences Goran Bogicevic/Shutterstock.com 8.3 564 8.4 Other Convergence Tests 8.5 Power Series 8.6 Representations of Functions as Power Series 8.7 Taylor and Maclaurin Series 585 592 598 604 Laboratory Project ■ An Elusive Limit 618 Writing Project ■ How Newton Discovered the Binomial Series 8.8 Applications of Taylor Polynomials 631 Vectors and the Geometry of Sp ace 9.1 9.2 9.3 9.4 627 628 Focus on Problem Solving Three-Dimensional Coordinate Systems Vectors 639 633 J 634 The Dot Product 648 The Cross Product 654 Discovery Project ■ The Geometry of a Tetrahedron 9.5 Equations of Lines and Planes 9.7 Functions and Surfaces 673 Cylindrical and Spherical Coordinates Laboratory Project ■ Families of Surfaces Review 662 663 Laboratory Project ■ Putting 3D in Perspective 9.6 618 619 Applied Project ■ Radiation from the Stars Review csp/Shutterstock.com 575 672 682 687 688 Focus on Problem Solving 691 VII viii CONTENTS 10 Vector Functions 693 10.1 Vector Functions and Space Curves 10.2 Derivatives and Integrals of Vector Functions 701 Arc Length and Curvature 707 Motion in Space: Velocity and Acceleration 716 10.3 10.4 Applied Project ■ Kepler's Laws 10.5 Parametric Surfaces Review 733 Partial Derivatives 11.1 726 727 Focus on Problem Solving h 694 735 737 11.2 Functions of Several Variables Limits and Continuity 749 738 11.3 Partial Derivatives 11.4 Tangent Planes and Linear Approximations 11.5 11.6 The Chain Rule 780 Directional Derivatives and the Gradient Vector 11.7 Maximum and Minimum Values 756 770 789 802 Applied Project ■ Designing a Dumpster 811 Discovery Project ■ Quadratic Approximations and Critical Points 11.8 Lagrange Multipliers 813 Applied Project ■ Rocket Science 820 Applied Project ■ Hydro-Turbine Optimization Review Focus on Problem Solving 12 Multiple Integrals 827 829 12.1 12.2 Double Integrals over Rectangles Iterated Integrals 838 12.3 Double Integrals over General Regions 844 Double Integrals in Polar Coordinates 853 Applications of Double Integrals 858 Surface Area 868 Triple Integrals 873 12.4 12.5 12.6 12.7 830 Discovery Project ■ Volumes of Hyperspheres 12.8 821 822 883 Triple Integrals in Cylindrical and Spherical Coordinates Applied Project ■ Roller Derby 889 Discovery Project ■ The Intersection of Three Cylinders 890 883 812 CONTENTS 12.9 Change of Variables in Multiple Integrals Review 891 899 Focus on Problem Solving Vector Calculus 903 905 13.2 Vector Fields Line Integrals 13.3 The Fundamental Theorem for Line Integrals 13.1 906 913 13.4 Green’s Theorem 13.5 Curl and Divergence 941 Surface Integrals 949 Stokes’ Theorem 960 13.6 13.7 934 Writing Project ■ Three Men and Two Theorems 13.8 The Divergence Theorem 13.9 Summary Review 967 973 974 Focus on Problem Solving Appendixes 977 A1 D Precise Definitions of Limits E A Few Proofs H Polar Coordinates I Complex Numbers J Answers to Odd-Numbered Exercises Index A51 925 A2 A3 A6 A22 A31 966 ix APPENDIX J ANSWERS TO ODD-NUMBERED EXERCISES dz _ 3xz — 2y 45 2z — 3x y ' dy dx 2z — 3xy "• ^ 1+ = -z dx + y + y 2z 2’ dy + y + y 2z 49 (a) /'(*), g'(y) (b) f '( x + y ) , f '( x + y) 51 / , , = 6x y s + x 2y ,fxy = 15.v2.y4 + 8a-1 = fyx,fyy = 20a 1.)-’ 53 wu„ = v 2/(u2 + v2)}/2, wiw = — uv/(ir + v2)212 = wm, * - - ^ 7 s » * ): 47 — = 13 62 15 7, «U _ du_dx_ du dy du du d.x du dy dr dx dr dy dr ’ ds dx ds dy ds ’ du _ du dx du dy dt dx dt dy dt jg dw_ _ dw_ dr_ ^ dw ds dw dl dx dr dx ds dx dt dx’ dw _ ,dwdr dw ds dw dt dy dr dy ds dy dt dy 21.85,178,54 23 f, f 25 36,24.30 27 sin(.r - y) + e y 2g 3yz - 2x 3xz - 2v sin(A* - y) - xey ‘ 2z - 3xy ’ 2z - 3at W„ = u-/(u2 + tr)1/2 55 59 61 63 67 85 zIX = ~ x / (\ + a 2)2, zxy = = zyx, zyy = -2 y / (\ + y 2) 2 x y,7 x y 24 sin(4A + 3y + 2z), 12 sin(4A + 3y + 2z) e r0(2 sin + d cos + rd sin 6) 65 6_vr2 «12.2, =16.8, «23.25 79 R 2/ R ] No 87 a = + /, y = 2, z = - 2f 89 - 91- (a) i i « s a4)’ + 4a y - v v - 4.ry /3, minimum —2/^3 II Maximum yf?>, minimum 13 Maximum /( i , i, i, i) = Z minimum / ( —l, — —i —i ) = —2 15 Maximum / ( 1, >/2, —s/2) = + >/2, minimum / ( 1, —y/2, s/2) = \ — 2s/2 17 Maximum i, minimum \ 19 Maxima/(± /7 , +1/(2>/2)) = e l/\ minima/(± l /v 2, ± l/( v/2 )) = e 1/4 27-37 Sec Exercises 35-49 in Section 11.7 39 ¿7(3 v/3) 11 (a) —3.5°C/m, -3.0°C /m (b) ~ °C /m b y Equation 11.6.9 (Definition 11.6.2 gives —1.1 °C/m.) (c) -0 13 f x = 1/y j x + y 2, f y = y / y / x + y 15 gu = tan“'i;, gv = u/( + v2) 17 Tp = In(q + er), Tq = p/(q + eT), Tr = pe7(^ + er) 19 / , , = 24x, / , v = - y = / , , , /w = - x 21 = k(k - 1)x k~2y lz'\ /,,- = klxk~]y ,~*zm = / , v, /,_- = kmxk~iy ,zm~] = /,,, /n = /(/ - \)xky'-2zm, fy: = lmxky ‘~'zm~l = /-v, /:.- = m(m - 1)*A>’7"'~2 25 (a) z = 8x + 4y + _ 27 (a) l x — 2_v — 3z = 29 (b) 31 33 35 37 43 47 (b) — -— = — o = -— — —1 , v — y + z —1 (b) — - ^ -— (a) jc — y — 2z = x = + 8/, y = — 2t, z = — At (2, i, - l ) , ( - , - , l) 60* + j y + j z - 120; 38.656 2xy'(1 + 6p) + 3x~y2(pe'1 + e'') + 4z'(p cos p + sin p ) -4 , 108 (2xey:\ x 2z 2ey:\ x 2yzey:‘) 45 V 145/2, (4, l) 49 knot/m i APPENDIX J Minimum / ( —4, 1) - -11 Maximum /(1, 1) = 1; saddle points (0, 0), (0, 3), (3, 0) Maximum /(1, 2) = 4, minimum / (2 , 4) = —64 Maximum / ( —1,0) = 2, minima /(1, ±1) = —3, saddle points (—1, ± 1), (1,0) 59 Maximum/(± -/2 /3 , 1/75) = 2/(373), minimum / ( ± 2/3, —1/75) = —2/(3-/5) 61 Maximum 1, minimum —1 63 (± ‘ 1/4, 3"'/JV2, ± '/J), (±3~'/4, - _1/\/2 , ±3'/4) 65 P{2 - 75), P(3 - /3 )/6 , />(275 - 3)/3 ANSWERS TO ODD-NUMBERED EXERCISES A45 51 53 55 57 13 Type I: D = {(.v, y) | =£ v =£ 1.0 =s y type II: D = {(.v, y) | =Sy =S l y S v i I };( FOCUS ON PROBLEM SOLVING ■ PAGE 827 L2W \ \L2W 75/2 ,3 /7 (a) r = w/3, base = w/3 (b) Yes CHAPTER 12 EXERCISES 12.1 ■ PAGE 837 (a) 288 (b) 144 (a) tt2/ = 4.935 (b) (a) (b) - U < V < L (a) = (b) =15.5 11 60 13 15 1.141606, 1.143191, 1.143535, 1.143617, 1.143637, 1.143642 39 13.984.735,616/14,549.535 41 EXERCISES 12.2 ■ PAGE 843 I 500>’\3 v 10 261,632/45 II 13 n 15 y 17 In 19 H73 - l) - ISTr 21 - 3) 7 /(.v y) dx dy 43 J; ^ f(x y) dy dx y In 45 j;;n: l;./(-v.y)rf.vr/v 25 y 27 y / 29 31 y 47 - H 49 (in 55 ( tt/ ) c u , , =s J ' l ; , ' 63 a 7» + !al)1 65 m r h 9- EXERCISES 12 ■ PAGE 857 T I,;'” ' J,t f { > ' co s 35 | 37 39 Fubini’s Theorem does not apply The integrand has an infinite discontinuity at the origin EXERCISES 12.3 ■ PAGE 850 32 y, e - 7r r sin 0)r dr d0 l'1, ( / " ;'(.v y) dy dx A46 APPENDIX J ?7rsin9 ANSWERS TO ODD-NUMBERED EXERCISES 11 (ir/2)(l — e 4) '15 'i-rr - 17 | 7t 19 177-a3 23 (8tt/3)(64 - 24^3) 29 2^2/3 35 H 25 31 180077 ft3 37 (a) -v/ tt/ 13 ¿¡772 21 (2tt/3)[ i - ( l / ^ ) ] tt/12 27 J ttU - cos 9) 33 2/(a + d) (b) V ^ /2 29- J -2 J EXERCISES 12.5 ■ PAGE 866 1.fC I, (f,0) c y dy dx dz 31 J’!, J4, Jo“'v/2/(x, y, z) dz dy dx = Jo ¡-f, j r /2/(*- y z) dz dx dy = Jo J T X’ y- z) dx dXdz = Jo j r - '/2 J'-Vv f (x’ X’ z) dz dy = J -2 J o "72 J J ^ ' / u x z>rfx dz ^ = Jo2J ^ J4"27 U , y, ) dy dx dz 33- Jo J7 Jd~7(*> x z) dz dy dx = Jo Jo J o ^ /U y, z) dz d r dy = Jo Jo - So f (x, y z) dx dy dz = Jo Jo’2' Jo’/(■*’ X’ z) dx dz dy = Jo Jo"^ JTt7 (* X’ z) dy dz dx EXERCISES 12.6 ■ PAGE 871 VỴ477 3->/Ỵ4 72/6 (2t7/3)(2v/2 - l) 11 Airbib - J b - a ) 15 (a) 24.2055 (b) 24.2476 17 4.4506 = Jo j r ’’ JJ/T /U X’ z) ¿X dx dz 13 13.9783 35 Jo C Jo/(*• X z) dz dx dy = J0‘ J’J J0'/(*> X z) dz dy dr = Jo Jr JT/ (x>X- z) dx dy dz = Jo J3 J’J f(x , y, z) dx dz dy 19 4{ J U + {fln[(ll>/5 + 3770)7(375 + 77Ô)] = Jo Jo JT/(*> y, z) dy dz dx = J„ (.' J*/(x, y, z) dy dr dz 37 g, ( i , g, 55J) 39 a 5, (7a/12, 7a/12, 7a/ 12) 41 /, = /v = / = \kL? 45 (a) m = J'33 43 (7 kh a A J,5_v >J x + y dz dy dx (b) (3c, y, z), where x = (l/m) JJ, J,5_-vX'Jx2 + y dz dy dx y = (1/ ot)JT, JT'3Pf J,5" vx7 * + y dz dy dx Z = (1/w) J jT^pp J,5"-vZ'Jx + y dz dy dx (c) J2" J/” 736 sin4« cos2v + sin4« sin2t> + cos2« sin2« du dv 25 47 (a) /5 77 + a 27 77 9“ t t EXERCISES 12.7 ■ PAGE 880 2y 13 8/(3