Chapter 1. Infinite Sequences and Series. 1.1 Sequences. A sequence is a set of numbers written in a definite order

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1 hpter Ite Sequeces d Seres. Sequeces A sequece s set o umers wrtte dete order,,,... The umer s clled the rst term, s clled the secod term, d geerl s th clled the term. Deto.. The sequece {,,...} s usull deoted, { } or { }. A sequece o rel umers c e cosdered s ucto rom N to R. Emple.. Some sequeces c e deed gvg ormul or the th term. () + +,,,..., 4 + () ( ) ( + ) ( ) ( + ) { } (c) { } { } (d) π cos 6 { }

2 Emple.. Some sequeces do ot hve smple deg equto. () The sequece { p } where p s the populto o the world s o Jur o er () The Focc sequece { } s deed recursvel the codtos The rst ew terms re,, + ( ) {,,,,5,8,,... } Deto..4 A sequece { } hs the lmt L d we wrte lm L or s or ever ε > there s correspodg teger N such tht I L < ε wheever > N. lm ests, we s tht the sequece coverges (or s coverget). Otherwse, we s tht the sequece dverges (or s dverget). Theorem..5 I lm ( ) L d ( ) the lm L. Theorem..6 e two coverget sequeces wth lmts A d B respectvel. The Let { } d { } () the sequece { + } s coverget d the lmt s A + B, () the sequece { } s coverget d the lmt s AB..e., lm A d lm B, the d lm( + ) A + B lm( ) AB

3 Theorem..7 (Squeeze Theorem) Gve three sequeces { }, { } d { } c, there s such tht c or ll d the lm lm c lm L L Theorem..8 I lm, the lm. Proo. Let d c. The Sce the Squeeze Theorem, c or ll lm lm d lm c lm lm. Deto..9 A sequece { } s clled cresg + or ll. It s clled decresg or ll. It s clled mootoc t s ether cresg or decresg. + Emple.. The sequece + 5 s decresg. Emple.. The sequece s decresg. +

4 Deto.. A sequece { } s ouded ove there s umer M such tht M or ll. It s ouded elow there s umer m such tht m or ll. I t s ouded ove d elow, { } s clled ouded. Theorem.. Ever ouded, mootoc sequece s coverget.. Seres Gve sequece { } we dd the terms together, we get epresso o the orm whch s clled te seres d s deoted. Sometmes t s mpossle to dd up seres: However, whe we tr to dd up the seres we see tht the cumultve sums s s s s gets closer to the umer s gets gger. So t s resole to s tht the sum o the te seres s d we wrte

5 Deto.. Gve sequece { } the sequece o prtl sums o { } s the sequece { } s s + s + + s deed s s. k k Deto.. Gve seres , the sequece o prtl sums { } coverges to umer s, the the seres s sd to e coverget d we wrte s o { } s or s The umer s s clled the sum o the seres. I seres s ot coverget, the t s sd to e dverget. Emple.. The geometrc seres + r + r + r +... r +... r s coverget r < d dverget otherwse. Emple..4 Fd the sum o the seres, where <. Emple..5 Show tht the seres ( + ) s coverget d d ts lmt.

6 Theorem..6 I the seres coverges, the lm. Remrk..7 The coverse o Theorem..6 s ot true. Emple..8 Show tht the seres s dverget orollr..9 I lm does ot ests or lm, the the seres dverges. Theorem.. I d re coverget seres, the so re c, ( + ) d ( ) d () c c () ( + ) (c) ( ) +. Itegrl Test Theorem.. Suppose s cotuous, postve, decresg ucto o [, ), d (). The the seres s coverget d ol the mproper tegrl () s coverget (equls to te umer)..e.,

7 () I () s coverget, the s coverget. () I () s dverget, the s dverget. Emple.. Test the seres + or covergece. Emple.. The p-seres p s coverget ol p >..4 The omprso Test Theorem.4. (omprso Test) Suppose d re seres wth postve terms. () I s coverget d () I s dverget d or ll, the s lso coverget. or ll, the s lso dverget. Emple.4. Determe whether the seres s coverget or ot. Emple.4. Test the seres l or covergece.

8 Theorem.4.4 (Lmt omprso Test) Suppose d re seres wth postve terms. () I lm c >, the ether oth seres coverge or oth dverge. () I lm (c) I lm, d coverges, the lso coverges., d dverges, the lso dverges. Emple.4.5 Test the seres l or covergece usg Lmt omprso Test..5 Altertg Seres A ltertg seres s seres whose terms re ltertel postve d egtve. For emple: Theorem.5. I the ltertg seres () ( ) + or ll () lm, the the seres coverges ( > ) stses 4

9 .6 Asolute overge, Root Test d Rto Test Deto.6. A seres s sd to e solutel coverget t s coverget the seres s coverget A seres s sd to e codtoll coverget t s coverget ut ot solutel coverget.... Emple.6. The seres ( ) s solutel coverget. Emple.6. The seres ( ) s ot solutel coverget, ut t s codtoll coverget. Theorem.6.4 I seres s solutel coverget, the t s coverget. Theorem.6.5 (Rto Test) Gve seres, + () lm L <, the the seres coverges solutel, lm the the seres dverges. + () L > ( or ) Emple.6.6 Test the covergece o the seres ( ).

10 Emple.6.7 Test the covergece o the seres! Theorem.6.8 (Root Test) Gve seres, () lm L <, the the seres coverges solutel, () L > ( or ) lm the the seres dverges..7 Power Seres A power seres s seres o the orm where s vrle d 's re costts clled the coecets o the seres. For ech ed vlue o, the seres s seres o costts (whch we re mlr wth), d we m ppl the tests or covergece s eore. A power seres m coverge or some vlues o or dverge or other vlues o. The sum o the seres s ucto ( ) Emple.7. I we tke or ll, the the power seres ecomes the geometrc seres Ths seres coverges to whe < d dverges whe (c. Emple..).

11 Emple.7. For wht vlues o does the seres coverge? Theorem.7. I power seres pot, < p. Proo coverges t pot p, the t coverges solutel t ever Sce the seres p s seres o costt umers coverges, Theorem..6, the sequece { p } coverges to zero d hece t s ouded. Thereore there ests M > such tht p < M. osequetl, s pot such tht < p, the p p Mr where r <. Sce the geometrc seres p coverges the omprso Test. Mr s coverget, lso orollr.7.4 I power seres dverges t pot p, the t dverges t ever pot, > p. Deto.7.5 Let seres e power seres. A umer R s clled the rdus o covergece o the or ech umer r, such tht r < R, the seres o costts r coverges d or ech umer p, p > R, the seres

12 dverges. p Deto.7.6 The tervl o coverge o the seres s the set o ll vlues or whch the power seres coverges. I R s the rdus o covergece, the tervl o covergece s oe o the R, R, R, R, R, R or R, R. ollowg: ( ) [ ] ( ] [ ) Emple.7.7 Seres Rdus o overgece Itervl o overgece (, ) [,).8 Mclur's Seres Theorem.8. I the power seres hs rdus o covergece R >, the the ucto deed s deretle o the tervl ( ) ( ) R, R d () ( ) () ( ) d The rd o covergece o seres () & () re oth R.

13 Emple.8. Show tht ( ) ( ) < d l ( < ) Emple.8. I ucto hs power seres represetto the ts coecets re gve ( ) ( R) < ( ) ()!. The seres ( ) ( ) ( ) ()! () () + +! ()! +... s clled the Mclur's seres o. Emple.8.4 Fd the Mclur's seres o the ucto e ( ) d ts rdus o covergece. Emple.8.5 Fd the Mclur's seres o s. For wht vlues does t coverge?

14 hpter Three-Dmesol Altc Geometr d Vectors. Three-Dmesol oordte Sstems To locte pot ple, two umers re ecessr. A pot the ple m e represeted s ordered pr (, ) o rel umers. Where s the -coordte d s the -coordte. To locte pot spce, three umers re requred. A pot,, c o rel umers. spce m e represeted ordered trple ( ) I order to represet pots spce, we rst choose ed pot O (the org) d three drected les through O tht re mutull perpedculr. These les re clled the coordte ces d re leled -s, -s, d z -s. Usull, we thk o the - d - es re eg horzotl d z -s eg vertcl. I three dmesol geometr, equto,, d z represets surce R. Emple.. Descre d sketch the surce () z () R represeted the equto Proposto.. The dstce P P etwee the pots P (, z ) d (, z ) P, P s, ( ) + ( ) + ( z ) P z Proposto.. A equto o sphere wth cetre ( h k, l), d rdus r s ( h) + ( k) + ( z l) r I prtculr, the cetre s the org O, the equto o the sphere s + + z r

15 Emple..4 Show tht + + z z s the equto o sphere, d d ts cetre d rdus.. Vectors Deto.. A two-dmesol vector s order pr, o rel umers. A three dmesol vector s ordered trple,, o rel umers. The umers, d re clled the compoets o the vector. Deto.. () The legth o two dmesol vector, s gve + () The legth o three dmesol vector,, s gve + + Deto.. I, d,, the the vector + s deed + Smlrl, or three dmesol vectors, + +,, ,,,,, Deto..4 I, d c s sclr, the the vector c s deed c c, c Smlrl, or three dmesol vectors, c, c, c, c,

16 There re three vectors R tht pl specl role. Let,, j,, k,, The, j, d k re vectors tht hve legth d pot the drecto o the postve -, -, d z -es. I,, the we c wrte,,,, + + j + k,,,, + +,,,, +,, A ut vector s vector whose legth s. For stce, j, d k re ll ut vectors. I, the the ut vector tht hs the sme drecto s s ˆ Emple..5 Fd the ut vector the drecto o the vector j k. The Dot Product Deto.. I,, d,,, the the dot product o d s the umer gve + + Theorem.. I,, d c re vectors the three dmesol spce, d c sclr, the... ( + c) + c 4. ( c) c( ) ( c) 5.

17 Theorem.. I θ s the gle etwee the ozero vectors d, the cosθ orollr..4 I θ s the gle etwee the ozero vectors d, the Emple..5 cosθ Fd the gle etwee,, d 5,, Deto..6 Two vectors re perpedculr or orthogol the gle etwee them s π. Theorem..7 Two vectors d re orthogol d ol.4 The ross Product Deto.4. I,, d,,, the cross product o d s the vector j k Emple.4. I,, 4 d,7, 5 d. ( ) ( ) j + ( )k Theorem.4. The vector s orthogol to oth d.

18 Theorem.4.4 I θ s the gle etwee d ( θ π ), the Theorem.4.5 Two ozero vectors sθ d re prllel d ol Theorem.4.6 I,, d c re vectors, d c sclr, the.. ( c ) ( c ) ( c). ( c) c ( ) c c c ( c) ( ) c 6. ( c) ( c) ( )c.5 Equto o Les d Ples.5. Les A le L three-dmesol spce s determed pot (,, z) r e the posto vector o (, z ) drecto. Let epressed s r + tv, or some t, where v s the drecto o L. Thereore the equto s vector equto o L. P o the le d ts P the ever pot o L c e r r + tv (.) I the drecto vector v,, c, the t v,, c. We c lso wrte r,, z d r,, z, the the vector equto. ecomes Thus,, z + t, + t z + ct, + t, + t, z + ct (.) z These re clled prmetrc equtos o the le L.

19 Emple.5. Fd vector equto d prmetrc equto or the le tht psses through (,, ) prllel to + j k. 5 d s Aother w o wrtg the equto o le L s to elmte the prmeter t rom Equtos.. I oe o,, or c s zero, we c ot z z c These equtos re clled the smmetrc equtos o L..5. Ples A ple s determed pot (,, z ) to the ple. Ths orthogol vector s clled orml vector. Let P (, z) the ple d let r d r e the posto vectors o to the ple d thus or P the ple d vector tht s orthogol ( r r ), e pot o P d P. The vector r r s prllel (.) r (.4) Ether o Equtos. or.4 s clled vector equto o the ple. I we wrte,, c, r,, z, d r,, z the the vector equto.4 ecomes,, c,, z,, c z r,, Thus + + cz d (.5) where d + + cz. Ths s clled the sclr equto o the ple through P, z wth orml vector,, c. ( ), oversel,,, d c re ot ll zero, the Equto.5 represets ple wth orml vector,, c. Two ples re prllel ther orml vectors re prllel. I two ples re ot prllel, the the tersect strght le d the gles etwee the two ples s deed s the cute gle etwee the two ormls. Emple.5. Fd the gle etwee the ples + + z d + z.

20 Emple.5. Fd ormul or the dstce D rom pot (, z ) P to the ple + + cz d.,.6 Vector Fuctos d Spce urves A vector vlued ucto or vector ucto s ucto whose dom s set o rel umers d rge s set o vectors. We re most terested vector uctos r whose vlues re three-dmesol vectors. Ths mes tht or ever umer t the dom, there s uque vector the three-dmesol spce deoted r () t. I () t, g() t d h() t re the compoets o r () t, the, g d h re rel-vlued uctos clled the compoet uctos o r d we c wrte r () t () t, g() t, h() t () t + g() t j + h()k t The lmt o vector ucto s deed tkg the lmts o ts compoet uctos s r t t, g t, h t, the ollows: I () () () () lmr t () t lm () t,lm g() t, lmh() t t t provded the lmts o the compoet uctos ests. t Deto.6. Fd r() t t lm where r() t ( + t ) + te j + k t st t Deto.6. A vector ucto r s cotuous t t () t ( ) lm r r. Deto.6. Suppose tht, g d h re cotuous rel-vlued uctos o tervl I. Let deote,, z spce where the set o ll pots ( ) () t g() t z h() t (.6) d t vres throughout the tervl I, the s clled spce curve. The equtos.6 re clled the prmetrc equtos o d t s clled prmeter.

21 Remrk.6.4 A cotuous vector ucto r dees spce curve. Emple.6.5 Sketch the curve whose vector equto s r () t cost + stj + tk Deto.6.6 The dervtve r o vector ucto r s deed s dr r dt () t r lm h ( t + h) r( t) h Let P e pot o the curve wth posto vector r () t. We ote tht r () t s vector d t s clled the tget vector to the curve t the pot P. The ut tget vector t P s the vector () t () t r T () t. r The ollowg theorem gves us coveet method or computg the dervtve o vector ucto r : just derette ech compoet o r. Theorem.6.7 I r() t () t g() t, h() t () t + g() t j + h()k t the,, where, g, d h re deretle uctos, () t () t, g () t h () t () t + g () t j + h ()k t r, Emple.6.8 t t Fd the dervtve o r() ( + ) + j + s k t. t te t. Fd the ut tget vector t the pot Emple.6.9 Fd prmetrc equtos or the tget le to the hel wth prmetrc equtos cost st z t t the pot,, π.

22 Theorem.6. Suppose u d v re deretle vector uctos, c s sclr, d s rel-vlued ucto. The Emple.6. Theorem.6. d dt () [ u () t + v() t ] u () t + vu() t d () cu () t cu t dt d dt [ ] () () [ () t v () t ] () t u() t + () t u () t d dt (4) [ u () t v() t ] u () t v() t + u() t v () t d dt (5) [ u () t v() t ] u () t v() t + u() t v () t Theorem.6. d (6) [ u ( () t )] () t u ( () t ) h Rule dt Emple.6.4 Show tht r () t c ( costt), the r () t s orthogol to r () t or ll t. The dete tegrl o cotuous vector ucto r () t c e deed the sme mer s the cse o rel vrle ucto ecept the tegrl s vector. Deto.6.5 Suppose r() t () t g() t, h() t () t + g() t j + h()k t, the r () () + () j + t dt t dt g t dt h() t dt k

23 .7 Arc Legth d urvture.7. Arc Legth For ple curve deed the prmetrc equtos () t g() t, t Where d g re cotuous, we c prove tht the rc legth o the curve s [ () t ] + [ g () t ] dt L. (.7) r, t, or For spce curve tht s deed the vector equto () t () t, g() t, h() t equvletl, the prmetrc equtos Where r () t s oe-to-oe, the the rc legth s () t g() t z h() t t [ () t ] + [ g () t ] + [ h () t ] dt L d d dz + + dt dt dt dt Note tht oth Equtos.7 d.8 c e put to the more compct orm L r ( t) dt (.8) Suppose s pecewse-smooth curve gve vector ucto r () t () t + g() t j + h() t k, t, d r s oe-to-oe, we dee ts rc legth ucto Thus () t sdes, we hve s () t r () t t s s the legth o the prt o the curve etwee r () d r() t ds dt r Emple.7. Re-prmetrze the hel r() t t + stj + tk drecto o cresg t. () t dt. I we derette oth cos wth respect to rc legth rom (,,) the

24 .7. urvture I s smooth curve deed the vector ucto r. Recll the ut tget vector T() t s gve T () t r r d dctes the drecto o. We ote tht T chges drecto ver slowl s rl strght ut t chges more rpdl eds more shrpl. We dee the curvture o to e the mgtude o the rte o chge o the ut tget vector wth respect to the rc legth. () t () t Deto.7. The curvture o curve s Where T s the ut tget vector. κ dt ds Proposto.7. κ T r () t () t Emple.7.4 Show tht the curvture o crcle o rdus s /. Theorem.7.5 The curvture o the curve gve the vector ucto r s gve Emple.7.6 κ () t r () t r () t r () t Fd the curvture o the twsted cuc () t t, t, t r t geerl pot d t (,,). Emple.7.7 Fd the curvture o the prol t the pots (,), (, ), d (,4).

25 hpter Prtl Dervtves. Fucto o Severl Vrles Deto.. Let D. A ucto o two vrles s rule tht ssgs to ech pot ( ) D uque rel umer deoted ( ) s the set o vlues tkes o,.e., { (, ) : (, ) D}. R,,. The set D s clled the dom o d the rge We ote wrte z (, ) to mke eplct the vlue tke o t the pot ( ),. The vrles d re clled depedet vrles d z s clled the depedet vrle. Emple.. Fd the dom o the ollowg uctos d evlute (,) () (, ) + + l () (, ) ( ) Deto.. I s ucto o two vrles wth dom D, the grph o s the set {(,, z) R : z (, ), (, ) D} Emple..4 Sketch the grph o the ucto (, ) 6. Emple..5 Fd the dom d the rge o the ucto g(, ) 9. Sketch ts grph.

26 Deto..6 The level curves o ucto o two vrles re the curves wth equto where k s costt the rge o. (, ) k, Emple..7 Sketch the level curves o the ucto (, ) 9 or,,, k. Deto..8 Let D R. A ucto o three vrles s rule tht ssgs to ech pot (,, z) D uque rel umer deoted (,, z). The set D s clled the dom o d the (,, z) : (,, z) D. rge s the set o vlues tke o,.e. { } Deto..9 Let D R. A ucto o vrles s rule tht ssgs to ech pot (,,, ) D uque rel umer deoted (,,, ). Emple.. I comp uses deret gredets mkg ood product, c s the cost per ut o th the gredet, d the ucto represetg the totl cost uts o th gredet (,,, ) s used.. Lmts d cotut Let e ucto o oe vrle. Itutvel, we s the lmt o ucto t pot s L,.e. lm ( ) L () gets ver er to L whe gets ver to. But how er s er? The swer s: s er s ou requre.

27 Deto.. Let e ucto o oe vrle deed o dom D R. Let R, we s tht the lmt o () s pproches s L or ever postve umer ε >, there ests correspodg δ > such tht ( ) L < ε wheever < < δ d we wrte lm ( ) L. Ths deto gves us resole descrpto o the lmt o ucto t pot : o mtter how smll mrg o error ε > tht s gve, we c d δ > such tht () s er to () L < < < δ. L ( ) wheever s er to ( ) I the cse o uctos o two vrles, the oto o eress s qut the Euclde orm. Deto.. Let e ucto o two vrles deed o dom D R. Let (, ) D, we s tht the lmt o (, ) s (, ) pproches (, ) ests there s umer L or ever umer ε >, there s correspodg umer δ > such tht d we wrte () L < ε wheever < ( ) + ( ) < δ ( or : () L < ε wheever < (, ) (, ) < δ ) lm (, ) (, ) (, ) L Ths deto reers ol to the dstce (, ) (, ) etwee (, ) d ( ) drecto o pproch. Thereore (, ) hs deret lmts s (, ) pproches ( ) deret drectos, the we s tht the lmt does ot est.., ut ot the, orollr.. I (, ) L s (, ) (, ) log pth d (, ) L s (, ) (, ) L the lm ( ) ( ) (, ) does ot est. log other pth where L,,

28 Emple..4 Fd lm ( ) ( ),, d lm ( ) ( ),, the est. Emple..5 Fd lm (, ) (, ) + t ests. Emple..6 Fd lm (, ) (, ) + t ests. Theorem..7 I lm ( ) ( ) (,,, ) d lm g ( ) ( ) (,, ) lm ( ) (, ), est, the { (, ) g(, ) }, lm ( ) ( ) { (, ) g(, ) } + lso est d,,, () lm ( ) ( ) { (, ) + g(, ) } lm ( ) ( ) (, ) lm (, ) (, ),,,, () lm ( ) ( ) { (, ) g(, ) } lm ( ) ( ) (, ) lm g ( ) ( ) (, ),,,,,, + g (, ) Emple..8 lm + + { } (, ) (, ) Deto..9 Let e ucto o two vrles deed dom cotuous t (, ) lm (, ) (, ) (, ) (, ). D R We s tht s cotuous o D t s cotuous t ech ( ) D,.. The s clled Emple.., + s cotuous. The ucto ( )

29 Emple.. A poloml o two vrles s cotuous. m (, ) c + c + c + c + c + c + c c p k Emple.. A rtol ucto p(, ) q(, ) ts dom (.e., o pots (, ) where (, ), whch s rto o two polomls p d q, s cotuous o q ). Emple.. Where s the ucto (, ) cotuous? + Emple..4 Let Is g cotuous t (,)? g (, ) + (, ) (,) (, ) (,) Emple..5 Let (, ) + (, ) (,) (, ) (,) Show tht s cotuous t (,) d hece cotuous o R.

30 . Prtl Dervtves Let e ucto o two vrles. I we vr whle keepg ed, s we re cosderg ucto o oe vrle,.e., () ( ) g,., the I g hs dervtve t, the we cll the dervtve the prtl dervtve o wth,. We kow tht respect to t ( ) so, d s deoted ( ) g () (, ) g lm h lm h ( + h) g( ) h ( + h, ) (, ) Smlrl, the prtl dervtve o wth respect to t (, ), deoted ( ) oted keepg ed t G, : () ( ) Deto.. h, s d dg the ordr dervtve t o the ucto (, ) lm h (, + h) (, ) I s ucto o two vrles, ts prtl dervtves re the uctos Notto.. z, we wrte I ( ) (, ) (, ) lm h lm h h ( + h, ) (, ) h (, + h) (, ) (, ) (, ) (, ) (, ) h z z d Remrk.. () To d () To d regrd s costt d derette ( ) regrd s costt d derette ( ), wth respect to., wth respect to.

31 Emple..4 I (, ) +, d (,) d (,). Emple..5 I ( ), s, clculte + d. Emple..6 Fd d z s deed mplctl s ucto o d the equto + + z + 6z Deto..7 (Prtl Dervtves or Fuctos o Three Vrles) I s ucto o three vrles,, d z, the ts prtl dervtve wth respect to s deed s (,, z) lm h ( + h,, z) (,, z) d t s oud regrdg d z s costts d derettg wth respect to. I geerl, s ucto o vrles, the ts prtl dervtve wth respect to ts th vrle s deed (,,..., ) lm h h (,..., + h,..., ) (,...,..., ) h Emple..8 Fd,, d (,, z) e l z. z Deto..9 I s ucto o two vrles, the ts prtl dervtves d re lso uctos o,,, d, two vrles, so we c cosder ther prtl dervtves: ( ) ( ) ( ) ( ) whch re clled secod prtl dervtves o. I z (, ), we wrte

32 ( ) ( ) ( ) ( ) z z z z The otto or mes tht we rst derette wrt the derette wrt. Emple.. Fd the secod dervtve o ( ), +. Theorem.. Suppose s deed o dom R D d ( ) D,. I oth d re cotuous t ( ),, the ( ) ( ),,.4 Tget Ples d Deretls We recll tht or ucto o oe vrle, the equto o the tget le t s gve () () or () ()( ) Suppose ow s ucto o two vrles, the equto o the tget ple to the surce ( ) z, t ( ) ( ),, s ( ) ( )( ) ( )( ),,, z + or smpl ( )( ) ( )( ),, z z + where ( ), z.

33 Emple.4. Fd the tget ple to the surce z + t the pot (,, ). Recll tht or ucto o oe vrle (), we dee the cremet o s d the deretl o s Deto.4. d Gve ucto o two vrles z (, ) ( + ) ( ) (). I d d, the the cremet o z s ( +, + ) ( ) z, The deretl o z clled the totl deretl, s deed (, ) + ( ) dz, re respectvel the chge As the cse o oe vrle, uder some crcumstces, the totl deretl dz s good ppromto or z whe d re smll. Theorem.4. Suppose.e., d ests o dom D d re cotuous t the pot (, ) D, the The reso we wt to use dz to ppromte lm ( dz z ) (, ) (, ) dz z whe d re smll. z s ecuse dz s eser to compute. Emple.4.4, +, d the deretl dz. I chges rom to. 5 d chges rom to. 96, compre the vlues o z d dz. I ( ) Emple.4.5 Use deretls to ppromte (.95) ( 8. ) 9 +

34 .5 The h Rule We recll tht the h Rule or uctos o oe vrle gves the rule or derettg composte ucto: I (), d g() t, where d g re deretle uctos o t, the d d d dt d dt For uctos o two or more vrles, the h Rule hs severl versos. Theorem.5. (h Rule (se )) Suppose tht z (, ) s deretle ucto o d, where g() t d h() t re oth deretle uctos o t. The z s deretle ucto o t d Emple.5. dz dt z d dt z + d dt I z + 4, where t e d st dz, d. dt The h Rule se s smplest cse. Suppose we re gve ucto z (, ) where oth d re uctos o two vrles s d t : g( s, t), d h( s, t). The z s ucto o two vrles s d t d we m d the dervtves z s z d. t Suppose we keep s ed d d the ordr dervtve o Theorem.5., z z z +. t t t z Smlrl, we m ot the sme rgumet. s z wrt t, the ccordg to Theorem.5. (h Rule (se )) Suppose tht z (, ) h( s, t) s deretle ucto o d the prtl dervtves g s, gt, s d t est. The z z z + t t t z z z + s s s d, where g( s, t) d

35 Emple.5.4 I z e s, where st d s t, d z z d. s t osder geerl stuto whch u s ucto o vrles,,...,, ech o whch s ucto o m vrles t, t,..., t m such tht ll dervtves j (,,..., m; j,,..., ) the u s ucto o m t t, t,..., t d or ech,,..., m. u t u t u + t u +... t Emple.5.5 Wrte dow the h Rule or the cse where w (,, z, t) d ( u, v), ( u, v) z z( u, v) d t t( u, v)., Emple.5.6 I u + 4 z, where r, s, t. t rse, rs e t, d z r sst, d the vlue o u s whe The h Rule m e ppled to ot dervtves o uctos tht re deed mplctl. Suppose we hve equto o the orm F (, ) tht dees mplctl terms o. From lculus I, we kow we c ppl mplct d deretto to d. However, usg the h Rule, we hve d d sce d F d, we hve F d d d d F + F F d d F F

36 Emple.5.7 Fd + 6. Now suppose tht z s deed mplctl s ucto o I F s deretle d d the equto F (, z) wrt, to ot (,, z) F. d equto o the orm est, the we c ppl the h Rule to derette F F F z + + z But we ote tht d (ecuse s depedet o ) so the ove equto ecomes F I, we ot z d smlrl, F F z + z F z F z F z F z Emple.5.8 z z Fd d + + z + 6 z.

37 .6 Drectol Dervtves d the Grdet Vector Recll z (, ) d d, the prtl dervtves (, ) (, ) lm h lm h re deed s ( + h, ) (, ) h (, + h) (, ) d the represet respectvel the rte o chge o z the - d - drectos,.e., the drectos o the ut vectors d j. h Suppose tht we wsh to d the rte o chge o z the drecto o rtrr vector u, the we hve to d the drectol dervtve o. Deto.6. The drectol dervtve o t ( ), ths lmt ests. D u (, ) the drecto o u, s lm h ( + h, + h) (, ) h Theorem.6. I s deretle ucto o d, the hs drectol dervtve the drecto o ut vector u, d Emple.6. Fd the drectol dervtve D (, ) D u (, ) (, ) + ( ) u d u s the ut vector gve gle, (, ) 4 + π 6 θ. Wht s (,) D u. Deto.6.4 I s ucto o two vrles d, the the grdet o s the vector (, ) (, ), (, ) + j Usg ths otto, we m rewrte the epresso Theorem.6. s deed

38 Emple.6.5 u (, ) ( ) u D, Fd the drectol dervtve o the ucto (, ) 4 t the pot (, ) drecto o the vector v + 5j., the For uctos o three vrles, we c dee drectol dervtves the sme mer. Deto.6.6 The drectol dervtve o t (, z ) D the lmt ests. u (,, z ) lm h the drecto o ut vector u,, c s, ( + h, + h, z + hc) (,, z ) h I we use vector otto, the drectol dervtves c e wrtte s D u ( ) lm h ( + hu) ( ) where, d,, z. h Deto.6.7 I s ucto o three vrles, d z, the the grdet o s the vector deed z (,, z), + j + k, d the drectol dervtve the drecto o u s u z (,, z) (, z) u D, Suppose S s surce wth equto F (,, z) k d let (, z ) P, e pot o S. Let e curve tht les o S d psses through the pot P. c e descred cotuous vector ucto r () t () t, () t, z() t. Let t e the vlue such tht r ( t ),, z. Sce les o S d ( () t () t, z() t ) S, tht s, ( () t, () t, z() t ) k F We ppl the h Rule to derette wrt t d we ot F d dt F + d dt F + z, must sts the equto o dz dt

39 other words, s r z F r F F, F, F d r () t () t, () t, z () t. I prtculr, whe t t, we hve ( t ),, z, so () t (, z ) ( t ) F r (.), Equto. ss tht the grdet vector t P, F(,, z ) vector r ( t ) to curve o S tht psses through P. I F(,, z ) vector, we dee the tget ple to the level surce F (,, z) k t (,, z ) ple psses through P d hs orml vector F(, z ) hpter, we c wrte the equto o ths ple s F s perpedculr to the tget, s ot the zero P s the. Usg the equto o ple (, z )( ) + F (,, z )( ) + F (,, z )( z z ), z Emple.6.8 Fd the equto o the tget ple t the pot (,, ) to the ellpsod 4 + z Mmum d Mmum Vlues I lculus I, we sw tht oe o the m pplctos o dervtves s to d the locl etrem. I ths secto we see how to use prtl dervtve to d the locl etrem o ucto o two vrles. Deto.7. A ucto o two vrles hs locl mmum t (, ) ( ) (, ) pots (, ) some dsk wth cetre (, ). The umer ( ) vlue. I (, ) (, ) or ll (, ) such dsk, ( ) vlue., or ll, s clled locl mmum, s clled locl mmum Theorem.7. I hs locl etremum t ( ) there, the, d the order o the rst order prtl dervtves est (, ) (, ).

40 Emple.7. Let (, ) , d locl mmum or ths ucto. Emple.7.4 Fd the etremum vlues o (, ). Theorem.7.5 (Secod Dervtve Test) Suppose the secod prtl dervtve o re cotuous dsk wth cetre (, ), d suppose tht (, ) d (, ). Let () I > D D [ ] (, ) (, ) (, ) (, ) D d (, ) >, the ( ) D d (, ) <, the ( ) () I > (c) I < D, the ( ), s locl mmum., s ot locl etremum., s locl mmum. Emple Fd the locl etrem o (, ) Emple.7.7 Fd d clss the crtcl pots o the ucto Also d the hghest pot o the grph o. 4 4 (, ) 5 4 Emple.7.8 A rectgulr o wthout ld s to e mde rom volume o such o. m crdord. Fd the mmum Theorem.7.9 (Etreme Vlue Theorem or Fuctos o Two Vrles) I s cotuous o closed, ouded set vlue (, ) D. D R, the tts solute mmum, d (, ) d solute mmum ( ) t some pots ( ), Algorthm.7. To d the solute mmum d mmum vlues o cotuous ucto o closed ouded set D :

41 . Fd the vlues o t the crtcl pots o D.. Fd the etreme vlues o o the oudr o D.. The lrgest o the vlues rom steps d s the solute mmum vlue; the smllest o these vlues s the solute mmum vlue. Emple.7. Fd the solute mmum d mmum vlues o the ucto (, ) + the rectgle D (, ),. { } o.8 Lgrge Multpler Algorthm.8. To d the mmum d mmum vlues o (, z) g (,, z) k (ssumg these vlues est): () Fd ll vlues o,, z d λ such tht d g (,, z) k () Evlute t ll pots (, z) (,, z) λ g(, z),, suject to the costrt, tht re oud rom step (). The lrgest o these vlues s the mmum vlue; the smllest o these vlues s the mmum vlue. Emple.8. A rectgulr o wthout ld s to e mde rom volume o such o. m crdord. Fd the mmum Emple.8. Fd the etreme vlues o the ucto ( ), + o the crcle +. Emple.8.4 Fd the etreme vlues o the ucto (, ) + o the dsk +.

42 hpter 4 Multple Itegrls I ths chpter we eted the de o dete tegrl to doule d trple tegrls o uctos o two or three vrles. 4. Doule Itegrls Over Rectgles We dee the doule tegrl o ucto o two vrles the mer s the oe vrle cse. Let R deote closed rectgle R [, ] [ c, d] (, ) We prtto the tervls [, ] d [ d] c < < { R, c d} c, to: < < <... < <... < The R m e prttoed to surectgles or Let R j < < <... < j m <... < d [, ] [, ] {(, ), } j j j j,..., m d j,...,. There re m such surectgles coverg R. the the re o R j s j j j j A j j * Net we choose pot ( ) * j, j j R d dee the doule Rem sum s m j * ( ) Deto 4.. The doule tegrl o over the rectgle R s R A * j, j j. m * * (, ) da lm ( j, j ) P j A j

43 where P deotes the logest dgol o ll the surectgles Rj Emple 4.. Fd ppromte vlue or the tegrl ( )da where {(, ), } R computg the doule Rem sum wth les d * d tkg ( ) to e ceter o ech rectgle. * j, j R Theorem 4.. I (, ) les ove R d uder the surce z (, ) d s cotuous o the rectgle R, the the volume V o the sold tht s V Emple 4..4 R (, )da Estmte the volume o the sold tht les ove the squre [,] [,] ellptc prolod * choose ( ) z R d elow the 6. Use the prtto o R to our equl squres d * j, j to e the upper rght corer o j rectgulr oes. R. Sketch the sold d the ppromtg 4. Iterted Itegrls It s usull dcult to evlute sgle tegrls rom rst prcples, the evluto o doule tegrls s eve more dcult. I ths secto we see how doule tegrl c e epressed s terted tegrl, whch c e evluted clcultg two sgle tegrls. Suppose s ucto o two vrles tht s tegrle over the rectgle R [, ] [ c, d] s oted keepg held ed d ( ). The prtl tegrto o wth respect to, deoted (, )d c, s tegrted wrt rom c to d. d Now (, )d s umer tht depeds o, so t s ucto o : c A d () ( )d I we tegrte A () rom to, we get A c,. d () d (, ) d d (4.) c The tegrl o the rght sde o Equto 4. s clled terted tegrl. Usull, the rckets re omtted d

44 Smlrl, the terted tegrl d c d c d ( ) dd (, ) d d, ( ) d, dd (, ) c s oted rst tegrte wrt (holdg ed) rom resultg ucto wrt rom c to d. c d d to the tegrte the Emple 4.. Evlute the terted tegrls () dd () dd Emple 4.. (Fu s Theorem) I s cotuous o the rectgle R [, ] [ c, d] (, ) d d (, ) da (, ) dd (, ) R c { R, c d} c dd, the Emple 4.. Evlute the doule tegrl ( )da R where R ( ) {,, } Emple 4..4 Evlute ( )da R s where [,] [,π ] R. Emple 4..5 Fd the volume o the sold tht les ove the squre [,] [,] prolod z 6. R d elow the ellptc

45 4. Doule Itegrls Over Geerl Regos For sgle tegrls, the rego over whch we tegrte s lws tervl. But, or doule tegrls, we wt to tegrte ot just over rectgles ut lso over regos o more geerl shpe. Suppose s ucto o two vrles deed over ouded rego (o geerl shpe). Let R e rectgle tht ecloses D. Dee ew ucto F s ollows: F ( ) (, ) (, ) (, ) s D, (4.) s R ut ot D I F s tegrle over R, the we s tht s tegrle over D d we dee the doule tegrl o over D ( ) da F(, )da, where F s deed s Equto 4.. D D The ove deto mkes sese ecuse R s rectgle d so F(, )da hs ee D prevous deed. A ple rego s sd to e tpe I t les etwee the grphs o two cotuous uctos o, tht s, To evlute (, )da D {(, ) :, g () g () } D., choose rectgle [ ] [ c, d] ucto gve Equto 4.. The Fu s Theorem, Note tht (, ) d ( ) da F(, ) da F(, )dd D, F < g () or g () d D, tht cots D. Let F e the > sce the ( ) ( ) g (, ) d F(, ) F d c g ( ) ecuse F (, ) (, ) whe () g () ormul. g c g g ( ) ( ), les outsde D. Thereore (, ) d. Thus we hve estlshed the ollowg Theorem 4.. I s cotuous o tpe I rego D such tht the {(, ) :, g () g () } D ( ) g (, ) da (, ) A ple rego s sd to e o tpe II t s o the orm D g ( ) dd

46 We c lso estlsh the ollowg: {(, ) : c d, h () h () } D. Theorem 4.. I s cotuous o tpe II rego such tht the {(, ) : c d, h () h () } D ( ) d h (, ) da (, ) dd. D c h ( ) Emple Evlute ( )da, where D s the rego ouded the prols D. + d Emple 4..4 Fd the volume o the sold tht les uder the prolod D the -ple ouded the le z + d the prol d ove the rego. Emple 4..5 Evlute da, where D s the rego ouded the le d the prol D + 6. Emple Evlute the terted tegrl s ( )dd Theorem 4..7 Suppose d g re uctos o two vrles deed o D d suppose tht d g re oth tegrle over D, the D [ (, ) + g(, ) ] da (, ) da + g(, ) I (, ) g(, ) or ll ( ) D D c D (, ) da c (, )da, the D D da

47 (, ) da g(, )da D I D D D where D d D re two o-overlppg regos, the D (, ) da (, ) da (, )da + I we tegrte the costt ucto (, ) I m (, ) M or ll ( ) Theorem 4..8 D D D over the rego D, we get the re o D : da D, D, the ma A ( D) ( D) (, ) da MA( D) The re o the surce wth equto z ( ), (, ) D cotuous, s Emple 4..9 D [ ] + () S [ (, ) ] + (, ) A da Fd the surce re o the prt o the surce,,, rego T the -ple wth vertces ( ) ( ), d ( ) D,, where d re z + tht les ove the trgulr,. Emple 4.. Fd the re o the prolod z + tht les uder the ple 9 z. 4.4 Doule Itegrls Polr oordtes A pot (, ) o the ple c e epressed s polr coordtes ( r,θ ) A polr rectgle s set o the orm r + r cosθ rsθ R {( r, θ ): r, α θ β} where ( r,θ ) s polr coordtes., where I order to compute ( )da sutervls: R, where R s polr rectgle, we prtto [ ] r < r <... < r < r <... < rm, to

48 d prtto [ α, β ] to sutervls α θ < θ <... < θ < θ <... < θ β The the polr rectgle R c e prttoed to polr surectgles r *, θ j j * Pck pot ( ) The re R R, where {( r, θ ): r r r, θ θ θ } j j r Aj o R j c e computed s * ( r + r ) θ ( θ + θ ) * j j j j A j r r θ * j where r r r d θ θ θ. The Rem sum o polr coordtes o the polr rectgle s m j j j j m * * * * * * ( j, j ) Aj ( r cosθ j, r sθ j ) j m j * * * * ( r cosθ, r sθ ) j j A * j r r θ j (4.) I we wrte ( r, θ ) r ( r cosθ, rsθ ) s g, the the Rem sum Equto 4. c e wrtte m j whch s the Rem sum o the tegrl Thereore, we hve R β α * ( θ ) g r *, r θ g j ( r, ) θ drdθ m * * * * (, ) da lm ( r cosθ j, r sθ j ) P j lm β α β α m P j g g ( r, θ ) * * ( r, θ ) drdθ ( r cosθ, rsθ ) rdrdθ j j r θ j * r r θ j

49 Theorem 4.4. I s cotuous o polr rectgle R gve β α π, the r, α θ β, where β (, ) da ( rcosθ, rsθ ) rdrdθ R α The ove theorem ss tht whe we covert rom rectgulr to polr coordtes doule tegrl wrtg r cosθ d rsθ, da s replced rdrd θ. Emple 4.4. Evlute ( 4 )da, where R s the rego the upper hl-ple ouded the R crcles + d + 4. Emple 4.4. Fd the volume o the sold ouded the ple z d the prolod z. We c lso tegrte over tpe I or tpe II polr rego s the rectgulr cse. Theorem I s cotuous o polr rego o the orm the {( r, θ ): r, g () r g () r } D θ () g r (, ) da ( rcosθ, rsθ ) R g () r rdrdθ Theorem I s cotuous o polr rego o the orm the D {( r, ): α θ β, h () θ r h () θ } θ ( ) β h θ (, ) da ( r cosθ, rsθ ) R α h ( θ ) rdrdθ

50 4.5 Trple Itegrls Just s we deed sgle tegrls or uctos o oe vrle d doule tegrls or uctos o two vrles, so we c dee trple tegrls or uctos o three vrles. osder ucto o three vrles deed o rectgulr o B [, ] [ c, d] [ r, s] (,, z) We prtto the tervls [, ], [ c, d], d [ s] c r z < < < z < < < z { R, c d, r z s} <... < <... < z r, to: <... < The B m e prttoed to lm su-oes The volume o where B j s jk j k < < < z k <... < j <... < <... < z [ ] [, ] [ z z ] B,, j j k V jk The we dee the trple Rem sum * * * where ( jk jk, zjk ) Bjk z j k z k l m z s d z j j j k k k l m j k * * * (,, z ) jk,. We dee the orm o the prtto P to e legth o the logest dgol o ll the oes B jk d we deote the orm P. jk jk V jk Deto 4.5. The trple tegrl o over the o B s lmt ests. B l m * * * (,, z) dv lm ( jk, jk, zjk ) P j k V jk Theorem 4.5. (Fu s Theorem or Trple Itegrls) I s cotuous o the rectgulr o B [, ] [ c, d] [ r, s], the s d (, z) dv (,, z)dddz B, r c

51 Emple 4.5. Evlute the trple tegrl B z dv where B s the rectgulr o gve {(,, z),, } B z Deto Let E e ouded rego, the trple tegrl o over the ouded rego E s deed where F (, z) dv F(,, z)dv E (,, z), E (,, z) (,, z) otherwse E Deto A sold rego E s sd to e o tpe I t les etwee the grphs o two cotuous uctos o d, tht s {(, z)(, ) D, φ (, ) z ( ) } E,, φ where D s the projecto o E oto the -ple. Theorem I E s tpe I rego, the Suppose urther tht D s tpe I rego the d E ( ) φ, (,, z) dv (,, z) D φ (, ) dz da {(, ) :, g ( ) g ( ) } D {(, z), g () g (), φ (, ) z ( ) } E,, φ E ( ) ( ) g φ, (,, z) dv (,, z) g ( ) φ (, ) dzdd

52 Emple Evlute zdv, where E s the sold tetrhedro ouded E,, d + + z 4.6 hge o Vrles Multple Itegrls I oe-dmesol clculus we ote use chge o vrle ( susttuto) to smpl tegrl where g() u d g() c g() c d () d ( g() u ) g ( u)du c,. Aother w o wrtg the ove s d d () d ( () u ) du (4.4) c du A chge o vrles c lso e useul doule tegrls. We hve see ths s Secto 4.4: coverso to polr coordtes. The ew vrles r d θ re relted to the old vrles d the equtos r cosθ rsθ d the chge o vrles ormul (4.4) c e wrtte s ( ) da ( r cosθ, rs )rda, θ R S where S s the rego the rθ -ple tht correspod to the rego R the -ple. More geerll, we cosder chge o vrles tht s gve trsormto T rom the uv -ple to the -ple: ( u, v) ( ) T, where d c e epressed s uctos o u d v We ssume tht the trsormto T s dervtves. ( u, v) h( u v) g,,.e., g d h hve cotuous rst order prtl

53 Deto 4.6. The Jco o the trsormto T gve g( u, v) d h( u, v) (, ) u v ( u, v) u v u v u u s Remrk 4.6. Let S e rectgle the uv -ple wth re trsormto hs re u v. The ts mge R uder the A (, ) ( u, v) u v Theorem 4.6. Suppose T s oe-to-oe trsormto wth ozero Jco tht mps rego S the uv -ple oto rego R the -ple. Suppose s cotuous o R d tht R d S re o tpe I or tpe II ple rego. The ( ) ( ( ) ( ) (,, da u, v, u, v ) ( u v) dudv, R S Emple osder the trsormto T rom the rθ -ple to the -ple gve The Jco o T s ( ) ( r, θ ) r r Thus Theorem 4.6. gves θ θr r cosθ rsθ cosθ sθ rsθ r cos θ + rs θ r > r cosθ, R ( ) ( ) (,, dd r cosθ, rsθ ) ( r, θ ) S β ( r cosθ, rsθ ) rdrdθ α drdθ

54 Emple Use the chge o vrles u v, uv to evlute the tegrl R da where R s the rego ouded the -s d the prols Emple Evlute the tegrl (,), (,), (, ) d (, ) R e. 4 4 d ( + )/( ) da, where R s the trpezum wth vertces

55 hpter 5 Vector lculus 5. Vector Felds I hpter, we studed uctos tht mp sets o rel umers to sets o vectors. I ths chpter, we stud tpe o ucto, clled vector eld, whose dom s set o pots R or R d whose rge s set o vectors the two- (or three-) dmesol spce. ( ) Deto 5.. Let D e set R. A vector eld o F, the two-dmesol spce. D vector ( ) R s ucto F tht ssgs to ech pot (, ) F s two-dmesol vector, we c wrte t terms o ts compoet uctos P d Q s ollows: Sce (, ) or short, F (, ) P(, ) + Q(, ) j P(, ), Q(, ) F P + Qj Note tht P d Q re sclr uctos o two vrles d re sometmes clled sclr elds. Deto 5.. Let D e set, z R. A vector eld o (, ) D vector (,, z) F the three-dmesol spce. R s ucto F tht ssgs to ech pot As the two-dmesol cse, we c lso epress vector eld compoet uctos P, Q d R. F (,, z) P(,, z) + Q(,, z) j + R(,, z)k R terms o ts Emple 5.. Newto s Lw o Grvtto sttes the mgtude o the grvttol orce etwee two ojects wth msses m d M s mmg F r

56 where r s the dstce etwee the ojects d G s the grvttol costt. Assumg the oject wth mss M s locted t the org. The grvttol orce o the oject wth mss m s drected towrds the org. Wrte dow the vector equto. Emple 5..4 I s sclr ucto o two vrles, recll rom Secto.6 tht ts grdet vector Thereore s rell vector eld o (, ) + ( )j., R d s clled grdet vector eld. Lkewse s sclr ucto o three vrles, ts grdet s vector eld o (,, z) + (,, z) j + (, z)k z, R gve Deto 5..5 A vector eld F s clled coservtve vector eld t s the grdet o some sclr ucto,.e., there ests ucto such tht F. I ths stuto, s clled the potetl ucto o F. 5. Le Itegrls I ths secto, we dee tegrl tht s smlr to sgle tegrl ecept tht sted o tegrtg over tervl [, ], we tegrte over curve (le). Such tegrls re clled le tegrls. Gve curve tht s deed the prmetrc equtos A prtto o [, ] : the pots { t,..., } { P (, ): () t, () t, } () t () t t, (5.) t < t <... < t : determes prtto P o the curve the pots,...,. These pots P dvde to surcs wth legths s, s,..., s.. Let P deote the legth o the logest surc uder the prtto * * * P. We choose pot ( ) * [ t t ] t, P the th surc. (Ths correspods to pot, ). We the orm the Rem sum o log the rc wth prtto P

57 R * (,, P) ( *, ) s Deto 5.. I s deed o smooth curve gve Equto 5., the the le tegrl o log s ths lmt ests. * ( ) ds lm (, ) s, (5.) * p Remrk 5.. It c e show tht s cotuous ucto, the the lmt Equto 5. lws ests d the ollowg ormul c e used to evlute the le tegrl Remrk 5.. d d (, ) ds ( () t, ( t ) + dt (5.) dt dt, o, the (, )ds s the re o the ol whose se s d whose I ( ) heght ove the pot (, ) s ( ),. Emple 5..4 Evlute ( + )ds, where s the upper hl o the ut crcle +. Remrk 5..5 Suppose s ot smooth ut pecewse-smooth;.e., s uo o te umer o smooth curves,,..., the we dee the tegrl o log s ollows: (, ) ds (, ) ds + (, ) ds +... (, )ds + Emple 5..6 Evlute ds where cossts o the rc o the prol ollowed the vertcl le segmet rom (,) to (,). rom (,) to (, )

58 Deto 5..7 Usg the otto s Equto 5., the le tegrl o log wth respect to s where. * (, ) d lm (, ) * P Smlrl, the le tegrl o log wth respect to s where. * (, ) d lm (, ) * P We cll the orgl le tegrl (, )ds the le tegrl o wth respect to rc legth. Remrk 5..8 () It c e show tht d (, ) d ( ( t) () t ) ( t)dt, (, ) d ( () t, () t ) ( t)dt () Sometmes le tegrls wrt d occur together. Whe ths hppes t s customr to wrte P (, ) d Q(, ) d P(, ) d + Q(, )d + Emple 5..9 Evlute () d + d, where () s the rc o the prol s the le segmet rom ( 5, ) to (,) 4 rom ( 5, ) to (,)., d Suppose s curve deed the prmetrc equtos () t, () t, t

59 wth tl pot A d terml pot B. The the curve deed the prmetrc equtos ( t + ), ( t t + ), t hs the sme set o pots s ecept t trverse t the opposte drecto wth tl pot A d terml pot B. We deote ths curve. It c e show tht d However, s s s lws postve, ut ( ) d (, )d, ( ) d (, )d, ( ) ds (, )ds d, chge sg whe we reverse the oretto o. 5.. Le Itegrls Spce Suppose tht s smooth spce curve gve the prmetrc equtos () t, () t, z z() t, t I s ucto o three vrles cotuous over rego cotg, the we dee the le tegrl o log mer smlr to tht or ple curves: * * * (,, z) ds lm (,, z ) P s We evlute the le tegrl o usg the ollowg ormul: d d dz, dt dt dt (, z) ds ( () t, () t, z() t ) + + dt (5.4) Remrk 5.. () Oserve tht Equtos 5. d 5.4 c e rewrtte s ( r () t ) r ( t)dt () I the specl cse where (,, z) where L s the rc legth o. ds r, we get () t dt L

60 Emple 5.. Evlute s z ds, where s the crculr hel gve the equtos cost, st, z t, t π 5.. Le Itegrls o Vector Felds Deto 5.. Let F e cotuous vector eld deed o smooth curve gve vector ucto r t, t. The the le tegrl o F log s deed s () ( () t ) r ( t)dt F dr F r (5.5) Remrk 5.. Recll the ut tget vector d T () t ds dt Susttutg these to Equto 5.5, we hve F dr () t r r () t dt () t r. ( () t ) r () t dt F Tds F r Emple 5..4 Fd the work doe the orce eld F (, ) + j movg prtcle log the semcrcle r () t cost + stj, t π. Emple 5..5 Evlute dr F, where F(, z) zj zk, + + d s the twsted cuc gve t, t, z t

61 Theorem 5..6 where F P + Qj + Rk. F dr Pd + Qd + Rdz 5. The Fudmetl Theorem or Le Itegrls Recll the Fudmetl Theorem o lculus c e wrtte s F () d F() F(). The ollowg theorem s verso o the udmetl theorem or the le tegrls. Theorem 5.. Let e smooth curve gve the vector ucto r () t, t. Let e deretle ucto o two or three vrles whose grdet vector s cotuous o. The dr ( r() ) ( r( ) Remrk 5.. Theorem 5.. ss tht we c evlute the le tegrl o coservtve vector eld (the grdet vector eld o the potetl ucto ) kowg the vlue o t the edpots o. Emple 5.. Fd the work doe the grvttol eld mmg () F movg prtcle wth mss m rom the pot (,4,) to the pot (,,) pecewse smooth curve. log 5.. Idepedece o Pth Suppose d re two deret pecewse smooth curves (whch re clled pths) tht hve the sme tl pot A d terml pot B. We kow geerl F dr But oe cosequece o Theorem 5.. s tht F dr

62 dr dr wheever s cotuous. Deto 5..4 I F s cotuous vector eld o dom D, we s tht the le tegrl F dr s depedet o pth F dr F dr or two pths d. Remrk 5..5 A cosequece o Theorem 5.. s tht le tegrls o coservtve vector elds re depedet o pths. Deto 5..6 A curve s clled closed ts terml pot cocdes wth ts tl pot, tht s, r r. () () Theorem 5..7 dr F s depedet o pth D d ol F dr or ever closed pth D. Let D e rego R ( or R ), we s tht D s ope or ever pot P D, there ests dsk (or respectvel ll) wth cetre P tht les completel D. We s tht D s coected two pots D c e joed pth D. We see tht ll coservtve vector eld re pth depedet. Is the coverse true?.e., Are ll pth depedet vector elds coservtve? Theorem 5..8 Suppose F s vector eld tht s cotuous o ope coected rego D. I F dr s depedet o pth D, the F s coservtve vector eld o D ; tht s, there ests ucto o D such tht F. The ove theorem clsses coservtve vector elds. However, t s hrd to use t to determe whether vector eld s coservtve or ot. The ollowg theorem gves us smple w to det those vector elds tht re ot coservtve.

63 Theorem 5..9 I F(, ) P(, ) + Q(, )j, s coservtve vector eld where P d Q hve cotuous rst order dervtves o dom D, the throughout the dom we hve P Q The coverse o Theorem 5..9 s ot true geerl. However, we mpose stroger codto o the dom, the coverse c e cheved. A curve s sd to e smple curve t does ot tersect tsel where etwee the edpots. A rego D s sd to e smpl coected t s coected d ever smple closed curve D ecloses ol pots D. A smpl coected rego cots o holes. Theorem 5.. Let F P + Qj e vector eld o ope smpl coected rego D. Suppose P d Q hve cotuous rst-order dervtves d P Q throughout D, the F s coservtve vector eld. Emple 5.. Determe whether or ot the vector eld s coservtve. (, ) ( ) + ( )j F Emple 5.. Determe whether or ot the vector eld s coservtve. F (, ) ( + ) + ( )j Emple 5.. () I F(, ) ( + ) + ( )j, d ucto such tht F. () Evlute the le tegrl F dr, where s the curve gve t t () t e st + e costj, t π r.

64 Emple 5..4 z e e, d ucto such tht F. z z I F(,, ) + ( + ) j + k 5.4 Gree s Theorem Gree s Theorem gves the reltoshp etwee le tegrl roud smple closed curve d doule tegrl over the ple rego D ouded. We s tht smple closed curve s postvel oreted the curve trverses couterclockwse s t creses. Theorem 5.4. (Gree s Theorem) Let e postvel oreted, pecewse smooth smple closed curve the ple d let D e the rego ouded. I P d Q hve cotuous prtl dervtves o ope rego tht cot D, the Notto 5.4. The otto Q P Pd + Qd da. D Pd + Qd s sometmes used to dcte tht the tegrl s clculted usg the postve oretto o smple closed curve. Emple 5.4. Evlute 4 d + d, where s the trgulr curve cosstg o the le segmets rom (,) to (,), rom (,) to (,), d rom (,) to (,). Emple s 4 Evlute ( e ) d + ( 7 + ) d, where s the crcle + 9. We see tht geerl, the doule tegrl s eser to compute. I some stutos, the le tegrl s eser to compute. For stce, P (, ) Q(, ) o the curve, the

65 Q P da Pd + Qd o mtter wht vlues P d Q ssume the rego D. D Aother pplcto o the Gree s Theorem s computg re. We kow tht the re o D s da, thus we wsh to choose P d Q such tht D Q P There re severl possltes, we choose P ( ) d ( ), A d. I we choose P (, ) d Q ( ) I we choose P(, ) d (, ) Thus we hve, A, the we hve d Q, the A A d d d d d Emple Fd the re eclosed the ellpse +. Q,, the we hve d Emple Evlute d + d, where s the oudr o the sem-ulr rego D the upper hl-ple etwee the crcles + d + 4.

66 Emple I + j F (, ), + show tht F dr π or smple closed curve tht ecloses the org.