MATHEMATICAL ANALYSIS

Mri Predoi Trdfir Băl MATHEMATICAL ANALYSIS VOL II INTEGRAL CALCULUS Criov, 5

CONTENTS VOL II INTEGRAL CALCULUS Chpter V EXTENING THE EFINITE INTEGRAL V efiite itegrls with prmeters Problems V 5 V Improper itegrls 9 Problems V 9 V Improper itegrls with prmeters Problems V Chpter VI LINE INTEGRALS VI Curves Problems VI 7 VI Lie itegrls of the first tpe 9 Problems VI 4 VI Lie itegrls of the secod tpe 44 Problems VI 5 Chpter VII MULTIPLE INTEGRALS VII Jord s mesure 56 Problems VII 6 VII Multiple itegrls 6 Problems VII 77 VII Improper multiple itegrls 8 Problems VII 88 V

Chpter VIII SURFACE INTEGRALS VIII Surfces i R 9 Problems VIII 97 VIII First tpe surfce itegrls 99 Problems VIII VIII Secod tpe surfce itegrls 4 Problems VIII VIII4 Itegrl formuls Problems VIII4 7 Chpter IX ELEMENTS OF FIEL THEORY IX ifferetil opertors 9 Problems IX 7 IX Curvilier coordites Problems IX 9 IX Prticulr fields 4 Problems IX 5 Chpter X COMPLEX INTEGRALS X Elemets of Cuch theor 55 Problems X 66 X Residues 68 Problems X 85 INEX 88 BIBLIOGRAPHY VI

CHAPTER V EXTENING THE EFINITE INTEGRAL V EFINITE INTEGRALS WITH PARAMETERS We cosider tht the itegrl clculus for the fuctios of oe rel vrible is kow Here we iclude the idefiite itegrls (lso clled primitives or ti-derivtives s well s the defiite itegrls Similrl, we cosider tht the bsic methods of clcultig (ectl d pproimtel itegrls re kow The purpose of this prgrph is to stud etesio of the otio of defiite itegrl i the sese tht beod the vrible of itegrtio there eists other vrible lso clled prmeter efiitio Let us cosider itervl A R, I = [, b] R d f : A IR If for ech A ( is clled prmeter, fuctio t f(, t is itegrble o [, b], the we s tht F : A R, defied b b F( = f(, tdt is defiite itegrl with prmeter (betwee fied limits d b More geerll, if isted of, b we cosider two fuctios φ, ψ : A [, b] such tht φ( ψ( for ll A, d the fuctio t f(, t is itegrble o the itervl [φ(, ψ(] for ech A, the the fuctio G( = ( ( f(, tdt is clled defiite itegrl with prmeter (betwee vrible limits The itegrls with vrible limits m be reduced to itegrls with costt limits b chgig the vrible of itegrtio: Lemm I the coditios of the bove defiitio, we hve: G( = [ψ( φ(] f(, φ( + θ[ψ( φ(]d θ Proof I the itegrl G( we mke the chge t = φ( + θ [ψ( φ(], dt for which = ψ( φ( } d Reltive to F d G we'll stud the properties cocerig cotiuit, derivbilit d itegrbilit i respect to the prmeter Theorem If f : A I R is cotiuous o A I, the F : A R is cotiuous o A

Chpter V Etedig the defiite itegrl Proof If A, the either Å, or is ed-poit of A I cse there eists η > such tht K η = {(, t R : η, A, t[, b]} is compct prt of A I Sice f is cotiuous o A I, it will be uiforml cotiuous o K η, ie for ε > there eists δ > such tht f(', t' f(", t" < ( b wheever (', t', (", t" K η d d((', t', (", t" < δ Cosequetl, for ll A for which < mi { η, δ } we hve b F( F( f(, t f(, t dt (b < ε, ( b which mes tht F is cotiuous t } 4 Corollr If the fuctio f : A I R is cotiuous o A I, d φ, ψ : A [, b] re cotiuous o A, the G : A R is cotiuous o A Proof Fuctio g : A [, ] R, defied b g(, θ = f(, φ( + θ[ψ( φ(], which ws used i lemm, is cotiuous o A [, ], hece we c ppl theorem d lemm } 5 Theorem Let A R be rbitrr itervl, I = [, b] R, d let us ote f : A I R If f is cotiuous o A I, d it hs cotiuous prtil derivtive f, the F CR (A, d F'( = Proof We hve to show tht t ech A, there eists F( F( f lim (, t dt b b f (, tdt For this purpose we cosider the followig helpful fuctio f (, t f (, t if h(, t = f (, t if O the hpothesis it is cler tht h is cotiuous o A I, hece we c use theorem for the fuctio b H( = b h(, tdt = f (, t f (, t F( F( dt

O this w, the equlit H( =, d b F'( = lim V efiite itegrls with prmeters H( shows tht F is derivble t f (, tdt The cotiuit of F' is cosequece of the cotiuit of f, b virtue of the sme theorem } 6 Corollr If, i dditio to the hpothesis of the bove theorem, we hve φ, ψ C R (A, the G C R (A d the equlit G'( = ( ( f (, tdt + f(, ψ( ψ'( f(, φ( φ'( holds t A Proof Let us cosider ew fuctio L : A I I R, epressed b v L(,u,v = u we hve f(, tdt Accordig to the bove theorem, for fied u d v v L f (, u, v (, t dt O the other hd, the geerl properties u L L of primitive led to (, u, v = f(, u d (, u, v = f(, v u v Becuse ll these prtil derivtives re cotiuous, L is differetible o A I I Applig the rule of derivig composite fuctio i the cse of G( = L(, φ(, ψ(, we obti the ouced formul The cotiuit of G' follows b usig theorem } 7 Theorem If f : A I R is cotiuous o A I, the F : A R is itegrble o compct [α, β] A, d b F ( d f (, t ddt Proof Accordig to theorem, F is cotiuous o [α, β], hece it is lso itegrble o this itervl It is well kow tht the fuctio Ф( = F(d is primitive of F o [α, β] We will show tht

Chpter V Etedig the defiite itegrl b Ф( = For this purpose let us ote U(, t = The, '( = f (, t ddt f(, td d ( = U (, t = f(, t, hece ccordig to theorem 5, we hve b b U(, tdt f(,tdt Cosequetl, the equlities '( = F( = Ф'( hold t [α, β], hece Ф( ( = c, where c is costt Becuse Ф(α = (α =, we obti c =, ie Ф = I prticulr, Ф(β = (β epress the required equlit } 8 Corollr If, i dditio to the coditios i the bove theorem, the fuctios φ, ψ : A[, b] re cotiuous o A, the G( d g(, dd where g(, θ = f(, φ( + θ[ψ( φ( ] [ψ( φ( ] (s i corollr 4 Proof Accordig to Lemm, we hve G( = g (, d, so it remis to use theorem 7 } 9 Remrk The formuls estblished i the bove theorems d their corollries (especill tht which refers to derivtio d itegrtio re frequetl useful i prctice for clcultig itegrls (see the problems t the ed of the prgrph I prticulr, theorem 7 gives the coditios o which we c chge the order i iterted itegrl, ie b b f (, t dtd f (, t ddt 4

/ PROBLEMS V Clculte l( si t dt, where > Hit eotig the itegrl b F(, we obti F'( = Usig the substitutio tg t = u, we obti F'( = F( = π l( + + c I order to fid c, we write V efiite itegrls with prmeters / si, d so c = F( π l( + = / = si t l l dt πl( + = / = si t l dt l Tkig here, it follows c = πl Clculte I = f( = f(d, where f : [, ] R hs the vlues Hit Notice tht f( = eists l t if (,, if, dt t dt t [,, d t the ed poit, there lim f (, so ol t this poit f differs form cotiuous fuctio o [, ] Cosequetl I = [ t dt ]d = t ddt l Clculte si t tg t e e dt lim dt 5

Chpter V Etedig the defiite itegrl Hit This is idetermitio; i order to use L'Hospitl rule we eed the derivtives reltive to, which is prmeter i the upper limits of itegrls, so the limit reduces to lim si cos cos si tg e e tg t ( t e t e dt t dt d 4 Clculte I =, where < b <, d deduce the vlues of b cos I = d cos, K = b cos ( b cos d d L = l( b cos d Hit The substitutio tg = t is ot possible i I becuse [, π is crried ito [, Sice the itegrl is cotiuous o R, we hve I = l d lim, b cos l d this lst itegrl c be clculted usig the metioed substitutio More ectl, l d l tg dt rctg tg b cos b t ( b b hece I = b b b To obti K, we derive I reltive to b Fill, rctg rctg 5 Clculte I = d b derivig I( = d, Hit Substitutio = cos θ gives / l L =I d d I'( = ( cos Becuse the substitutio tg θ = t crries [, ito [,, d the substitutio tg = t leds to complicted clcultio, we cosider 6

d I'(= lim cos l l V efiite itegrls with prmeters If we replce tg θ = t i this lst itegrl, the we obti l d cos = Cosequetl I'( = tgl dt t rctg tgl, hece I( = l( + + c Becuse I( = it follows tht c =, hece I = I( = l(+ rctg rctg d 6 Clculte I = d usig the formul Hit Chgig the order of itegrtio we obti I = d d d d ( so the problem reduces to I'( from problem 5 bsi d 7 Clculte K= l, > b > bsi si Hit Usig the formul obti Sice si bsi l bsi K = b d b b d b si d si d b K = b rcsi b b b b b d it follows tht d si si d we 7

Chpter V Etedig the defiite itegrl 8 Show tht I + ( = d Usig this result, clculte ( Hit erive I ( reltive to I' (, where I ( = ( d, N *, 9 Use Theorem 7 to evlute I = f ( d, where si(l, if d f ( l, if or d >, > Hit Itroduce prmeter t d remrk tht I = t dt si(l d Chge the order of itegrtio to obti I = t dt si(l d dt = ( t The result is I = rctg ( ( 8

V IMPROPER INTEGRALS I the costructio of the defiite itegrl, oted b f ( t dt, we hve used two coditios which llow us to write the itegrl sums, mel: (i d b re fiite (ie differet from + ; (ii f is bouded o [, b], where it is defied There re still m prcticl problems, which led to itegrls of fuctios ot stisfig these coditios Eve defiite itegrls reduce sometimes to such "more geerl" itegrls, s for emple whe chgig the vribles b tg = t, the itervl [, π] is crried ito [, ] The im of this prgrph is to eted the otio of itegrl i the cse whe these coditios re o loger stisfied efiitio The cse whe b = If f : [, R is itegrble o [, β] for ll β >, d there eists L = lim f ( t dt, the we m s tht f is improperl itegrble o [,, d L is the improper itegrl of f o [, I this cse we ote f ( t dt = lim f ( t dt, d we s tht the improper itegrl is coverget Similrl we discuss the cse whe = The cse whe f is ubouded t b Let f : [, b R be ubouded i the eighborhood of b, i the sese tht for rbitrr δ > d M > there eists t (b δ, b such tht f (t > M If f is itegrble o [, β ] for ll < β < b, d there eists L = lim b f ( t dt, the we s tht f is improperl itegrble o [, b, d L is clled improper itegrl of f o [, b If L eists, we ote b f ( t dt = lim f ( t dt, d we s tht the b improper itegrl is coverget We similrl tret the fuctios which re ubouded t Remrks I prctice we ofte del with combitios of the bove simple situtios, s for emple f ( t dt f ( t dt lim ot R f ( t dt, 9

Chpter V Etedig the defiite itegrl b f ( t dt lim f ( t dt, where < α < β <b b b The itegrl f ( t dt c be improper becuse f is ubouded t some poit c (, b, i which cse we defie b f t dt ( = ( t dt lim lim f f ( t dt c c b c c b From the geometricl poit of view, cosiderig improper itegrls m be iterpreted s mesurig res of ubouded subsets of the ple The eistece of the bove cosidered limits shows tht we c spek of the re of ubouded set, t lest for sub-grphs of some rel fuctios c I spite of the diversit of tpes of improper itegrls, there is simple, but essetil commo feture, mel tht the itegrtio is relied o o-compct sets I fct, compct set i R is bouded d closed, hece [,, (, b], (,+ re o-compct becuse the re ot bouded, while [, b, (, b], etc re o-compct becuse of o-closeess Obviousl, other combitios like (,, (, c (c, b], etc re possible Becuse improper itegrl is defied b limitig process, whe provig some propert of such itegrls it is sufficiet to cosider ol oe of the possible cses dt Emples The itegrl I(λ = t (λ R is coverget for λ >, whe I(λ = (λ, d diverget for λ I fct, ccordig to the bove defiitio, I(λ = lim t ( t dt l Fill, it remis to remember tht lim dt, where if if if if if

V Improper itegrls dt b The itegrl I(μ = t (μ > is coverget for μ <, whe it equls I(μ = ( μ, d it is diverget for μ Figures V, respectivel b, suggest how to iterpret I(λ d I(μ s res of some sub-grphs (htched portios t Fig V The usul properties of the defiite itegrls lso hold for improper itegrls, mel: 4 Propositio The improper itegrl is lier fuctiol o the spce of ll improperl itegrble fuctios, ie if f, g : [, b R re improperl itegrble o [, b, d λ, μ R, the λf + μg is improperl itegrble o [, b d we hve: b ( g( t dt f ( t dt b f g( t dt b The improper itegrl is dditive reltive to the itervl, ie b c f ( t dt = f ( t dt + b c b f ( t dt c The improper itegrl is depedet o the order of the itervl, mel b f ( t dt = t b f ( t dt 5 Theorem (Leibi-Newto formul Let f : [, b R be (properl itegrble o compct [, β ]icluded i [, b, d F be the primitive of f o [, b The ecessr d sufficiet coditio for f to be improperl itegrble o [, b is to eist the fiite limit of F t b I this cse we hve: b

Chpter V Etedig the defiite itegrl b f ( t dt = lim F( F( b 6 Theorem (Itegrtio b prts If f, g stisf the coditios: (i f, g C R([, b] (ii there eists d is fiite lim ( fg( b (iii b b b f ( t g' ( t dt is coverget the f '( t g( t dt is coverget too, d we hve b f '( t g( t dt = lim ( fg( b b b f(g( f ( t g' ( t dt 7 Theorem (Chgig the vrible Let f : [, b R be cotiuous o [, b, d let φ : [', b' [, b be of clss C R([', b'], such tht φ(' = d lim ( b b ' ' b b If f ( t dt is coverget, the the itegrl b' ' is lso coverget, d we hve b' ' f ( ( ' ( d f ( ( ' ( d b = f ( t dt The bove properties (especill theorems 5 7 re useful i the cses whe primitives re vilble If the improper itegrl c't be clculted usig the primitives it is still importt to stud the covergece For developig such stud we hve severl tests of covergece, s follows: 8 Theorem (Cuch's geerl test Let f : [, b R be (properl b itegrble o [, β] [, b The ε > there eists δ > such tht b', b" (b δ, b implies f ( t dt is coverget iff for ever b" b' f ( t dt

V Improper itegrls Proof Let F : [, b R be defied b F( = f ( t dt The f is improperl itegrble o [, b if F hs fiite limit t b, which mes tht for ever ε > we c fid δ > such tht b', b" (b δ, b implies F(b' F(b" < ε It remis to remrk tht F(b' F(b" = b" b' f ( t dt } The bove Cuch's geerl test is useful i reliig logies with bsolutel coverget series s follows: 9 efiitio If f : [, b R, the we s tht the itegrl b bsolutel coverget iff b f ( t dt is f ( t dt is coverget, ie f is improperl itegrble o [, b Remrk I wht cocers the itegrbilit of f d f, the improper itegrl differs from the defiite itegrl: while f itegrble i the proper sese implies f itegrble, this is ot vlid for improper itegrls I fct, there eist fuctios, which re improperl itegrble without beig bsolutel itegrble For emple, let f : [, R be fuctio of ( vlues f ( =, d f (t = if t (, ], where N * This fuctio is improperl itegrble o [,, d f ( t dt ( l, but it is ot bsolutel itegrble sice f ( t dt The et propositio shows tht the opposite implictio holds for the improper itegrls: Propositio Ever bsolutel coverget itegrl is coverget Proof Usig the Cuch's geerl test, the hpothesis mes tht for ever ε > there eists δ > such tht for β', β" (b δ, b we hve " ' f ( t dt < ε Becuse f is properl itegrble o compct from [, b, d

Chpter V Etedig the defiite itegrl " ' f ( t dt " ' f ( t dt 4 " = ' f ( t dt it follows tht f is improperl itegrble o [, b } Theorem (The compriso test Let f, g : [, b R be such tht: f, g re properl itegrble o compct from [, b for ll t [, b we hve f(t g(t b g ( t dt is coverget b The f ( t dt is bsolutel coverget " Proof Becuse ( t dt ' " f g( t dt holds for ll, b, b, ', we c ppl the Cuch's geerl test } Remrk Besides its utilit i estblishig covergece, the bove theorem c be used s divergece test I prticulr, if f(t g(t for b ll t [, b, d f ( t dt is diverget, the g ( t dt is diverget too b I prctice, we relie compriso with fuctios like i emple, ie o [,, t o [, b, q ( b t t o [,, etc The compriso with such fuctios leds to prticulr forms of Theorem, which re ver useful i prctice We metio some of them i the followig theorems 4-8 4 Theorem specil form # I of the compriso test (Test bsed o lim t t f ( t Let f : [, R + be itegrble o compct from [, d let us ote = lim t f ( t t If λ > d <, the If λ d <, the b f ( t dt is coverget f ( t dt is diverget Proof If (,, the for ever ε > there eists δ > such tht t > δ implies < ε < t λ f(t < + ε, ie

V Improper itegrls f ( t t t If, the the itegrl of o [δ, is diverget, so the first iequlit from bove shows tht λ >, the t the itegrl t f ( t dt is diverget too Similrl, if is itegrble o [δ,, d the secod iequlit shows tht f ( t dt is coverget The cses = d = re similrl discussed usig sigle iequlit from bove } 5 Theorem specil form # II of the compriso test (Test bsed o lim( b t tb f ( t Let f : [, b R + be itegrble o compct from [, b, d let us ote = lim( b t f ( t, where λ R tb If λ < d <, the If λ d <, the b b f ( t dt is coverget, d f ( t dt is diverget The proof is similr to the bove oe, but uses the testig fuctio ( b t o [, b } The bove two tests hve the icoveiet tht the refer to positive fuctios The followig two theorems re cosequeces of the compriso test for the cse of o-ecessril positive fuctios 6 Theorem specil form # III of the compriso test (Test of (t itegrbilit for f(t = t o [, Let f : [, R, where >, (t be fuctio of the form f(t = t where: φ is cotiuous o [, There eists M > such tht ( tdt M for ll α > 5

Chpter V Etedig the defiite itegrl The f ( t dt is coverget, wheever λ > Proof B hpothesis, for Φ( = ( tdt we hve α [, Sice λ + >, it follows tht d ( M for ll is coverget So, ( ccordig to theorem, d is bsolutel coverget Itegrtig b prts we obti ( t dt t '( t t dt ( t dt t which shows tht f is itegrble o [, } 7 Theorem specil form # IV of the compriso test (Test of itegrbilit for f(t = (b t λ φ(t o [, b Let f : [, b R, where b R, be fuctio of the form f(t = (b t λ φ(t If φ is cotiuous o [, b there eists M > such tht ( tdt M for ll α [, b, b the the itegrl f ( t dt is coverget for λ > Proof Let us remrk tht Φ( = ( tdt verifies the iequlit b ( ( b M ( b d Sice λ <, b is coverget, hece ( bsolutel coverget It remis to itegrte b prts b ( b t ( t dt ( b t ' ( t dt b b ( b t b ( t ( ( b dt d d use the form of f } is 6

7 V Improper itegrls The followig test is bsed o the compriso with the prticulr fuctio g : [, R, of the form g( = q, where q > d > (see lso problem V 8 Theorem specil form # V of the compriso test (The Cuch's root test Let f : [, R, where >, be itegrble o compct from [,, d let us suppose tht there eists = If <, the If >, the f ( t dt is bsolutel coverget, d f ( t dt is ot bsolutel coverget / t lim f ( t t Proof B the defiitio of, we kow tht for ever ε > there eists δ > such tht t > δ implies f(t /t < ε, ie ε < f(t /t < + ε If <, let us ote q = + ε < If t > δ, we hve f(t < q t So, it remis to see tht q t is itegrble o [δ, sice q < Becuse f is itegrble o the compct [, δ ], it will be itegrble o [, too The secod cse is similrl led b otig q = ε >, whe q t diverget, d f(t > q t } The covergece of some improper itegrls c be reduced to the covergece of sequeces d series 9 Theorem (Test of reductio to series If f : [, R + is decresig fuctio, itegrble o [, b] [,, the the followig ssertios re equivlet: f ( t dt is coverget b The sequece of terms u = f ( t dt, N, is coverget c The series N f ( is coverget Proof implies b becuse if there eists = lim f ( t dt, the lim f ( t dt = too b b The writte itegrls eist becuse decresig fuctios re itegrble o compct itervls dt is

Chpter V Etedig the defiite itegrl b c follows from the iequlit f (t f ( + o [ +, + ], which leds to b k f ( k f ( t dt Fill, c becuse from k k k f ( t dt f( + k it follows tht f ( t dt f ( k for ll b [, + ] } Remrks Betwee improper itegrls d series there re still sigifict differeces For emple, the covergece of f ( t dt does ot geerll impl lim f (t = (see problem 6 t b The otio of improper itegrl is sometimes used i more geerl sese, mel tht of "priciple vlue" (lso clled "Cuch's pricipl vlue", deoted s pv B defiitio, pv pv b f ( t dt lim f ( t dt, d c b f ( t dt lim f ( t dt f ( t dt c where c (, b is the poit roud where f is ubouded Of course, the coverget itegrls re lso coverget i the sese of the pricipl vlue, but the coverse implictio is geerll ot true (see problem 7 8

V Improper itegrls PROBLEMS V Show tht q t dt, where >, q > is coverget for q < d it is diverget for q Hit If q =, the d is diverget Otherwise q si Stud the covergece of the itegrls d Hit Use theorems 4 d 5 for b si d l d [q b q ] l q d l d si Show tht d is coverget but ot bsolutel coverget si Hit Becuse lim, the itegrl is improper ol t the upper limit We c ppl theorem 6 (specil form # III to φ ( = si, for λ = The itegrl is ot bsolutel coverget becuse for > we hve si si, d which is diverget si d d d 4 Estblish the covergece of (cos cos d, for λ (, Hit Appl theorem 7 (specil form # IV for φ ( = cos dt t t 5 Ale the covergece of the itegrls si si cos, sice 9

Chpter V Etedig the defiite itegrl I = where N * d, d J = d, ( Hit Use theorem 8 (specil form # V For I, lim, hece I is (bsolutel coverget For the (positive fuctio i J we hve lim, so J is diverget for > I the cse =, we hve lim, hece J is diverget 6 Show tht t cost dt is coverget eve if lim cos does't eist Is this situtio possible for positive fuctios isted of cos? Hit Use theorem 6 for φ ( = cos d λ =, sice t cost dt si si Accordig to theorem 4, the swer to the questio is egtive, ie positive fuctios which re itegrble o [, must hve ull limit t ifiit I fct, o the cotrr cse, whe lim f ( does't eist or is differet from ero, we hve i the metioed test, it would follow tht lim f (, hece tkig λ = d f ( t dt is diverget 7 Stud the pricipl vlues of the itegrls I = e t si tdt, J = t dt, where [] is the etire prt of,

V Improper itegrls dt K = cos tdt, d L = t Solutio I is (bsolutel coverget; J is diverget, but pvj = ; K is diverget i the sese of pv; L is diverget, but pvl = l 8 Stud the covergece of the itegrls I = d K = cos d, where N Hit lim e e d, J = si d, for N, hece pplig theorem 4, I is coverget J, J, K, K re diverget ccordig to the defiitio I J d K, for we m replce = t, d use theorem 6 9 Show tht the followig itegrls hve the specified vlues: I = e d! b J = e d! Hit Estblish the recurrece formul I = I b Replce = t i the previous itegrl Usig dequte improper itegrls, stud the covergece of the series:, R * ; b l, R ; c, R (l Hit Use theorem 9 I itegrl d (l b l d we c itegrte b prts I the we c chge l = t All these itegrls (d the correspodig series re coverget iff α >

V IMPROPER INTEGRALS WITH PARAMETERS We will recosider the topic of V i the cse of improper itegrls efiitio Let A R, I = [, b R, d f : A I R be such tht for ech A, the fuctio t f(, t is improperl itegrble o [, b The F : AR, epressed b b F( = f (, t dt ; f (, t dt ; f (, t dt ; etc is clled improper itegrl with prmeter Remrk Accordig to the defiitio of improper itegrl, F is defied s poit-wise limit of some defiite itegrls, ie F( p lim f (, t dt b More ectl, this mes tht for A d ε >, there eists δ(, ε > such tht for ll β (b δ, b, we hve f (, t dt F( M times we eed stroger covergece, like the uiform oe, which mes tht for ε >, there eists δ(ε > such tht for ll A d β (b δ, b, we hve the sme iequlit: f (, t dt F( I this cse we s tht the improper itegrl uiforml coverges to F, d we ote F( u b lim f (, t dt The followig lemm reduces the covergece of the itegrl to the covergece of some fuctio sequeces d series Lemm Let us cosider A R, I = [, b R, d f : A I R fuctio, such tht for ech A, the mp t f(, t is itegrble o ech compct from I The followig ssertios re equivlet: (i The improper itegrl b f (, t dt, with prmeter, is uiforml (poit-wise coverget o A to F ; (ii For rbitrr icresig sequece (β N for which β = d lim b, the fuctio sequece (F N, where F : A R hve the vlues F (= f (, t dt, is uiforml (poit-wise coverget o A to F

V Improper itegrls with prmeters (iii For rbitrr icresig sequece (β N such tht β = d lim b, the fuctio series u, of terms u : A R, where u ( = f (, t dt, is uiforml (poit-wise coverget o A to F The proof is routie d will be omitted, but we recommed to follow the scheme: (i (ii (iii 4 Theorem (Cuch's geerl test Let A R, I = [, b R, d f : A I R be such tht the mp t f(, t is itegrble o ech b compct from I, for rbitrr A The the improper itegrl f (, t dt with prmeter, is uiforml coverget o A iff for ever ε >, there eists δ(ε > such tht for rbitrr A d b', b" (b δ, b, we hve Proof If F( u b" b' lim b b' f (, t dt b" b' f (, t dt f (, t dt, the we evlute f (, t dt F( b" f (, t dt F( s we usull prove Cuch coditio Coversel, usig the bove lemm, we show tht the sequece (F N, where F ( = f (, t dt, β =, β < β +, d lim b, is uiforml Cuch o A I fct, for ε > we hve F ( F m ( = f (, t dt, wheever β, β m (b δ, b, ie m, > (δ N } m Usig this geerl test we obti more prcticl tests: 5 Theorem (Compriso test Let A, I d f be like i the bove theorem Let lso g : I R + be such tht:

Chpter V Etedig the defiite itegrl f(, t g(t for ll (, t A I b g ( t dt is coverget b The f (, t dt is uiforml coverget o A Proof I order to ppl the bove geerl test of uiform covergece we evlute b" b' b" b" f (, t dt f (, t dt g( t dt The lst itegrl c be b' mde rbitrril smll for b', b" i pproprite eighborhood of b, sice g is itegrble o [, b } 6 Remrk If compred to theorem,, we see tht the uiform boudedess reltive to, f(, t g(t, leds to the uiform covergece o A Cosequetl, prticulr tests similr to theorems 48 i V re vlid, if the hpothesis re uiforml stisfied reltive to A As i V, we re iterested i estblishig the rules of opertig with prmeters i improper itegrls 7 Theorem (Cotiuit of F Let f : A I R be cotiuous o A I, b where A R, d I = [, b R If the itegrl coverget o A, the F : A R, epressed b F( = cotiuous o A Proof Accordig to Lemm, F u b' lim F f (, t dt is uiforml b f (, t dt is O the other hd, F re cotiuous o A (see theorem i Cosequetl, F is cotiuous s uiform limit of cotiuous fuctios } 8 Theorem (erivbilit of F Let A R, I = [, b R, d f : A I R be such tht: f is cotiuous o A I f is cotiuous o A I b f (, t dt is poit-wise coverget o A to F : A R bf 4 (, t dt is uiforml coverget o A 4

V Improper itegrls with prmeters f The F is derivble o A, its derivtive is F'( = (, t dt, d F' is cotiuous o A Proof Let us ote F ( = sequece for which b = d lemm, F= lim F b b f (, t dt, where (b N is icresig lim b b Accordig to the previous poit-wise O the other side F is derivble s defiite itegrl with prmeter (see theorem 5,, d b F '( = f (, t dt Now, usig the sme lemm for uiforml coverget itegrls, we obti ll the climed properties of F } The opertio of itegrtio m be relied either i the proper sese (s i defiite itegrls, or i the improper sese 9 Theorem (The defiite itegrl reltive to the prmeter Let us cosider A = [α, β] R, I = [, b R, d f : A I R be such tht: f is cotiuous o A I b f (, t dt is uiforml coverget o A = [α, β] to F b The F is itegrble o [α, β] d F( d f (, t d dt Proof Let (b N be icresig sequece such tht b = d lim b b Accordig to Lemm, F u lim F, where F : [α, β] R re epressed b F ( = b f (, t dt O the other hd, ccordig to theorem, V, F re cotiuous fuctios, hece F is cotiuous too So, we deduce tht F is itegrble o [α, β], d F( d lim F ( d Now it remis to use theorem 7, V, i order to clculte b F ( d f (, t d dt, d to ppl lemm gi } 5

Chpter V Etedig the defiite itegrl Theorem (The improper itegrl reltive to the prmeter Let us cosider A = [α, β R, I = [, b R, d f : A I R be such tht: f is positive d cotiuous o A I b b f (, t dt is uiforml coverget to F: AR o compct from A f (, t d is uiforml coverget to G : I R o I 4 G ( t dt is coverget The F is improperl itegrble o [α, β, d F ( d = G ( t dt Proof Accordig to the previous theorem, for ech η [α, β, the fuctio b F is itegrble o [α, η], d F( d f (, t d dt Let us ote b φ : [α, β] [, b R the fuctio of vlues φ(η, t = f (, t d if t, G( t if t The third hpothesis of the theorem shows tht φ is cotiuous o the set [α, β] [, b O the other hd, if we ote b Φ: [α, β] R the fuctio Φ(η = b (,tdt, we obti Φ(η = F ( d for ll η [α, β Now, the problem reduces to etedig this reltio for η = β I fct, becuse f is positive, for ll η [α, β d t [, b we hve b f (, t d f (, t d, b ie φ(η, t G(t Sice G ( t dt is coverget, the compriso test shows tht b (,tdt is uiforml coverget to Φ Addig the fct tht φ is cotiuous, theorem 7 shows tht Φ is cotiuous o [α, β], hece there 6

eists lim Φ(η = Φ(β, ie Φ(β = 7 V Improper itegrls with prmeters F ( d Replcig Φ d φ b their vlues, we obti the climed formul } Remrks Theorems 9 d estblish the coditios whe we c chge the order of itegrtio, ie b b f (, t dtd f (, t ddt b The coditio f to be positive i theorem is essetil For emple, if t f : [, [, R is epressed b f(, t =, the f(, t ( t s well s f(, t t for ll (, t [, [,, hece f is itegrble o [, reltive to t, d lso reltive to B direct clcultio we fid F( = d G(t = Cosequetl, F d G re lso ( ( t itegrble o [,, but G ( t dt F( d Eceptig the coditio of beig positive, f stisfies ll coditios of theorem The itegrls with prmeter re useful i defiig ew fuctios The Euler's Γ d B fuctios re tpicl emples i this sese: efiitio The fuctio Γ : (, (, epressed b Γ( = t is clled Euler's gmm fuctio The fuctio B: (, (, (, of vlues B(, = t e t dt ( t dt is clled Euler's bet fuctio This defiitio mkes sese becuse: Propositio The itegrls of Γ d B re coverget Proof The itegrl which defies Γ is improper both t d Becuse t e t t for t [, ], d t is itegrble if >, it follows tht the itegrl of Γ is coverget t This itegrl is coverget t becuse t e t is itegrble o [, for ll N

Chpter V Etedig the defiite itegrl The itegrl which defies B is lso improper t d t, d, i dditio, it depeds o two prmeters The covergece of this itegrl follows from the iequlit t ( t [t + ( t ], which holds for t [, ], > d > (see the compriso test This iequlit m be verified b cosiderig two situtios: If t [/,, d >, the t, so tht i this cse t ( t ( t [t + ( t ]; b If t (, /], the ( t [/,, d sice > too, we hve ( t, d similr evlutio holds } 4 Theorem Fuctio Γ hs the followig properties: (i it is cove d idefiitel derivble fuctio; (ii Γ( + = Γ( t > ; (iii Γ( + =! for ever N, ie Γ geerlies the fctoril Proof (i It is es to see tht f(, t = t e t stisfies the coditios i theorem 8, hece t Γ'( = t e l tdt B repetig this rgumet we obti Γ (k ( = t 8 e t k l tdt for k N *, ie Γ is idefiitel derivble Its coveit follows from Γ"( > for ll > (ii Itegrtig b prts we obti we obti Γ(+ = t e t dt = lim t e t + t t e t dt = Γ( (iii Accordig to (ii, Γ( + = Γ( = ( Γ(, d Γ( = e t dt = 5 Theorem Fuctio B hs the properties: (i B(, = B(,, ie B is smmetric; (ii For (, (, (, we hve B(, = (iii It hs cotiuous prtil derivtives of order Proof (i Chgig t = θ, B(, becomes B(, v v (ii Replcig t = i B, we obti B(, = v ( v other hd, chgig t = ( + vu i Γ, it follows tht ( ( ; ( dv O the

Γ( = ( + v u 9 e V Improper itegrls with prmeters u( v Writig this reltio t + isted of, we hve du u( v Γ(+ u e du ( v Amplifig b v d itegrtig like i B, we obti u( v Γ(+B(, = v u e dudv Usig theorem we chge the order of the itegrls d we obti u uv Γ( + B(, = u e v e dvdu = u = u e u ( d = = Γ( u e u du = Γ( Γ( (iii This propert results form the similr propert of Γ, tkig ito ccout the bove reltio betwee Γ d B } e t 6 Remrkble itegrls Γ( = dt d t (lso clled Euler-Poisso itegrl I fct, B(, = Γ ( = replcig = si t e u du d, which turs out to be π, if ( The secod itegrl follows from Γ( b tkig t = u m b The biomil itegrl I = d, >, b >, p > m + > p ( b m be epressed b elemetr fuctios ol if p is iteger m is iteger (positive p m is iteger (positive

Chpter V Etedig the defiite itegrl I fct, otig b = u d k = I = k u m Aother chge of vribles, mel I = kv m m p p b p m ( u du u u ( v dv = k B(, we obti = v, leds to m m, p = m m p = k ( p This formul shows tht i geerl, I is epressed b Γ; i the metioed cses Γ reduces to fctorils, so I cotis ol elemetr fuctios m We recll tht i the cse whe is iteger, we mke the substitutio + b = t s, where s is the deomitor of the frctio p m Similrl, if p is iteger, the evlutio of the itegrl m be mde b the substitutio + b = t s

V Improper itegrls with prmeters PROBLEMS V Show tht F( = si t e t dt is coverget for [, d t F(=rctg Hit The itegrl is improper t ; the covergece is cosequece of si t the compriso test, if g(t=, t (see lso theorem 6, V B t theorem 8, F'( =, hece F( = rctg + C Tke = Clculte I(r = ( r cos r d, where r < Hit The substitutio t = tg i I'(r gives r cos I'(r = r cos r r r d 4 r ( t t ( t dt A where = > Brekig up ( t ( t t t where A = B =, we obti 4 dt I'(r = ( r t ( t Cosequetl, I(r = C, but I( =, hece I(r = too B, Show tht Φ( = si t dt (Poisso t e t si t dt rctg, d deduce tht t Hit Usig the result of problem, Φ( = F( = rctg = rctg Aother method cosists i itegrtig two times b prts i Φ'(, d obtiig Φ'( = Φ'(, wherefrom it follows tht Φ( = rctg + C

Chpter V Etedig the defiite itegrl For we deduce C = Fill, the Poisso's itegrl is Φ( b e e cos cosb 4 Clculte I = d, d J = d, where < < b b b b t Hit I = t b e dt d e ddt dt l t b b t si si t J = tdt d d dt si = (b, where d = is the Poisso's itegrl (see problem idepedetl of t > 5 Let f : (, ] (, ] R be fuctio of vlues Show tht dd ( ( wh these itegrls hve differet vlues Hit Theorem does ot work sice f chges its sig, d d 6 Use the fuctios bet d gmm to evlute the itegrls I = b J = p p e m q ( d, p, q, m > ; q d, p > -, q > Hit Chge the vrible m = t, d evlute I = m t p m b Replce q = t, d clculte J = q t p q ( t e q t dt dt p = B, q m m p = q q t f (, t ( t, d epli

CHAPTER VI LINE INTEGRAL We will geerlie the usul defiite itegrl i the sese tht isted of fuctios defied o [, b R we will cosider fuctios defied o segmet of some curve There re two kids of lie itegrls, depedig of the cosidered fuctio, which c be sclr or vector fuctio, but first of ll we must precise the termiolog cocerig curves (there re plet mterils i the literture VI CURVES We le the otio of curve i R, but ll the otios d properties c be obviousl trsposed i R p, p N \ {, }, i prticulr i R efiitio The set γ R is clled curve iff there eists [, b] R d fuctio φ : [, b] R such tht γ = φ ([, b] I this cse φ is clled prmeteritio of γ Tpes of curves The poits A = φ( d B = φ(b re clled edpoits of the curve γ ; if A = B, we s tht γ is closed We s tht γ is simple (without loops iff φ is ijective Curve γ is sid to be rectifible iff φ hs bouded vritio, ie there eists b V sup ( ti ( ti, i where δ = {t = < t < < t = b} is divisio of [, b] The umber b L = V is clled legth of γ We s γ is cotiuous (Lipschite, etc iff φ is so Let us ote φ(t = ((t,(t,(t for t [, b] If φ is differetible o [, b], d φ' is cotiuous d o-ull, we s tht γ is smooth curve This mes tht there eist cotiuous derivtives ', ' d ', d ' (t + ' (t + ' (t, t [, b] The vector t ( / (t, / (t, / (t is clled tget to γ, t M ((t,(t,(t For prcticl purposes, we frequetl del with cotiuous d piecewise smooth curves, ie curves for which there eists fiite umber of itermedite poits C k γ, k =,, where C k = φ(c k for some c k (, b, such tht φ is smooth o ech of [, c ], o [c k, c k+ ] for ll k =,,, d o [c, b], d φ is cotiuous o [, b] The imge of restrictio of φ to [c, d] [, b] is clled sub-rc of the curve γ, so γ is piece-wise smooth iff it cosists of fiite umber of smooth sub-rcs

Chpter VI Lie itegrl Remrks The clss of rectifible curves is ver importt sice it ivolves the otio of legth Geometricll spekig, the sum i ( t, i ( t i from the bove defiitio of the vritio b V, represets the legth of broke lie of vertices φ(t i Pssig to fier divisios of leds to loger broke lies, hece is rectifible iff the fmil of these iscribed broke lies hs u upper boud for the correspodig legths Without goig ito detils, we metio tht fuctio f :[, b] R hs bouded vritio if it hs oe of the followig properties: mooto, Lipschit propert, bouded derivtive, or it is primitive, ie f ( ( t dt, [, b] (for detils, icludig properties of the fuctios with bouded vritio, see [FG], [N--M], etc The bove defiitio of the rectifible curves is bsed o the followig reltio betwee bouded vritio d legth of curve: 4 Theorem (Jord Let = (, : [, b]r be prmeteritio of ple curve The curve is rectifible if d ol if the compoets, d of hve bouded vritio We omit the proof, but the reder m cosult the sme bibliogrph 5 Corollr If is smooth curve, the it is rectifible, d its legth is L ( t ( t dt b / A similr formul holds for curves i R d R Becuse ll the otios from bove re bsed o some prmeteritio, it is importt to kow how c we chge this prmeteritio, d wht hppes whe we chge it These problems re solved b cosiderig the followig otio of "equivlet" prmeteritios of smooth curve 6 efiitio The fuctios φ : [, b] R d ψ : [c, d] R re equivlet prmeteritios iff there eists diffeomorphism σ : [, b] [c, d] such tht σ'(t for ll t [, b], d φ = ψ σ I this cse we usull ote φ ψ, d we cll σ itermedite fuctio 7 Remrks (i Reltio from bove is rell equivlece I dditio, this equivlece is pproprite to prmeteritios of curve becuse equivlet fuctios hve ideticl imges Whe we re iterested i studig more geerl th smooth curves, the "itermedite" fuctio σ (i defiitio stisfies less restrictive coditios, s for emple, it c ol be topologicl homeomorphism / 4

VI Curves (ii Becuse σ : [, ] [, b] defied b σ(t = tb + (t, is emple of itermedite (eve icresig fuctio i defiitio, we c lws cosider the curves s imges of [, ] through cotiuous, smooth or other fuctios Aother useful prmeteritio is bsed o the fct tht the fuctio σ : [, b] [, L], defied b σ(t = t ' ( ' ( ' ( d stisfies the coditios of beig itermedite fuctio I this cse s = σ(t represets the legth of the sub-rc correspodig to [, t], d L is the legth of the whole rc γ If s is the prmeter o curve, we s tht the curve is give i the coicl form (iii From pure mthemticl poit of view curve is clss of equivlet fuctios I other words we must fid those properties of curve, which re ivrit uder the chge of prmeters More ectl, propert of curve is itrisic propert iff it does ot deped o prmeteritio i the clss of equivlet fuctios (the sese of the cosidered equivlece defies the tpe of propert: cotiuous, smooth, etc For emple, the properties of curve of beig closed, simple, cotiuous, Lipschite, d smooth re itrisic Similrl, the legth of curve should be itrisic propert, so tht the followig result is ver useful: 8 Propositio The propert of curve of beig rectifible d its legth do ot deped o prmeteritio Proof Beig mootoic, σ relies : correspodece betwee the divisios of [, b] d [c, d], such tht the vritio of the equivlet fuctios o correspodig divisios re equl It remis to recll tht the legth is obtied s supremum } The fct tht either σ' > or σ' < i defiitio llows us to distiguish two subclsses of prmeteritios which defie the oriettio of curve 9 Orietted curves To oriette curve mes to split the clss of equivlet prmeteritios ito two subclsses, which cosist of prmeteritios relted b icresig itermedite fuctios, d to choose which of these two clsses represet the direct oriettio (sese, d which is the coverse oe B covetio, the direct (positive sese o closed, simple d smooth curve i the Euclide ple is the ti-clockwise oe More geerll, the closed curves o orietted surfces i R re directl orietted if the positive orml vector leves the iterior o its left side whe ruig i the sese of the curve Altertivel, isted of cosiderig two seses o curve, we c cosider two orietted curves More ectl, if γ is orietted curve (ie the itermedite diffeomorphism i defiitio is lso 5

Chpter VI Lie itegrl icresig of prmeteritio φ : [, b] R, the the curve deoted γ of prmeteritio ψ : [, b] R defied b ψ(t = φ ( + b t is clled the opposite of γ Aother w of epressig the oriettio o curve is tht of defiig order o it More ectl, we s tht X = φ (t is "before" X = φ (t o γ iff t t o [, b] Usig this termiolog, we s tht A = φ ( is the first d B = φ (b is the lst poit of the curve If o cofusio is possible, we c ote γ = AB d γ = BA Cotrril to the divisio of curve ito sub-rcs, we c costruct curve b likig together two (or more curves with commo ed-poits efiitio Let γ i, i =, be two curves of prmeteritio φ i : [ i, b i ] R such tht φ (b = φ ( The curve γ, of prmeteritio φ : [, b + (b ] R, where ( t if t, b ( t is clled coctetio ( t b if t b, b ( b (uio of γ d γ, d it is oted b γ = γ γ Propositio The coctetio is ssocitive opertio with curves hvig commo ed-poits, but it is ot commuttive The proof is routie, d will be omitted If γ γ mkes sese, the the coctetio γ γ is possible, but geerll γ γ is ot Propositio The smooth curves hve tget vectors t ech M γ, cotiuousl depedig o M The directios of tget vectors do ot deped o prmeteritios I coicl prmeteritio, ech tget t = ('(s, '(s, '(s is uit vector Proof If fuctio φ : [, b] R, of vlues φ(t = ((t, (t, (t is prmeteritio of γ, the M M = ((t (t, (t (t, (t (t Sice φ is differetible, M M ('(t (t t, '(t (t t, '(t (t t, with equlit whe t t Cosequetl the directio of t is give b ('(t, '(t, '(t B chgig the prmeter, t = σ(θ, this vector multiplies b σ'(θ, hece it will keep up the directio For the coicl prmeteritio we hve Δ s = Δ + Δ + Δ, hece the legth of the tget vector is ' (s + ' (s + ' (s = } 6

VI Curves PROBLEMS VI Is the grph of fuctio f : [, b] R curve i R? Coversel, is curve i R grph of such fuctio? Hit Ech fuctio f geertes prmeteritio φ : [, b] R of the form φ(t = (t, f(t The circle is curve, but ot grph Show tht the coctetio of two smooth curves is cotiuous piecewise smooth curve, but ot ecessril smooth Hit Use defiitio 7 of coctetio Iterpret the grph of, where [, +], s coctetio of two smooth curves Let γ i, i =, be two curves of prmeteritio φ i : [ i, b i ] R with commo ed-poits, ie φ ( = φ ( d φ (b = φ (b Show tht both γ γ d γ γ mke sese d the re cotrril orieted closed curves 4 Fid the tget of ple curve implicitl give b F(, = I prticulr, tke the cse of the circle Hit If = (t, = (t is prmeteritio of the curve, from F((t, (t o [, b], we deduce df =, hece F' ' + F' ' = Cosequetl, we c tke t = ('(t, '(t = λ(f', F' 5 If the ple curve γ is implicitl defied b F(, =, we s tht M γ is criticl poit iff F' (M = F' (M = Stud the form of γ i the eighborhood of criticl poit ccordig to the sig of Δ = " " " F F F Emple = +, d M = (, Hit M is sttior poit of the fuctio = F(,, d γ is the itersectio of the ple O with the surfce of equtio = F(, I this istce F( + h, + k F" (, h + F" (, hk + F" (, k, hece Δ < leds to isolted poit of γ, Δ > correspods to ode (double poit, d Δ = is udecided (isolted poit I the emple, M is isolted for <, it is ode for > ; it is cusp for = 6 Fid the legth of the logrithmic spirl φ(t = (e t cos t, e t si t, e t, where t Solutio L = ' ' ' dt 7

Chpter VI Lie itegrl 7 Estblish the formul of the legth of ple curve which is implicitl defied i polr coordites, r = r(θ Use this formul i order to fid the legth of the crdioid r = ( + cos θ Hit Followig Fig VI, we hve Δs = (rδ θ + (Δr dr r d r r r s b Fig VI The legth of the crdioid (sketched i Fig VIb is L = r r' d cos d 4 cos d 8 8 Fid the legth of the curves defied b the followig equtios: si r, [, ]; b r si, [, ] Aswer (8 ; b 8 9 Fid the legth of the curve of equtio r, r [, ] r Hit Estblish formul similr to tht i the bove Problem 7 The legth of the curve is l 8

VI LINE INTEGRALS OF THE FIRST TYPE I this prgrph we cosider the lie itegrl of sclr fuctio Such itegrls occur i the evlutio of the mss, ceter of grvit, momet of ierti bout is, etc, of mteril curve with specified desit The costructio of the itegrl sums Let γ be smooth d orietted curve i R, of ed-poits A d B B divisio of γ we uderstd set δ ={M k γ : k =,,, } such tht M = A, M = B, d M k < M k+ i the order of γ, for ll k =,,, The orm of δ is δ = m M k M k k If γ k = M km k deotes the sub-rc of the ed-poits M k d M k+ o γ, we write Δs k for the legth of γ k, k =,,, O ech sub-rc γ k we choose poit P k betwee M k d M k+ i the order of γ The set S = {P k k : k =,,, } represets the so clled sstem of itermedite poits B P k M k+ B = M R M k s k M k+ Pk f A M k A = M Fig VI Now we cosider tht γ is etirel cotied i the domi o which the sclr fuctio f is defied (see Fig VI Uder these coditios, we c clculte S γ, f (δ, S = k f ( P k s k, 9

Chpter VI Lie itegrl which is clled itegrl sum of the first tpe of f o the curve γ, correspodig to the divisio δ, d to the sstem S of itermedite poits efiitio We s tht f is itegrble o the curve γ iff the bove itegrl sums hve (fiite limit whe the orm δ, d this limit is ot depedig o the sequece of divisios with this propert, d o the sstems of itermedite poits If this limit eists, we ote lim S γ, f (δ, S = 4 fds, d we cll it lie itegrl of the first tpe of f o the curve γ Remrk The bove defiitio of the lie itegrl mkes o use of prmeteritios, but cocrete computtio eeds prmeteritio i order to reduce the lie itegrl to usul Riem itegrl o R I fct, if φ : [, b] R is prmeteritio of γ, the to ech divisio δ of γ there correspods divisio d of [, b], defied b M k = φ(t k for ll k =,, Of course, d iff δ Similrl, to ech sstem S = {M k γ k : k =,,, } of itermedite poits of γ, there correspods sstem T = {θ k [t k, t k+ ] : k =,,, } of itermedite poits of [, b] The vlues f(p k m be epressed b (f φ(θ k, such tht k f S γ, f (δ, S = ( f ( k sk k tk ( ( k, ( k, ( k ' ( t ' ( t ' t k ( t dt Fill, usig the me theorem for the bove itegrls, we obti S γ, f (δ, S = f ( ' ( ' ( ' ( ( t t, ( k k k k k k which looks like itegrl sum of simple Riem itegrl Thus we re led to the followig ssertio: 4 Theorem Let γ be (simple smooth curve i R, d let f : R be cotiuous sclr fuctio The there eists the lie itegrl of f o γ, d for prmeteritio φ : [, b] R of γ we hve fds b = ( f ( t ' ( t dt I prticulr, the lie itegrl does ot deped o prmeteritio Proof Let us ote F(t = (f φ(t φ'(t, d let ( k k k k σ F (d, T = f ( ' ( ( t t k k

VI Lie itegrls of the first tpe be the Riem itegrl sum of F o [, b] Becuse γ is smooth, it follows tht F is cotiuous, hece there eists F( t dt lim σ F (d, T More b d ectl, for ever ε > there eists η > such tht for ever divisio d of [, b], for which d < η, we hve σ F (d, T b F ( t dt < (* O the other hd, f φ is uiforml cotiuous o the compct [, b], hece for ε > there eists η > such tht for ll t', t" [, b] for which t' t" < η, we hve (f φ(t' (f φ(t" <, where L is the L legth of γ If d is divisio of [, b] such tht d < η, the S γ, f (δ, S σ F (d, T = = f ( k ( f ( k '( k ( tk tk sk (** L k k Cosequetl, if d is divisio of [, b] for which d < η = mi {η, η }, the usig (* d (** we obti S γ, f (δ, S b F ( t dt S γ, f (δ, S σ F (d, T + σ F (d, T b F ( t dt < ε, ie b F ( t dt is the limit of the itegrl sum of f o γ The lst sttemet of the theorem follows from the fct tht the itegrl sums S γ, f (δ, S do ot deped o the prmeteritio, d the prmeteritio used i the costructio of F is rbitrr } The geerl properties of the lie itegrl of the first tpe re summried i the followig : 5 Theorem (i The lie itegrl of the first tpe is lier fuctiol, ie for smooth curve γ, cotiuous f, g, d λ, μ R, we hve ( f g ds fds gds (ii The lie itegrl is dditive reltive to the rc, ie fds = fds + fds, wheever γ = γ γ (iii The lie itegrl of the first order does ot deped o the oriettio o the curve, ie fds = fds The proof is directl bsed o defiitio, d will be omitted 4

Chpter VI Lie itegrl PROBLEMS VI Clculte ( + + ds, where γ (spirl hs the prmeteritio φ : [, π] R, φ(t = (cos t, si t, t Aswer π Evlute the itegrl ( ds, where is the curve of equtio ( (, Hit Recogie the lemiscte i polr coordites r the prmeteritio cos cos, cos si, 4 4 The swer is cos, d use Clculte the mss of the ellipse of semi-es d b, which hs the lier desit equl to the distce of the curret poit up to the is Hit The recommeded prmeteritio is give b φ : [, π] R, where φ(t = (cos t, bsi t We must clculte where e = b ds = b b + rcsi e, e is the e-cetricit of the ellipse 4 etermie the ceter of grvit of hlf-rc of the homogeeous ccloid = (t si t, = ( cos t, where t [, π] Hit G = M ρ(, sds, G = M ρ(, sds, where M is the mss of the wire I this cse G = G = 4 5 Fid the momet of ierti, bout the is of the first loop of the homogeeous spirl = cos t, = si t, = bt Hit I = ( + ρ(,, ds = π b 4

VI Lie itegrls of the first tpe 6 A mss M is uiforml distributed log the circle + = i the ple = Fid the force with which this mss cts o mss m, locted t the poit V(,, b Mm Hit Geerll spekig, F k r I the prticulr cse F = (,, F, r where F = km ( (,, t kmmb ds r ( b 7 Let be rc of the stroid i the first qudrt, whose locl desit equls the cube of the distce to the origi Fid the force of ttrctio eerted b o the uit mss plced t the origi Hit A prmeteritio of the stroid is cos t, si t Up to costt k, which depeds o the chose sstem of uits, the compoets of the force hve the epressios: F k ds 4 k = k sit cos t dt = ; 5 F k ds = si 4 k k t cost dt = 5 8 Show tht if f is cotiuous o the smooth curve γ, of legth L, the there eists M * γ such tht the me vlue formul holds fds = Lf(M * Hit Usig prmeteritio of γ, we reduce the problem to the me vlue formul for Riem itegrl / 9 Show tht if f is cotiuous o the smooth curve γ, the fds f ds Hit Use theorem 4 4

VI LINE INTEGRALS OF THE SECON TYPE The mi object of this prgrph will be the lie itegrl of vector fuctio log curve i R The most sigifict phsicl qutit of this tpe is the work of force The costructio of the itegrl sums Let γ R be smooth orietted curve, d let F : R be vector fuctio We suppose tht γ, d tht F hs the compoets P,Q, R : R, ie for ever (,,, we hve F (,, = (P(,,, Q(,,, R(,, Altertivel, usig the coicl bse { i, j, k } of R (see Fig VI, we obti r = i + j + k d F = Pi + Q j + R k B T k M k+ F i k j M k A Fig VI If δ = {M k γ : k =,, } is divisio of γ, we ote r k for the positio vector of M k For ech sstem of itermedite poits S = {T k = (ξ k, η k, ζ k M km k : k =,, } we costruct the itegrl sum S, F ( δ, S = F( Tk, r k 44 k = [P(ξ k, η k, ζ k ( k+ k + Q(ξ k, η k, ζ k ( k+ k + R(ξ k, η k, ζ k ( k+ k ] k where <, > is the Euclide sclr product o R These sums re clled r itegrl sums of the secod tpe of F log the curve γ k

VI Lie itegrls of the secod tpe efiitio We s tht F is itegrble o γ iff the itegrl sums of the secod tpe hve (fiite limit whe the orm of δ teds to ero, d this limit is idepedet of the sequece of divisio which hve δ, d of the sstems of itermedite poits I this cse we ote the limit b lim S, F ( δ, S = < F, d r > = F d r = Pd + Qd + Rd d we cll it lie itegrl of the secod tpe of F o γ Remrk The mi problem is to show tht such itegrls re lso idepedet of the prmeteritio of γ, d to clculte them usig prmeteritios We will solve this problem b reducig the itegrl of the secod tpe to itegrl of the first tpe, which is kow how to be hdled I order to fid the correspodig sclr fuctio, we modif the form of the itegrl sums b usig prmeteritio φ : [, b] R of γ I fct, if φ (t = ((t, (t, (t, the ccordig to Lgrge's theorem, o ech [t k, t k+ ] we hve (t k+ (t k = '(θ k (t k+ t k (t k+ (t k = '(θ k (t k+ t k (t k+ (t k = '(θ k (t k+ t k, where θ k, θ k, θ k (t k, t k+ Cosequetl, S (δ, S becomes k, F [P(φ(θ k '(θ k + Q(φ(θ k '(θ k + R(φ(θ k '(θ k ](t k+ t k, (* where φ(θ k = P k, k =,,, re the itermedite poits of δ r Let us ote the uit tget vector t curret poit of γ b C r More ectl, if M = φ(θ, θ [, b], the '( i '( j '( k C( M ' ( ' ( ' ( Let us cosider the sclr fuctio f = < F, >, which hs the itegrl sums of the first tpe (see remrk i, F (f φ( θ k r '( k (t k+ t k (** S (δ, S = k B comprig the itegrl sums of F d f, we turll clim tht the lie itegrl of the secod order of F reduces to the lie itegrl of the first order of f I fct, this reltio is estblished b the followig 4 Theorem Uder the bove ottios, if F is cotiuous o γ, the F is itegrble o γ, d we hve F d r = f ds 45

Chpter VI Lie itegrl Proof If F is cotiuous, the f is cotiuous too, sice C is cotiuous for smooth curves Cosequetl, ccordig to theorem 4 i, f is itegrble o γ It remis to evlute S f ds S, F, F (δ, S (δ, S S γ, f (δ, S + S γ, f (δ, S f ds The lst modulus is less th for δ < η, hece it remis to fid upper boud of the other modulus I fct, usig (* d (** we obti : S (δ, S S γ, f (δ, S k + k P ( k r' Q ( k r', F ['(θ k r '(θ k '(θ k r '( k ](t k+ t k + ['(θ k r '(θ k '(θ k r '( k ](t k+ t k + R ( k ['(θ k r '(θ k '(θ k r '( k ](t k+ t k r' + k Usig the uiform cotiuit of the fuctios P φ, Q φ, R φ, r ' (which lso is differet from ero!, d ', ', ' o [, b], this epressio is lso less th for δ < η } 5 Corollr The lie itegrl of the secod order of cotiuous fuctio o "smooth curve " does ot deped o the prmeteritio (up to sig, which is determied b the oriettio! Proof Becuse C =, f does ot deped o prmeteritio, hece it remis to ppl theorem 4 i, which epresses similr propert of the lie itegrls of the first tpe } 6 Corollr For prmeteritio φ : [, b] R of γ, we hve: b = F d r = [P((t, (t, (t'(t + Q((t, (t, (t'(t + R((t, (t, (t'(t]dt Proof Usig theorem 4 i, for f = F C, we obti F d r = b f ds = (f φ(t φ'(t dt = 46

b = b = (( F r ' φ(t VI Lie itegrls of the secod tpe r φ'(t dt = '( t [( P φ(t'(t + (Q φ(t'(t + (R φ(t'(t] dt, where we remrked tht r '(t = φ'(t } The geerl properties of the lie itegrl of the secod tpe c be obtied from the similr properties of the lie itegrl of the first tpe (formulted i theorem 5, 7 Theorem The lie itegrl of the secod order hs the properties: (i Lierit reltive to the fuctios: (λ F + μg d r = λ F d r + μ G d r (ii Additivit reltive to the uio of curves F d r = F d r + F d r (iii Oriettio reltive to the sese o the curve F d r = F d r Proof Properties (i, (ii re direct cosequeces of (i, (ii of theorem 5, Reltive to (iii, it is ecessr to remrk tht eve if the lie itegrl of the first tpe is the sme o γ d γ, fuctio f i the formul estblished i the bove theorem 4 depeds o the sese chose o γ I fct, if φ : [, b] R is prmeteritio of γ, the C (φ(θ = C (ψ(t t ech φ(θ = ψ(t γ } 8 Remrk B clcultig lie itegrls of the secod tpe, we c see tht sometimes the result does ot deped o the curve but ol o the edpoits (see problem I prctice this is importt cse, for emple, whe the itegrl represets the work of force, so it must be crefull led This propert of the lie itegrl will be studied i terms of "totl differetils" More ectl, F d r is cosidered to be totl differetil iff there eists differetible fuctio U : R such tht du = F d r = Pd + Qd + Rd Altertivel, F d r is totl differetil iff F = grd U, ie F derives from potetil 9 Theorem (i If R is ope set, d U : R is differetible fuctio such tht F = grd U, the for smooth curve γ, of ed-poits A d B, we hve 47

Chpter VI Lie itegrl F d r = U(B U(A, ie the lie itegrl of F is ot depedig o γ (ii Coversel, if R is ope d coected set, d F : R is cotiuous vector fuctio for which the lie itegrl depeds ol o the ed-poits of the curves, the F d r is totl differetil Proof (i If φ : [, b] R is prmeteritio of γ, d ccordig to the hpothesis P = b = F d r = = U [ b U U, Q = Pd + Qd + Rd = U ( φ(t '(t + U, R =, the U U d+ U d+ d = U (φ(t '(t + ( φ(t '(t] dt = (U φ'(t dt = U(φ(b U(φ( = U(B U(A (ii We hve to costruct U, for which F = grd U With this im we fi A = (,,, d we let B = (,, free i Becuse is ope d coected, it will lso be coected b rcs, hece there eists smooth curve γ of ed-poits A d B Cosequetl, we m defie fuctio U : R b formul B U(,, = F d r, A where we metio ol the poits A d B becuse, b hpothesis, the cosidered lie itegrl does ot deped o the curve, which hs these edpoits It remis to show tht = P, = Q, = R, t U U U poit B = (,, I fct, U( + h,, U(,, = h F d r, where γ h is curve (i prticulr stright segmet betwee (,, d ( + h,, Usig the prmeteritio φ h (t = ( + th,, of γ h, we obti U( + h,, U(,, =h P( + th,, dt 48

49 VI Lie itegrls of the secod tpe Applig the me-vlue theorem to the lst itegrl, it follows tht there eists θ (, such tht P( + th,, dt = P( + θh,,, hece U ( h,, U (,, = P( + θh,, h Sice P is cotiuous (s compoet of F, it follows tht there eists U U ( h,, U (,, (,, = lim = P(,, h h Similrl we evlute the other prtil derivtive of U } Remrk (i Beod the eistece of the potetil U, the bove theorem cotis formul, which gives U cocretel, mel U(,, = (,, (,, Pd + Qd + Rd More th this, becuse this itegrl is idepedet of the curve, we c chose it such tht to obti the most coveiet clcultio I prctice, it is frequetl prefered broke lie γ = [(,,, (,, ] [(,,, (,, ] [(,,, (,, ], whe the lie itegrl reduces to three simple (Riem itegrls, ie U(,, = P(t,, dt + Q(, t, dt + R(,, tdt This formul provides U up to costt which correspods to the choice of (,,, d equls U(,, A prcticl ke of correct clcultio is the reductio of the "mied" terms, which re evluted t (,,, (,,, etc (ii The bove formuls for clcultig U c be cosidered s rules of determiig fuctio whe its differetil is kow; i other words this mes fidig ti-derivtives (or primitives of give fuctio Simple emples show tht ol prticulr triplets of fuctios (P, Q, R represet prtil derivtives of fuctio U, so it is ver importt for prcticl purposes to kow how to idetif these cses efiitio We s tht the field F C R ( is coservtive iff its compoets P, Q, R stisf the coditios P Q Q R R P,, t ech poit of Isted of "coservtive" m uthors use the term "irottiol" which derives from the otio of "rottio" More ectl, the rottio of F = (P, Q, R, oted rot F, is defied s vector formll epressed b the determit

Chpter VI Lie itegrl i rot F =, The ssertio " F is coservtive" rell reduces to "rot F = " Theorem Let R be ope d str-like set, d let P F C R ( be vector field A ecessr d sufficiet coditio for F to derive from potetil is to be coservtive We remid tht domi R is sid to be str-like if there eists M such tht MM holds for ll M Proof If F = grd U for some U : R, the F hs the compoets U U U P =, Q =, R = Becuse P, Q, R C R (, we c ppl Schwrt' theorem (o mied secod order prtil derivtives to U, d so we esil see tht F is coservtive Coversel, let F be coservtive o Sice is str-like there eists M such tht for other M we hve MM A prmeteritio of this segmet is φ(t = ( + t(, + t(, + t(, t [,] Let us defie U : R, b U(,, = F d r 5 j Q M M Usig the prmeteritio φ i the formul estblished i corollr 6, we obti U(,, = k R [( P φ(t( + (Q φ(t( + (R φ(t( ]dt Accordig to theorem V5, cocerig the derivtio reltive to prmeter i defiite itegrl, we hve U (,, = P Q R = [ (φ(tt( + (P φ(t + (φ(tt( + (φ(tt( ]dt The hpothesis of beig coservtive llows us to epress this itegrl ol b the prtil derivtives of P, ie U (,, = [t(p φ'(t + (P φ(t]dt =

VI Lie itegrls of the secod tpe = [t(p φ]'(tdt = (P φ( = P(,, U Cosequetl, for (,, we hve (,, = P(,, U U Similrl, we prove tht = Q, d = R } Simple emples show tht the bove coditio o to be str-like is essetil for coservtive field to derive from potetil Emple O the ope (but ot str-like set = R \ {(,} we cosider the field F (, =, Obviousl, we hve F C P R R (, d, hece F is coservtive o Now, let γ be the uit circle i the ple trced couter clockwise Sice F d r = π it is cler tht F c ot derive from potetil 4 Coclusio I prctice, whe we hve to clculte lie itegrl of the secod tpe, it is useful primril to check whether the correspodig vector field is coservtive or ot If it is't coservtive we must fid prmeteritio of the curve d ppl the most geerl formul (s i corollr 6 If the field is coservtive (d the domi is str-like!, we ppl the formul i theorem 9 (i, whe U m be obtied s i remrk, (i Fill, we metio other pplictio of the lie itegrl of the secod tpe (i dditio to the work of force 5 Propositio Let γ be simple, smooth d closed cotour, trced oe time couter-clockwise, d hvig the propert tht prllel to the o d to o is meets the curve t most twice The the re bouded b γ is epressed b A = d d Proof We c cosider γ = γ γ s i Fig VI (, d ltertivel γ = γ γ 4 s i Fig VI (b B iterpretig d d d like res of sub-grphs, we obti A = d 5

Chpter VI Lie itegrl 4 b Fig VI Similrl, A = d It remis to dd the two epressios of A } This formul of A is be prticulr cse of the Gree-Riem formul (see lter VII d There eist m similr formuls of the re, which ivolve o-euclide coordites I prticulr: 6 Emple Let us s we eed the formul of the re of ple domi, which is bouded b closed curve, eplicitl epressed i polr coordites b the equtio r = φ(θ, where φ : [θ, θ ] R + I this cse we hve to evlute A = ( d I prticulr, the domi cotied iside Beroulli's lemiscte (of equtio r = cos θ hs the re A = 5

VI Lie itegrls of the secod tpe PROBLEMS VI Evlute the work of the forces F = i + j d G = i j i the process of movig mteril poit log ellipse of hlf-es d b i the o ple Hit We clculte F d r d G d r, where γ hs the prmetric equtios = cos t, = b sit, t [, π Clculte d + d, where O = (,, A = (,, for differet OA rcs OA i the ple (stright lie, prbols, broke lies (of edpoits O d A Fid the work of the force F (,, = (,, b movig poit log the screw lie γ of prmeteritio = cos t, = b si t, = bt, t [, π Solutio w Fdr = π( + b ( d d 4 Clculte the itegrl, where γ is the right-hd loop of the lemiscte r = cos α, trced couter-clockwise Hit A prmeteritio of γ, i polr coordites, is: r cos cos cos,, r si si cos 4 4 The itegrl is ull 5 Fid the ti-derivtive U if the differetil is: (i du = ( + d + ( 4d (ii du = e [( + + d + ( d] (iii du = d + d (iv du = d + d (v du = d d (vi du = d + ( + d + d (vii du = d d d 5

Chpter VI Lie itegrl Hit Verif tht the correspodig field is coservtive (ie the problem is correctl formulted, idetif the domi d clculte U usig the formul i remrk, (i 6 Fid the ti-derivtives of the itegrds d clculte: (i (ii (iii (, (, (, (, ( 4 + 4 d + (6 5 4 d ( d d, where does ot itersect the stright lie of ( equtio = d (,, (,, d 7 Evlute the lie itegrls of the totl differetils: (i (ii (iii (,, (,, (,, (,, (,, (,, d + d + d d d d d d d, where the itegrtio curve is situted i the first octt 8 Fid the work of the Newtoi force F r, which is ecessr to r move mteril poit from A(,, to B(,, log rbitrr curve of these edpoits, such tht (,, Hit F derives from the sclr potetil t dt t dt U (,, = + ( t ( t (, Cosequetl, W F dr = U (,, B A, (,, + ( t dt t, where U r 54

VI Lie itegrls of the secod tpe 9 Show tht if f : R + R is cotiuous, d γ is closed piecewise smooth cotour, the f ( + ( d + d = Hit Cosider Φ(r = itegrl becomes dv r f(tdt d V(, = Φ( +, such tht the A circle of rdius r is rollig without slidig log fied circle of R rdius R d outside it Assumig tht is iteger, fid the re r bouded b the epiccloid (hpoccloid determied b some poit of the movig circle Ale the prticulr cse of the crdioid, (where R = r, d steroid, (whe R = 4r Hit A prmeteritio of the epiccloid is R r R r = (R + r cos t r cos t, = (R + r sit r si t, where t[, π is the gle betwee two rdiuses of the fied circle, oe correspodig to the strtig commo poit, d the other to rbitrr curret poit The prmeteritio of the hpoccloid is obtied b replcig r b r Aswer: π(r + r(r + r d d d Evlute I = ( / where Γ is smooth curve of edpoits (,, d (,, Hit V = ( + + / ( i j k derives from potetil U, hece I = U(B U(A 55

CHAPTER VII MULTIPLE INTEGRALS I this chpter we'll eted the otio of itegrl defied o itervl I R, d tht of lie itegrl log curve, b cosiderig itegrls o domis i R, R d geerll i R p These re clled "multiple itegrls" becuse of the higher dimesio of the cosidered domis The whole theor is bsed o the otios of "re" d "volume" which eted the otio of "legth" Becuse ll these otios re prticulr cses of "mesures", for the begiig we hve to clrif some topics cocerig the Jord's mesure i R p, p N * VII JORAN'S MEASURE It is well kow tht i the process of clcultig res d volumes, we strt out with simple figures like rectgles d rectgulr prllelepipeds, which re lter used for pproimtig other figures ( sigifict emple is the re of sub-grph This method c be uitril pplied i order to mesure bodies i R p, for rbitrr p N * efiitio If P = [, b ] [,b ] [ p, b p ] is closed rectgulr prllelepiped (lso clled pdimesiol itervl, or "prlleloid", the the umber p v(p = b k k k is clled pvolume of P, or mesure of P A fiite uio of such closed rectgulr prllelepipeds, ech pir of them hvig o commo iterior poit, is med elemetr bod The pvolume (or mesure of elemetr bod is the sum of the pvolumes of ll prllelepipeds which form the bod Remrk (i The ide of cosiderig fiite fmilies of prllelepipeds i the so clled elemetr bod is specific to the mesure theor i Jord's sese The ltertive is the Lebesgue's poit of view of tkig coutble fmilies of prllelepipeds i the elemetr bodies We will develop here the Jord's mesure theor becuse it is simpler, d it is sufficiet for studig the Riem multiple itegrls However, we metio tht the simplicit of Jord's mesure is couter-blced b some disdvtges (for emple, see lter the otio of mesurble set (ii The uio of two elemetr bodies is elemetr bod, d the sme for the dherece of the differece (ot for the differece itself 56

VII Jord s mesure (iii The mesure of elemetr bod does ot deped o its decompositio ito rectgulr prllelepipeds; there re still problems whe the sides re o loger prllel to the is efiitio Let A R p be rbitrr bouded set, d let P', P" deote elemetr bodies i R p The the umber μ i (A = sup {v(p' : P' A} is clled Jord's iterior (or "iterl" mesure of A, d μ e (A = if {v(p" : P" A} is clled Jord's eterior (or "eterl" mesure of A These otios mkig sese sice A is bouded If μ i (A = μ e (A we s tht A is mesurble i Jord's sese, d the commo vlue, deoted μ(a = μ i (A = μ e (A is clled the Jord's mesure of A 4 Emples (i The set A = { : N * } is mesurble i R, d μ(a = There re still coutble sets (eg N, Q [,], etc, which re ot mesurble, hece this propert depeds o the positio of the terms (ulike the Lebesgue's mesure, which is ull for coutble set (ii The elemetr bodies d their iteriors re mesurble sets, d we lws hve μ(b = v(b = μ( B (iii If A d B re mesurble sets i R p, the A B, A B, A \ B re lso mesurble (for more detils see Theorem 6 below I order to evlute mesures, the followig lemm is helpful: 5 Lemm Let A be bouded set i R p For A to be mesurble is ecessr d sufficiet tht for ech ε > there eist some elemetr bodies P' A d P" A such tht μ(p" μ(p' < ε Proof Becuse lws μ(p" μ(p', it eough to epress the coditio sup {μ (P' : P' A} = if { μ (P" : P" A} i terms of ε > } Now we c estblish some properties of the Jord's mesure: 6 Theorem If A, B R p re mesurble sets, the: (i A B implies μ (A μ (B (ii If A B cotis o prlleloid of o-ull pvolume, the μ (AB = μ (A + μ (B (iii If A B the μ (B \ A = μ (B μ (A Proof (i Is obvious (ii is bsed o the fct tht for two prlleloids P, Q with o commo iterior poits we hve μ (P Q = μ(p + μ(q (iii We m ppl (ii to B = A(B \ A } 57

Chpter VII Multiple itegrls Becuse the sets with ull mesure pl importt role i mesure theor, we distiguish them b specil term: 7 efiitio We s tht set A R p is egligible iff it is mesurble d μ(a = 8 Remrks (i The fiite sets re egligible The fiite uios of egligible sets re lso egligible A subset of egligible set is egligible (ii Sice we lws hve μ i (A μ e (A, it follows tht A is egligible iff μ e (A = I other terms, A is egligible iff for ever ε > there eists elemetr bod B, such tht A B d v(b < ε (iii If A is bouded i R p, d p < q, the A is egligible sets i R q I prticulr, the segmet [, b] R is egligible i R More geerll, the fct tht the smooth curves d surfces re egligible i R is cosequece of the followig theorem: 9 Theorem Let A R p be bouded If f : A R q, where p < q, is Lipschite (ie there eists c > such tht f( f( < c for ll, A, the f(a is egligible i R q Proof Let K R p be pcube of side h such tht A K B dividig ech side ito equl prts, the cube breks up ito p cubes of side h We clim tht if ω is such smll cube, the f(a ω is cotied i cube of side c p h i R q I fct, if A ω = Ø or cosists of sigle poit, the ssertio is obvious If A ω cosists of more th two poits, the we fi A ω, d for other A ω, we obti f( f( c cp h, ie f(a ω S ( f(, cp h It is sufficiet to remrk tht this sphere is icluded i cube of side cp h If P is the uio of ll the cubes which coti sets of the form f(a ω, the f(a P, d μ(f(a μ(p p cph = (cph q q p From q > p it follows tht lim q =, hece f(a is egligible } p The followig theorem shows tht the egligible sets re ver useful i estblishig the mesurbilit of other sets q 58

VII Jord s mesure Theorem Let A R p be bouded set A ecessr d sufficiet coditio for A to be mesurble is tht FrA be egligible (i the sese of the Jord's mesure Proof Let A be mesurble The for ε > there eists the elemetr bodies P, Q such tht P A Q d μ(q μ(p < ε Becuse FrA Q \ P d μ(p = μ( P, we obti μ e (Fr (A μ(q \ P = μ(q μ(p < ε, which shows tht FrA is egligible Coversel, let us suppose tht FrA is egligible, ie for ε > there eists elemetr bod B such tht FrA B, d μ(b < ε It is es to see tht A \ B is ope d Fr (A \ B Fr B Let us ote P = A \ B, Q = A B We clim tht: (i P is elemetr bod, (ii Q is elemetr bod too, (iii P A Q, d (iv μ(q μ(p < ε These properties re sufficiet to coclude tht A is mesurble I fct, to prove (i we remrk tht sice A \ B is ope, for ech A \ B there eists prlleloid P such tht P A \ B Let us remrk tht Fr (A \ B A I fct, o the cotrr cse, if Fr (A \ B d Fr A, the we deduce tht for eighborhood V of we hve V (A {B I dditio, V B holds for some of these eighborhoods (sice Fr A B, which is impossible Cosequetl, for Fr (A\B there eists prlleloid P such tht P A I coclusio, the fmil { P : A \ B} { P : F (A \ B}{ B } is ope cover of A B hpothesis A is bouded, hece A is compct, so there eists fiite subfmil { i P : i =,, } { P : j =,, m}{ B } j which lso covers A I prticulr, removig B, this subfmil covers P= A \ B, so it remis to modif the prlleloids of this cover such tht P to pper s elemetr bod (ii is immedite if we ote tht Q = P B Similrl, i (iii, A Q is obvious, d P A is bsed o the fct tht Fr (A \ B A 59

Chpter VII Multiple itegrls Fill, for (iv we evlute μ(q μ(p = μ( ( A \ B B μ(p = =μ(p B μ(p = μ(b < ε } The lst two theorems hve useful cosequeces i the stud of the mesurbilit For emple: Corollr Let A be bouded (i prticulr compct set i R p, p N * If Fr (A cosists of fiite umber of smooth imges of t most (pdimesiol mesurble sets, the A is mesurble Proof Smooth fuctios re Lipschite, hece, ccordig to theorem 9, Fr (Ais egligible The rest is sid b theorem } Emples The pdimesiol bll S(,r is mesurble sice its boudr is smooth Similrl, bouded polhedro is mesurble becuse its boudr cosists of fiite umber of (pdimesiol flt surfces I prticulr, the prllelepipeds hvig fces o-prllel to the es re mesurble too Evlutig their mesure, s well s the preservtio of the mesure uder isometries d other trsformtios, remi more complicted problems, which will ot be studied here 6

VII Jord s mesure PROBLEMS VII Show tht the sub-grph of bouded d icresig fuctio f : [, b] R is mesurble (i Jord's sese Hit To two elemetr bodies P', P" for which P' sub-grph f P" there correspod two divisios δ' d δ" of [, b] such tht for δ = δ' δ" = { = < < < = b} we hve k k k k k k v(p' f ( ( f ( ( v(p" k For sufficietl fie divisios will be rbitrr smll k δ, whe δ < η, the differece P P = f ( k f ( k ( k k k Show tht eve though the fuctio f : [,] R, ( t,si if t, f(t= t (, if t is ot Lipschite, the imge f([,]is egligible i R Hit f(t f(t is possible for rbitrr closed t', t" [, ] For ε > there eists η > such tht f([, η] be icluded i rectgle of re less th ε The remiig f([η, ] is egligible ccordig to theorem 9 Compre the mesures of I = (, R d f (I R, where the fuctio f : I R, is defied b f(c c c c 4 = (c c, c c 4 Are these o-egligible simple curves i R p, p? Hit f(i = (, (,, d f is : However, if deotes the mesure (re i R, we hve (I = d (f(i = Tke γ = f(i 4 Stud the mesurbilit (i Jord's sese of the followig sets i the ple: A = {(, [, ] [, ] :, Q} B = {(, [, ] [, ] :, R\Q} C = A B, d C Aswer Ol C is mesurble 6

VII MULTIPLE INTEGRALS The otio of multiple itegrl will be cosidered for rbitrr dimesio of the spce p, with prticulr stress o the cses p = d p = of the "double" d "triple" itegrls Strtig out with some prcticl problems, we discuss the methods bsed o rbou d Riem sums Prcticl problems (i Let be compct domi of the ple, bouded b piecewise smooth curve γ, d let f : R be bouded fuctio If we hve to clculte the volume of the sub-grph of f i R, we turll divide the set ito mesurble sub-domis k, k =,,, of res ( k, d we pproimte the sked volume b sums of the form k k k (if [f( k ] ( k, (sup [f( k ] ( k, or [f(ξ k ] ( k, where ξ k k I prticulr, the sub-domis k c be rectgles, which costitute elemetr bodies used i the process of obtiig the iterl d eterl mesure of (ii Let R be compct set bouded b piecewise smooth surfce If represets phsicl bod of desit f, the the mss of m be pproimted b sums of the form k f(ξ k v( k, where ξ k k, d v( k is the volume of k Usull k re prllelepipeds with o commo iterior poits, i fiite umber, icluded i The bove sums suggest how to defie the itegrl sums i the cse of the multiple itegrls, but first we must specif some terms: Termiolog d ottios The closure of domi (ope d coected set is clled closed domi A bouded closed domi i R p, p N * is clled compct domi If R p is mesurble (i Jord's sese compct domi (briefl mcd, the fiite fmil of sets, δ = {,,, }, which stisfies the coditios: (i ech k, k =,, is mcd, (ii = k, k (iii = Ø wheever k l, k l 6

6 VII Multiple itegrls is clled divisio of B orm of divisio δ we uderstd the umber δ = m{d( k : k =,, }, where d( k = sup { :, k } is the dimeter of k ivisio δ" is sid to be fier th δ' iff for ' δ' there eists " δ" such tht ' " If δ' d δ" re divisios of, the δ = {' " : ' δ' d " δ"} is clled supremum of δ' d δ" d it is deoted b δ = δ' v δ" The Jord mesure o R p will be deoted b μ The costructio of the itegrl sums Let R p be mcd, let f : R be bouded, d let δ = {,,, } be divisio of For ech k =,, we ote m k = if f ( k, d M k = sup f ( k The sum s f (δ = k is clled rbou iferior sum, while S f (δ = k m k μ( k M k μ( k is clled rbou superior sum If S = {ξ k : k =,, }, where ξ k k is sstem of itermedite poits, the the sum σ f (δ, S = k f(ξ k μ( k is clled Riemi sum Usig these sums we defie the "multiple" itegrls: 4 efiitio The umber I = sup s f (δ ( I if S f (δ is clled rbou iferior (superior itegrl of f o We s tht f is rbou itegrble o iff I = I I such cse, the commo vlue is deoted b I = I = I = fd d is clled the multiple itegrl of f o (i rbou' sese We s tht f is itegrble i Riem's sese o iff there eists L = lim σ f (δ, S d this limit is idepedet of the sequece of the divisios (with δ d of the sstems of itermedite poits If so, L is med the Riem multiple itegrl of f o, d it is lso deoted L = fd This coicidece of ottios is bsed o the followig:

Chpter VII Multiple itegrls 5 Theorem The followig ssertios re equivlet: (i f is itegrble o i rbou' sese (ii for ε > there eists divisio δ such tht S f (δ s f (δ < ε (iii f is itegrble o i Riem's sese The proof is the sme s for simple itegrls o R, d it will be omitted I prticulr, ssertio (i (ii represets the well kow rbou criterio of itegrbilit 6 Prticulr cses If p =, d = [, b] R we recogie tht I the cse p = we usull ote b fd = fd = f ( d f (, dd d we cll it double itegrl of f o Similrl, whe p =, we ote f (,, ddd fd = d we me it triple itegrl of f o Eve for p greter th the multiple itegrl is sometimes writte i the form fd = fd f (,, p d dp Usig the previous theorem we obti importt clss of itegrble fuctios: 7 Theorem Let R p be mcd, d let f : R be bouded fuctio If the set Δ = { : f is discotiuous t } is egligible, the f is itegrble o Proof The cse μ( = is trivil, so tht we'll cosider μ( > For the begiig we prove the theorem uder the hpothesis Δ = Ø, whe f (cotiuous o (compct is uiforml cotiuous, ie for ε > there eists η > such tht ' " < η implies f(' f(" < ( Now, if δ = {,,, } is divisio of, for which δ < η, we obti S f (δ s f (δ = k (M k m k μ( k = < ( k k μ( k = ε, [ f(' k f(" k ] μ( k < 64

VII Multiple itegrls where the eistece of ' k, " k k such tht f(' k = M k d f(" k = m k for ever k =,,, is ssured b the cotiuit of f o the compcts k Cosequetl, ccordig to theorem 5, f is itegrble o Fill, let us cosider the geerl cse whe Δ Ø Sice μ( =, for ε >, there eists elemetr bod B, such tht Δ B d μ(b <, where M = sup { f( : } The set \ B is mcd too, 4M o which f is cotiuous, hece, s before, f is itegrble o \ B I other terms, there eists divisio ~ = {,,, } of \ B such tht S f ( ~ s f ( ~ < Of course, δ = ~ {B} is divisio of, for which we hve S f (δ s f (δ = S f ( ~ s f ( ~ + (M B m B μ( B, where M B = sup {f( : B} d m B = if {f( : B} Cosequetl S f (δ s f (δ < + M = ε } 4M Bsicl the multiple itegrls hve the sme properties s the simple oes (defied o compct sets from R oes: 8 Propositio The itegrble fuctios o mcd hve the properties: f g d = α fd + β gd (lierit (i (ii fd = fd + fd, wheever 65 to the domis (iii If f g o, the fd gd (mooto Ø (dditivit reltive (iv If f is itegrble o, the f is lso itegrble o, d: fd f d (bsolute itegrbilit (v μ( if f( fd μ( sup f( (me-vlue propert The proof is directl bsed o defiitios d it is omitted 9 Propositio If f is cotiuous o the mcd R p, p, the there eists ξ such tht fd = f(ξ μ( (me-vlue itegrl formul Proof Becuse f is cotiuous o the compct, there eists, such tht if f( = f( d sup f( = f( If we ote

Chpter VII Multiple itegrls λ = ( fd, the propert (v i propositio 8 tkes the form f( λ f( Usig the fct tht is lso ope d coected, there eists cotiuous curve γ of ed-poits d If φ : [, b] is prmeteritio of γ, the g = f φ : [, b] R is lso cotiuous, hece it hs the rbou' propert I prticulr, becuse g( = f( λ f( = =g(b, it follows tht there eists t [, b] such tht λ = g(t = f(ξ, where ξ = φ(t Cosequetl, ( fd = f(ξ, for some ξ } Remrk Mil there re two methods for clcultig the multiple itegrls: oe uses the reductio of the dimesio b itertio; the other cosists i chgig the vribles We will le the first method strtig out with the simplest cse whe R p reduces to prlleloid P More ectl, we cosider Crtesi decompositio of P of the form P = P' P", which leds to the distictio of two compoets i P, mel = (u, v, where u = (,, m P' d v = ( m+,, p P" for some m p If f : P R, the we ote f( = f(u, v Theorem Let f : P R be itegrble fuctio o the prlleloid P = P' P" If for ech fied up' there eists I(u = f ( u, v dv, the I : P' R is itegrble fuctio o P', d the followig equlit holds: f ( d I( u du P Proof B dividig ech side of P' d P" ito equl prts, we obti the divisios δ' = {P',,P' m } of P', δ" = {P",,P" p m } of P", d δ = {P ij = P' i P" j ; i =,, m ; j =,, pm } of P Let us ote m ij = if f(p ij d M ij = sup f(p ij, so tht for ech u i P' i d v j P" j we hve m ij f (u i, v j M ij Becuse f is itegrble reltive to the vrible v o P", it will be itegrble reltive to v o P" j too, hece b itegrtig the bove iequlit we obti m ij μ"(p" j '' P j P' f ( u, v dv M ij μ"(p" j, where μ" is Jord's mesure o R pm Addig these reltios for ll j =,, pm, we obti i P" 66

pm j m ij μ"(p" j P" j f pm ( ui, v dv = I(u i j VII Multiple itegrls M ij μ"(p" j (* If μ' is Jord's mesure o R m, it is es to see tht μ'(p' i μ"(p" j = μ(p ij for ll i =,, m d j =,, pm, where μ is the mesure o R p Let us ote b S ' = {u,,u m } the sstem of itermedite poits, d let m σ I (δ', S ' = i I(u i μ'(p' i be the Riemi sums of I o P' B multiplig the reltios (* b μ'(p' i, d ddig ll the forthcomig reltios, we obti m pm m ij μ(p ij σ I (δ', S ' M ij μ(p ij (** i j i j Now we metio tht implies δ' d δ", s well s δ sice δ δ' + δ" Cosequetl, becuse f is itegrble o P, the first d the lst sums i the iequlit (** hve commo limit, mel * f ( u, v dv, so it follows tht the limit P" lim ' σ I (δ', S ' = m pm Id ', lso eists, d fd = Id ' } P P' Corollries (i I the coditios of the bove theorem we hve: f ( u, v dudv f ( u, v dv du P' P" P' P" I prticulr, whe f(u, v = g(uh(v, we c reduce the itegrl of f to product of itegrls of g d h, ie fd = g ( u du h( v dv P P' P" (ii Iterchgig u d v, if J(v = f ( u, v du is itegrble o P', the P P' P' fd = Jd ", or equivletl, f ( u, v dudv f ( u, v dv du P' P" P' ' P' (iii If m = p, ie P" = [ p, b p ], we hve P" 67

Chpter VII Multiple itegrls bp fd = f u p dp du (,, P P' p d b repetig the itertio, we obti b b bp fd = f (,,, d d p p d P p All these formuls re direct cosequeces of the bove theorem, so the eed o proof I prticulr, for double d triple itegrls we hve: b b f (, dd = f d d (,, P where P = [, b ] [, b ], d respectivel b b b f (,, ddd = f d d d (,,, P where P = [, b ] [, b ] [, b ] Remrk The bove formuls re rrel useful i prctice becuse the refer to ver prticulr form of the domi, mel tht of prlleloid I order to eted these formuls up to rbitrr mcd R p, p >, we itroduce the otio of "sectio" s geerlitio of the Crtesi decompositio of prlleloid More ectl, if u = (,, m for some m =,, p is fied, the the sectio of is defied b [u] = {v=( m+,, p : = (u, v } The set Pr m ( = {u=(,, m : [u] Ø} represets the mprojectio of Further we'll cosider tht the sectios d the mprojectios of re lso mc domis I prticulr, whe m = p we suppose tht [u] reduces to closed itervl; more ectl, we s tht is simple iff there eist two fuctios φ, ψ C R (Pr p ( such tht [u] = [φ(u, ψ(u] for ll u Pr p ( 4 Theorem Let R p, p >, be mcd d let f : R be itegrble o If for ech u Pr m ( there eists I(u= f ( u, v dv, the I : Pr m ( R is itegrble d fd = I ( u du Pr ( m [ u] Proof I order to reduce this theorem to theorem, let P be prlleloid which cotis, d let f * : P R be etesio of f, ie 68

f * ( = f ( if if P\ I this situtio, f * is itegrble o P, d f * d = fd P VII Multiple itegrls Becuse i the Crtesi decompositio P = P' P" we hve P' = Pr m (P d P' = P[u] for ll u P', theorem tkes the form fd = * I ( u du where I * (u = P" f * P' ( u, v dv Now, it remis to see tht I * I( u if u Prm ( (u =, if u P'\Prm ( d, becuse f * is ull outside, we hve I * (u = f ( u, v dv [ u] To coclude, we itroduce this epressio i the itegrl of f } I prctice this theorem is mil used for m = d m = p, whe it furishes the pricipl methods of itertio: 5 Method I of itertio (m = Let [, b ] = Pr ( be the projectio o is of, d let us suppose tht for [, b ] there eists f (,, d I( = [ ] p d p The I is itegrble o [, b ] d fd = I ( d f (,, d d b, ie b f (,,, d d d [ ] 6 Method II of itertio (m = p Let be simple mcd d let f : R be itegrble fuctio If for u = (,, p Pr p ( the fuctio p f(u, p is itegrble o [φ(u, ψ(u], the the fuctio I : Pr p ( R, defied b is itegrble o Pr p (, d (,, I(,, p = f (,,, d, fd = Prp (,, ( p p p p I (,, d d, p p p 69

Chpter VII Multiple itegrls ie f ( (,, p,, p d dp f (,, p, p dp d Pr p ( (,, p 7 Remrks (i The bove methods reduce the multiple itegrls to simple itegrls with vrible limits I the cse of p =, whe is simple mcd i the ple, methods I d II coicide (ii The bove methods of itertig the multiple itegrls c be ituitivel described s techiques of "sweepig" the domi b differet sectios For emple, if is simple mcd i the ple, the we m iterpret ( the clculus of the itegrl I( = ( f (, d d p s fidig double itegrl o thi bd B = Δ [φ(, ψ(] from (see the figure VII =ψ ( B =φ ( +Δ b Fig VII Fill, to obti the double itegrl, the bd B "sweeps" the domi b movemet betwee d b, which mes to clculte b I ( d f (, dd Similrl, we c sweep the domi usig horiotl bds, if llows (iii Besides itertio, there is other techique of clcultig multiple itegrls, which is bsed o the chge of vribles The formuls re similr to those cocerig the simple itegrls, but i the cse of the multiple itegrls we mil use the chge of the vribles i order to trsform the give domi ito simpler oe, for emple ito prllelepiped, if possible 7

VII Multiple itegrls The followig theorem of chgig the vribles i multiple itegrl turll eteds the rule of chgig the vrible i the simple itegrl o [, b] R We recll tht chge of vrible of the ope set A R P is : diffeomorphism T : A R P betwee A d B = T(A As usull, we ote = T(u = (φ (u,, φ p (u, where u A Accordig to the locl iversio theorem, if the Jcobi of T is o-ull t u, the T relies chge of vribles i eighborhood of u 8 Theorem Let, E R P, p N * be mcd, d let T : E be trsformtio such tht: (i T is : (ii T(E = (iii det J T (u t u E If f : R is cotiuous o, the f ( d ( f T ( u det J ( u du E Proof We m reso iductivel reltive to p N * For p = the propert reduces to the well kow theorem of chgig the vrible i the defiite simple itegrl Let us suppose tht the theorem is vlid up to p = I order to prove it for p =, we decompose the trsformtio T : E ito T = T T, where T : E R is defied b (v,,v = T (u, u,,u = (u, φ (u,u,, φ (u,,u d T : F T (E R is defied b (,, = T (v, v,,v = (φ T (v,,v, v,, v It is es to see tht T d T stisf coditios (i(iii if T does, so the problem reduces to prove the ssertio of the theorem for T d T So we clim tht f (,, d d ( f T ( v,, v det JT ( v,, v dv dv (* F I fct, if Pr ( = [,b ] ccordig to theorem 4 (method II, b f ( d f (,, d d d [ ] B chgig the vrible i the bove simple itegrl (the cse p = we obti b f (,,, d f ( (, ( v,,, '(,, ( v where Φ (,, (v = φ T (v,,,, d Φ (,, ([α, β] = [, b ] Becuse T dv 7

Chpter VII Multiple itegrls Φ' (,, (v = ( φ T (v,,, =det JT (v,,,, where [α, β] = Pr (F d [ ] = F[v ] we obti f ( d ( f T ( v,, vdet JT ( v,, v dv dv dv F[ v ] Pr ( F F ( f T ( v,, v det J T ( v,, v dv dv dv which proves (* Now we ote g = (f T det J T, d we clim tht g v dv ( g T ( u,, u det J ( u,, u du du (** F ( T E I fct, usig gi theorem 4 (method I, we c write g( v dv g( v,, v dv dv dv, F F[ v ] where F[v ] R Becuse the propert is supposed vlid for p =, we obti g( v, v,, v dv dv E[ u ] g( v, where v ( v, u,, u F[ v ],, ( v, u,, u det v ( u,, u du du (u,, u = (φ (v, u,,u,, φ (v, u,, u Cosequetl, det J v (u,, u = det J T (v, u,,u I fct, T preserves the first compoet which implies tht [α, β] = Pr (E, hece g( v dv ( g T ( u, u,, u det JT ( u, u,, u du du F Pr ( E E[ u ] E (g T (u detj T (u du, which is (** Fill, combiig (* d (** we obti : f ( d g( v dv ( g T ( u det J ( u du T F E [( f T T ]( u det E = J ( T u det J ( u du T T du 7

= ( f T ( u det JT ( u du, E VII Multiple itegrls which ccomplishes the proof } 9 Prticulr trsformtios (i The trsitios to polr coordites i the ple r cost r si t is trsformtio T : (, [, π R \ {(, }, for which det J T (r, t = r Cosequetl, for mcd R \ {(, } d cotiuous f : R we hve f (, dd f ( r cost, r si t rdrdt T ( (ii Similrl, pssig to the clidricl coordites i ope spce (,, (r,t,, r cost r si t represets trsformtio T : (, [, π R R \ {(,,} with det J T (r, t, = r Accordig to the previous theorem, for mcd R \ {(,, } d cotiuous f : R we c write f (,, ddd T ( f ( r cost, r si t, rdrdtd (iii The sphericl coordites i spce re itroduced b the formuls cossi si si cos Cosidered s trsformtio T : (, [, π [, π] R, with det J T (ρ, φ, θ = ρ si θ, the chge of vribles (,, (ρ, φ, θ i the triple itegrl of cotiuous fuctio f : R where R \ {(,, } is mcd, is relied b the formul f (,, ddd T ( f ( cos si, si si, cos siddd 7

Chpter VII Multiple itegrls Remrk The chge of the vribles i multiple itegrl formll reduces to the modifictio of the domi, d to the replcemet of the "differetils" ccordig to the formul d d p = det J T (u,,u p du du p This lst equlit m be cosidered correspodece betwee the mesures of the simplest elemetr bodies i the cosidered coordites More ectl, i Crtesi coordites u,, u p, the prlleloid of sides Δ,, Δ p, hs the mesure Δμ = det J T (u Δ,, Δ p It is es to see (Fig VII tht i the bove prticulr cses we hve: - Δ = rδrδt for the re i polr coordites i the ple; - Δv cl = rδrδtδ for the volume i clidricl coordites i spce; - Δv sph = ρ siθδ ρδ φδθ for the volume i sphericl coordites +Δ Δv cl Δ ( t Δt r Δr r Δt Δ Δv sph Δρ ρ siθ ρ ρδθ ρsiθδφ Δθ θ ρ (b φ Δφ ρδφ Fig VII 74

VII Multiple itegrls To close this prgrph we will le importt reltio betwee double itegrls d lie itegrls of the secod order, which is kow i the literture s Gree's formul Theorem Let γ R be simple, closed, piecewise smooth curve, which bouds the compct domi, whe it is trced oce couterclockwise If P, Q C R ( ~, where ~ ( ~ is ope, d hs fiite decompositios i simple sub-domis reltive to the s well s reltive to the es, the the Gree's formul holds: Q P Pd Qd dd Proof It is cler tht c be decomposed ito fiite umber of rectgles d sub-domis of the form,, d 4 s i the figure VII ( from below Cosequetl it is sufficiet to prove the formul for such simpler domis, eg for B B 4 A γ M A γ ( Fig VII (b I fct, usig the two equtios of, = φ(, where [, ], d = ψ(, where [, ], the double itegrl o becomes (see Fig VII, (b ( ( Q P Q P dd dd d d d d = = Q (, d Q(, d P(, ( d P(, d ( 75

Chpter VII Multiple itegrls = Qd Pd Qd Pd Pd Qd BM MA The other forms of the sub-domis re similrl discussed B ddig such formuls, the lie itegrls o the itervl segmets ccel ech other out, beig clculted i opposite seses } Corollr Uder the coditios cocerig i the bove theorem, the re of hs the epressio ( = d d Proof We c cosider P = d Q = i the bove theorem, d tke dd We recogie here the formul of ito cosidertio tht ( = Propositio 5,, Chpter VI, for more geerl shpe of the domi } To coclude this sectio, we metio iterestig pplictio of the double itegrls i mechics: Emple A bod of costt desit γ is obtied from sphere of rdius R b removig cocetric sphere of rdius r < R We c show tht the ttrctio of this solid o mteril poit lig i the iterior sphere is ull I fct, usig the sphericl coordites (ρ, φ, θ, the elemet of mss of, s ΔM = γ ρ si θδ ρ Δ φ Δθ, cts o the mss m with force of vlue m M ΔF = k, d where d = ρ r ρcosθ + r The compoet log o is cos r ΔF = ΔF cos(,d = ΔF d Cosequetl F = k γm R R si ( cos r d d d = ( rcos r / = d ( cos r d km r si, r ( rcos r / r where the itegrl reltive to θ c be computed b prts Fill F = 76

VII Multiple itegrls PROBLEMS VII epict the domis of itegrtio d evlute the followig iterted itegrls: (i (iii d d (ii d d (iv d / d ( d 77 d Aswers (i ; the itegrl breks up ito product of simple itegrls; 4 (ii ; the domi is rectgle, but the fuctio differs from product; (iii 4 9 ; the fuctio is product g(h(, but is ot rectgle; (iv 6 ; is qurter of disc d the itegrl is 8 from the volume of the uit sphere Chge the order of itegrtio i the followig double itegrls: 4 ( d f (, d (b d 48 Hit ( d /8 / f (, d f (, d ; (b Epress the itegrl s sum Evlute dd, where is: ( trigle with vertices O(,, A(,, B(, ; (b regio bouded b the stright lie pssig through the poits A(, d B(,, d b the rc of circle of ceter C(, d rdius r = Hit ( 6; (b 6 4 Clculte e dd, where is curvilier trigle bouded b the curves of equtios =, =, = Aswer

Chpter VII Multiple itegrls 5 Evlute the itegrl I = dd, where is bouded b the is of bscisss d rc of the ccloid = R(t si t, = R( cos t R ( Hit I = dd, d we chge the vrible i the simple itegrl R R( cost 5 reltive to, ie I = dr( cost dt R 6 Clculte I= f (,, ddd if = [, ] is the uit cube d: (i f(,, = e (ii f(,, = Hit (i I is product of simple itegrls (ii Use theorem 7 Clculte I= ddd if: (i is tetrhedro bouded b the ples + + =, =, = d = (ii is regio betwee the coe = prboloid = Hit (i I = d d d 78 7 where Pr ( = {(, R : + r }, d r = pss to clidricl (or polr coordites ; (ii Pr ( d the ddd, 5 Altertivel, 8 Evlute I = ddd, where is bouded b the ple = d: (i the upper hlf of the ellipsoid b c (ii the prmid + + =, Hit Use the formul of Method I, mel I = dd d, where the [ ] double itegrl represets the re of simple sectio

VII Multiple itegrls 9 Evlute: (i d d R (ii d d R R R R d 8 Aswers (i 4 ; (ii R 5 9 5 usig clidricl coordites ( d usig sphericl coordites Pssig to polr coordites, evlute I = dd, where is loop of the lemiscte ( + = (, Hit rw the correspodig domi bouded b r = cost i the ple (r, t, t [, π] Clculte dd, eteded over the regio, which is b bouded b the ellipse b Hit Use the geerlied polr coordites (r, t, defied b r cost r sit b Evlute I = ( [,] [,] 4 e ddusig the coordites u = + d v = Hit ivide the squre i the (u, vple ito two trigles Show tht there is ifiite re betwee two hperbols = r d = r, >, r >, r > Hit Use the chge = r cosh t, = r sih t 4 Idetif the domis d evlute their res: 79

Chpter VII Multiple itegrls (i d d (iii dt (ii rctg 4 8 sec t d d (iv dt rdr ( cost rdr 9 9 Aswers (i ;(ii ( ; (iii ; (iv ( 8 4 4 5 Fid the volume of the bod bouded b the ple, the sphere + + = d the clider + = Hit Idepedetl of the use of double or triple itegrl, the volume is epressed b the itegrl dd, where is the iterior of the disc + = Pssig to polr coordites, whe is bouded b r = cost, t,, it reduces to cost r rdrdt ( 9 4 6 Usig the Gree's formul, evlute I = d l( d, where γ cosists of the grphs of = cos d = si for betwee 4 d 5 4 5/ 4 si Hit I = dd d d / 4 cos 7 Evlute the lie itegrl I = times couter-clockwise, d: the origi is lig outside γ b the origi is lig iside γ d d, where γ is circle trced

VII Multiple itegrls Hit Accordig to Gree's formul, I = b The Gree's formul is ot vlid more, but direct clcultio of the lie itegrl gives I = π 8 Fid the mss d the ceter of grvit of the solid bod bouded b the prboloid + = 4 d the ple =, whose desit is Hit The mss is M = ρ(,, = ddd [ ] ddd, where [] is elliptic lmi of semi-es d, hece the double itegrl is ([] = π The coordites of the ceter of grvit re G = (,, ddd, M d G = G = (becuse of smmetr 9 A solid circulr coe hs the rdius of the bse equl to R, the ltitude h, d costt desit ρ Fid its momet of ierti reltive to dimeter of the bse Hit Tke the ple of the bse s o d the is of smmetr s o Evlute I = ( ddd hr result is I = (h R 6 usig clidricl coordites The Show tht the force of ttrctio eerted b homogeeous sphere o eterl mteril poit does ot chge if the etire mss of the sphere is cocetrted t its ceter Hit Let M (= 4 πr γ be the mss of the sphere of desit γ d rdius R Puttig the origi of the coordites i the ceter of the sphere, d the mss m o the ois, t the distce L to the origi, i clidricl coordites, the distce betwee m d the curret poit (r, t, of the sphere (r R will be d = r ( L The elemetr force hs the mv vlue ΔF = k d, where Δv = r Δr Δt Δ Becuse of smmetr, we re iterested i fidig the compoet of this force L ΔF = ΔF cos( d, = ΔF Evlutig the triple itegrl, it follows tht F = kmm d L 8

VII IMPROPER MULTIPLE INTEGRALS Up to ow we hve cosidered multiple itegrls of bouded fuctios o compct domis i R p These itegrls correspod to the defiite itegrl o R, d the re clled itegrls o compct domis As i the cse of simple itegrl o itervl of R, there re situtios whe we must evlute multiple itegrls of o-bouded fuctios, or o o-bouded sets All these situtios re icluded i the stud of itegrbilit o o-compct sets efiitio Let Ω R p be o-compct domi for which ech bouded prt of the frotier is egligible We s tht sequece ( N of mesurble compct domis (briefl mcd is ehustig Ω iff for compct K Ω there eists N such tht K for ll As for rbitrr sequeces of sets, we s tht ( N is icresig iff + for ll N Emples (i The domi Ω = R is ehusted b ech of the sequeces ( N, (E N d (F N of mcd, where = {(,, R : + + } E = {(,, R : + + } F = {(,, R : m{,, } } (ii The sequece of mcd ( N of the form = {(, R : + } is ehustig the o-compct domi Ω = S(, \ {(, }, which is the uit disk without ceter (iii I the ple (ρ, θ, the ifiite bd Ω = [, [, π] is ehusted b the sequece of mc domis of the form = {(ρ, θ R : ρ, θ π} efiitio Let Ω R p be o-compct domi, d let f : Ω R be itegrble o ech mcd Ω We s tht f is improperl itegrble o Ω iff for ever icresig sequece of mcd, ( N, which is ehustig Ω, the sequece fd is coverget (see lter tht its N limit does ot deped o the prticulrl chose sequece ( N I such cse we ote lim fd fd d we cll it improper itegrl of f o Altertivel we s tht the itegrl of f o Ω is coverget 8

8 VII Improper multiple itegrls The correctess of the bove defiitio is bsed o the followig propert: 4 Propositio If f is (improperl itegrble o Ω, the fd does ot deped o the prticulr icresig d ehustig sequece of mc domis ( N, for which we clculte the limit of umericl sequece fd N Proof Let ( N d (E N be two icresig sequeces of mc domis which ehust Ω B hpothesis, I = lim fd J = lim fd eist The problem is to show tht I = J E I fct, becuse both ( N d (E N re icresig d ehustig, for ech N there eists k N such tht E k Similrl, for k N there eists m N such tht E k m, d for m N there eists l N such tht m E l, etc O this w we obti icresig d ehustig sequece of mc domis E k m E l for which, ccordig to the hpothesis, the sequece of itegrls fd,, fd, fd, E k m fd, fd, is coverget Becuse this coverget sequece cotis subsequeces of the coverget sequeces fd d fd, N E k N it follows tht ll these sequeces hve the sme limit, hece i prticulr we obti the desiged equlit I = J } 5 Remrks (i Becuse R is complete metric spce, the sequece fd is coverget if d ol if it is fudmetl I dditio, N becuse ( N is icresig sequece, d the multiple itegrl is dditive reltive to the domis, the bove theorem m be formulted s follows: The itegrl fd is coverget if d ol if there eists E l d

Chpter VII Multiple itegrls icresig d ehustive sequece ( N of mcd, such tht for ε > there eists (ε N such tht for > (ε d m N we hve \ m fd < ε (ii Tkig s model the simple improper itegrls, the stud of the multiple improper itegrls c be doe i terms of umericl series with elemets of the form fd The geerl properties of the multiple \ itegrl remi vlid for improper itegrls: 6 Propositio (i If f, g : Ω R re improperl itegrble o Ω d α, β R, the αf + βg is lso itegrble o Ω d (ii Let Ω, Ω d Ω = Ω f g d fd ( gd (lierit Ω be o-compct domis for which Ø If f : Ω R is improperl itegrble o Ω d Ω, the it is itegrble o Ω d fd = fd + fd (dditivit reltive to the domis Proof (i The sme reltio holds o compct K Ω (ii If ( N d (E N re icresig d ehustig sequeces of mcds for Ω d Ω, the ( E N is icresig d ehustig for Ω, d fd = fd + fd holds for ll E N } I prticulr, the covergece of improper itegrls of positive fuctio c be esil studied: 7 Theorem (Boudedess criterio of covergece The positive fuctio f : Ω R + is improperl itegrble if d ol if there eists icresig d ehustig sequece ( N of mcds for which the sequece fd is bouded N Proof Becuse f is positive d ( N is icresig, it follows tht the sequece fd is icresig too, hece it is coverget if d ol N if it is bouded } E 84

VII Improper multiple itegrls 8 Propositio (i Let the fuctios f, g : Ω R + stisf f g If f d g re itegrble o Ω, the fd gd (ii If f : Ω R + is improperl itegrble o Ω, d o some subset Ω' Ω, the the followig iequlit holds fd fd ' Proof (i For ever mesurble compct domi K Ω, we hve fd gd K (ii Let us cosider h : Ω R, of vlues, h( = if ', if \ ' Of course hf f d h f = f o Ω ' Becuse f is itegrble o Ω, d hf is itegrble o Ω', ccordig to (i we obti hfd fd It remis to see tht hfd = fd } ' 9 Remrk I the cse of simple improper itegrl o domis I R, we hve see tht f ( d m be coverget without f ( d, so it I I mkes sese to discuss bout semi-covergece, d bsolute covergece This propert hs o logue i the theor of multiple itegrls I fct, ccordig to the followig theorem, the itegrls fd d f d re simulteousl coverget (respectivel diverget Cosequetl, it is osese to spek of semi-coverget improper multiple itegrls It is ot wrog to spek of bsolutel coverget itegrls, but this otio coicides with tht of simple covergece I order forms to stud the reltio betwee "covergece" d "bsolute covergece" for multiple itegrls for rbitrr f : Ω R, we will defie the positive d the egtive prt of f b f + = [ f + f ]; f = [ f f ] It is cler tht both f + d f re positive, but smller tht f I dditio, we obviousl hve f = f + f, d f = f + + f Theorem Let us cosider tht f : Ω R is itegrble o mcd Ω The f is improperl itegrble o Ω if d ol if f is K 85

Chpter VII Multiple itegrls Proof At the ver begiig we metio tht f is properl itegrble o mcd Ω iff f is, so the sttemet of the theorem essetill refers to the improper itegrbilit o the o-compct domi Ω I fct, for rbitrr ', " we hve f(" f(' f(" f(', hece f hs smller oscilltio th f o divisio of It remis to use the rbou criterio of itegrbilit Let us suppose tht f is itegrble Becuse f +, f f, ccordig to propositio 8, the itegrbilit of f implies tht of f + d f Usig the propert of lierit, it follows tht f is lso itegrble Coversel, let us suppose tht f is improperl itegrble o Ω, but f is ot itegrble Sice f, this mes tht for sequece ( N of mootoicll ehustig mcds i Ω, we hve lim f d B rerrgig the coveiet idices, if ecessr, we c cosider tht the successive terms d + re chose so tht f d > f d + for N eotig A = \, d usig the dditivit of the multiple itegrl, this iequlit becomes f d > f d + for ll N Becuse f (d lso f is properl itegrble o, it follows tht f + d f re lso properl itegrble, d sice f = f + + f, we obti f d = f d + f d A Now let us suppose tht I this cse A A A A fd f d (* A f d f d hece, ccordig to the previous iequlit, f d > f d + A A A Now, let B be closed prt of A o which f + = f, such tht f d = fd The fd > f d + B B A 86

VII Improper multiple itegrls Addig this iequlit to the obvious oe fd > f d, we obti B fd > Similrl, if isted of (* we dmit its cotrr, we would obti tht fd < B Fill, it remis to see tht (E N, where E = B is icresig d ehustig sequece of mcd, for which fd >, hece the sequece ( fd N cot be coverget } E E Remrks (i Becuse the stud of the improper itegrbilit of positive fuctio (like f is esier, the bove theorem simplifies the problem of covergece for the itegrl of fuctios which do ot miti the sig (iithe covergece of multiple itegrl is sometimes cosidered i the sese of the pricipl vlue This mes tht the icresig d ehustig sequece of mcd ( N cosists of "sphericl sets" More ectl: whe Ω = R p, we tke = { R p : }, d b whe Ω = K \ { }, whe K is compct domi for which K, the = K \ { R p : < r }, where r is chose i order to hve S(, r K (i Before clcultig improper multiple itegrl it is ecessr to check the covergece of the respective itegrl, sice prticulr w of crrig out the clcultio m led to coverget process, eve though the itegrl is diverget Therefore it is dvisble to use the methods of clcultig multiple itegrls (itertio, chge of vribles, etc just o compct domis, but ot o the whole o-compct domi I other terms, the simple itegrls, which occur whe usig some method of evlutig multiple itegrl, might be coverget eve for ocoverget multiple itegrls 87

Chpter VII Multiple itegrls (i Show tht I = e R ( PROBLEMS VII dd (ii Evlute I usig polr coordites; (iiieduce the vlue of J = e d is coverget; Hit Fuctio f : R R, epressed b f(, = hece it is sufficiet to show tht ll the itegrls I = e dd, e is positive, re bouded, where = {(, R : + }, N I fct, usig polr coordites (r, t we obti I = π e r rdr ( e (ii I = lim I = π (iii Itertig i Crtesi coordites we obti J = I, hece J = Show tht fuctio f : R R, of vlues f(, = si ( + is ot improperl itegrble o R Hit If we ote ={(, R : + π } d E = {(, R : + π + }, the, for N, we hve fd =, while fd = π E Stud the covergece of the itegrls dd I(α =, d J(α = ( ( where d ddd Ω ={(, R : + }, Ω = {(, R : + + }, α R, 88

VII Improper multiple itegrls Hit Both I(α d J(α refer to positive fuctios, hece we c ppl theorem 7 eotig = {(, Ω : + }, N *, d pssig to polr coordites, we obti I (α = ( dd = π [ ( r dr ], hece I(α is coverget for α >, d diverget for α Similrl, cosiderig E = {(, Ω : + + }, where N *, d usig sphericl coordites, we obti J (α = E ddd ( =4 π 4 [ d ], hece J(α is coverget for α >, d diverget if α cses α = i I, d α = i J, must be seprtel discussed The 4 Stud the covergece of the itegrls dd ddd I(β = (, d J(β = ( where Σ = {(, R : < + }, Σ = {(, R : + + } d β is rel prmeter Hit O compct K ={(, Σ : + }, N*, usig polr coordites, we hve dd = π[ (β ] I (β = K ( hece I is coverget for β < Similrl, for compct L ={(, Σ : + + }, N *, i sphericl coordites we obti: ddd 4 J (β = [ ], ( L hece J is coverget for β <, 89

Chpter VII Multiple itegrls 5 Test for covergece the improper double itegrl I = l dd, where Ω = {(, R : + } Hit Tke = {(, Ω : + }, N*, d use polr coordites i order to obti l I = l dd 4 R 6 Test for covergece the itegrls: I = e ( cos( dd, where, d J = l( ddd, ( where, Ω= {(, R : < + + } Hit I is coverget for α >, sice cos ( + J is diverget d for ll α < ; evlute it i sphericl coordites 7 Test for covergece the itegrl e evlute it usig its pricipl vlue Hit pricipl vlue is R dd d, hece we c ppl the compriso criterio The si 8 Show tht the itegrl I = dd is diverget R Hit The itegrl is ot bsolutel coverget (see Theorem, ie si si r dd = π dr r However, o prticulr domis like = {(, R we hve I = si dd = π si r dr r : + }, 9

CHAPTER VIII SURFACE INTEGRALS The surfce itegrls eted the otio of double itegrl i the sme mer i which the lie itegrls eted the simple itegrls o R We will cosider ol surfces i R, m spects beig similr for the higher dimesiol cse At the begiig, we hve to le the otio of surfce VIII SURFACES IN R From the mthemticl poit of view, the otio of surfce (s well s tht of curve reduces to clss of fuctios, which represet differet prmeteritios From the prcticl poit of view, the curves d the surfces re prticulr objects (sets i R d R, the problem of fidig the most dequte prmeteritio is of cpitl importce i clculus efiitio We s tht the set S R is surfce iff it is the imge of domi (usull ope d coected, but sometimes closed!, R through fuctio φ : R, clled prmeteritio of S, ie S = φ( More precisel, prmeteritio is vector fuctio of two vribles d three compoets, ie for ech (u, v, we ote the prmeteritio b φ(u, v = ((u, v, (u, v, (u, v R, so tht the surfce becomes S = {((u, v, (u, v, (u, v R, (u, v } Their specific clsses of prmeteritios describe the differet tpes of surfces Tpes of surfces We s tht the surfce S is simple iff its prmeteritio φ is : Similrl, S is clled smooth (cotiuous, Lipschite, etc iff φ C R ( (φ C R (, φ Lip R (, etc A smooth surfce S is sid to be o-sigulr, iff the rk of the Jcobi mtri of its prmeteritio φ is equl to two, ie: ( u, v ( u, v ( u, v rk J φ (u, v = rk u u u = ( u, v ( u, v ( u, v v v v Remrk I this chpter we will cosider ol simple, smooth d o-sigulr surfces, which will be clled regulr Becuse ech surfce dmits more prmeteritios, oe of the fudmetl problems i the 9

Chpter VIII Surfce itegrls stud of surfces is to fid the itrisic properties, ie those properties, which re idepedet of prmeteritio More ectl, propert of regulr surfce S is cosidered itrisic iff it is mitied b chge of prmeteritio relied b diffeomorphism of strict positive Jcobi It is cler tht the precise meig of this otio is obtied b defiig the clss of "equivlet" prmeteritios 4 efiitio Let φ : R d ψ : H R be two prmeteritios of the sme surfce S i R We s tht φ d ψ re equivlet d we ote φ ~ ψ, iff there eists diffeomorphism T : H of compoets u (, b,(, b H v (, b such tht ψ = φ T, d (, b (, b et J T = > (, b (, b b b t (, b H The diffeomorphism T is lso clled chge of prmeters o the surfce S 5 Remrks (i It is es to verif tht ~ is i fct equivlece To be more rigorous, we idetif the surfce S with its clss of equivlet prmeteritios (ii Whe we hve prmeteritio of surfce S, we cosider tht S is eplicitl give There re m prcticl cses whe the surfce is described b coditio of the form Φ(,, =, which is clled implicit equtio of the surfce The problem of fidig eplicit form (equtio, ie to write (,, = φ(u, v c be geerll solved ol locll, usig the implicit fuctio theorem (iii A prticulr, but ver coveiet prmeteritio of surfce S R is epressed b fuctio = f(, More ectl, = Pr, (, d f : R stds for the prmeteritio φ(, = (,, f (, 6 The tget ple If (u, v, the the correspodig poit M = φ(u, v S m be lso specified b its positio vector r = (u, v i + (u, v j + (u, v k, where {i, j, k } is the coicl bse of R The curve = {φ(u, v : (u, v } uu is clled curve of prmeter v o S (or coordite curve of tpe ucostt Similrl, = {φ(u, v : (u, v } vv 9

9 VIII Surfces i R is clled curve of prmeter u o S (respectivel, curve of tpe vcostt Obviousl, φ [u ] is prmeteritio of, while φ [v ] is prmeteritio of vv, where u u [u ] = {v R : (u, v } is the sectio of t u, d similrl, [v ] = { u R : (u, v } is the sectio of t v The vectors (which re well defied for regulr surfces r u i j k, d u u u r v i j k v v v represets the tget vectors to the curves of coordites u, respectivel v, t the curret poit (,, = φ(u, v S Sice S is o-sigulr, the orml vector r u r v is defied t poit of the surfce Usig it, the tget ple of the surfce is defied b ( r r, ie - - = u u u v v v Eve if the vectors r u d r v deped o prmeteritio, the tget ple is uiquel determied t ech poit of regulr surfce 7 Propositio The tget ple to S t M does ot deped o prmeteritio Proof Let φ ~ ψ be two prmeteritios of S, d let φ(m d ψ(m be the vectors orml to S t M S, epressed b the prmeteritios φ d ψ A direct clcultio shows tht φ(m = k ψ(m, where k = et J T, ie φ(m ψ(m 8 Corollr (i If S dmits prmeteritio = f (, o its projectio, the the orml to S hs the compoets = (p, q, +, where p = f ', d q = f ' For coveiece, if the sese of does t mtter, ie the surfce is o-orietted, the we c tke = (p, q, (ii If S is implicitl defied b the equtio Φ(,, =, the the orml tkes the form ' ' = (Φ ', Φ ', Φ ' sice p =, q = ' '

Chpter VIII Surfce itegrls The equtios of the tget ple t M (,, S will be = p( + q(, respectivel, ( Φ' + ( Φ' + ( Φ' = The proof reduces to simple clcultio d will be omitted Aother useful otio i the stud of surfce is tht of re, which is itroduced b the followig costructio: 9 efiitio Let S be regulr surfce of equtio = f (,, where the domi = Pr, (S of f is mesurble compct domi (mcd To divisio δ = {,, } of i mcd, we ttch divisio Σ δ = {S,, S } of S, where, for ll i =,, we hve I ech sub-domi k S i = {(,, f(, S : (, i } we choose poit ( k, k k, so tht M k ( k, k, f( k, k S k for ll k =,,, d we ote b π k the tget ple to S t M k I ech such tget ple we delimitte domi T k = {(,, π k : (, k }, k =,, which is mesurble (ie it hs re s imge of mcd k through Pr Let (T k deote the re of T k, for ll k =,, We s tht S hs re (is mesurble, etc iff there eists the (fiite umber A = 94 lim (, k which is the sme for ll sequeces of divisios for which δ, d for ll possible choices of this "itermedite" poits M k S k, k =,, I this cse we ote A = (S, d we cll it re of S For the evlutio of the re of surfce we metio: Theorem Let S be regulr surfce for which = Pr S is mcd i R, d = f (,, where f : R is the equtio of S The S hs re d it is epressed b the double itegrl (S = / T k / f f dd ( Proof Let θ k be the gle betwee the o es d the orml k t the poit M k S k S Usig θ k, c specif the reltio betwee the re ( k of k d tht of T k, mel ( k = (T k cos θ k We m fid of the vlue of cos θ k from the formul / k k = k k cos θ k, while gives cos θ k = [+ f (, f (, ] / for ll k =,, Cosequetl, k k / k k

VIII Surfces i R 95 (, (, ( ( / / k k k k k k k k f f T hs the form of Riemi sum of double itegrl o, s metioed i the theorem The eistece of this itegrl is ssured b theorem 7,, chpter VII, sice f hs cotiuous prtil derivtives o, d is mcd The re of surfce m be epressed b other formuls which mke use of some specific ottios More ectl, if φ : H R is prmeteritio of S, of compoets (u, v, (u, v, (u, v, d if we ote A =, (, ( v u, B =, (, ( v u, C =, (, ( v u the the orml becomes = Ai + B j +Ck, d v r u r = C B A holds t poit M S Other useful ottios re the Guss coefficiets: E = u u u r u F = v u v u v u r r v u G = v v v r v A direct computtio shows tht A + B +C = EG F Corollr Let S be smooth surfce for which = Pr (S is mcd, d let = f(, be the equtio of S If φ : H R is other prmeteritio of S, the the followig formuls hold: (S = H dudv C B A ( (S = H v u dudv r r ( (S = H dudv F EG (4 Proof Let T : H be trsformtio (diffeomorphism of compoets = α(u, v d = β(u, v, which reltes the prmeteritios More ectl, (,, = φ(u, v mes, (,, ( (, (, ( v u v u f v u v u

Chpter VIII Surfce itegrls for ll (u, v H Usig the prtil derivtives of, / / f f u u u / / f f v v v / A / B we obti f, d f f' Chgig the vribles (, (u, v C C i ( we obti / (S = f / f dd = A B C JT dudv, C et, which represets (, sice C = et J T Formul ( is simple trsformtio of ( becuse r u r v = Ai + B j +Ck Fill, (4 follows from ( s cosequece of the idetit r u rv = r u r v [ r u rv ] } More th the equivlece of the formuls (, (, ( d (4, the re of surfce is itrisic chrcteristic of the surfce, ie it is the sme for ll equivlet prmeteritios Theorem If S is regulr surfce which hs prmeteritio o the mcd = Pr (S, the other equivlet prmeteritio of S gives the sme vlue for the re of S Proof Let φ : H R d ψ : L R be equivlet prmeteritios of the regulr surfce S, of compoets φ(u, v = ((u, v, (u, v, (u, v, d ψ(, b = ( ~ (, b, ~ (, b, ~ (, b, d let A, B, C, respectivel A ~, B ~, C ~ be the correspodig coefficiets Accordig to the bove corollr, both double itegrls represet the sme H L (S = E A B C dudv, d ~ A ~ B C ddb ~ / / f f dd We metio tht the proof could be bsed o the reltios A ~ = AΔ; B ~ = BΔ; C ~ = CΔ, where Δ = et J V, d V : L H is diffeomorphism which relies the chge of prmeters (, b (u, v } 96

VIII Surfces i R PROBLEMS VIII Fid the re of the trigle cut out b the coordite ples from the ple, where, b, c R + b c Hit The projectio of S i ={(, R :,, } b d the equtio of the surfce hs the form = f(,, where f(, = c( b Cosequetl, (S = / f / f dd = b bc c Compute the re of the helicoidl surfce of polr equtios r cost r sit, t, < r b, where, b, k R * + kt Hit Pr (S = {(, R : + b,, } Evlute the double itegrl which represets the re i polr coordites Let C be the clider of equtio + =, d let S be the sphere of equtio + + = Evlute the re (S if: (i S is tht prt of C which is cut out b S (ii S is the prt of S iside C Hit (i Cosider = f(, o Pr (S = {(, R : +, > } (ii Tke = f(, o Pr (S = {(, R : ( ( / + 4 } 4 Clculte the re of the torus obtied b rottig the circle of ceter (R,, d rdius r, where < r < R, lig i the ople, roud the ois Hit S is the imge of = {(u, v R : u, v π} through φ of compoets = (R + rcos ucos v, = r si u, d = (R + rcos usi v, hece r u r = r(r + rcos u, d (S = 4 π Rr v 5 Compre the res of the prts of prboloid + = (circulr d = (hperbolic, cut out b the clider + = R Hit Use polr coordites The res re equl 97

Chpter VIII Surfce itegrls 6 Fid the re of ellipsoid of hlf es, b, c Hit S = φ(, where = {(u, v R : u [, π], v [, π]}, d φ(u, v = ( si u cos v, b si u cos v, c cos u Use the formul (S = A B C dudv 7 (Schwrt's emple Let S be the lterl surfce of clider of rdius r d ltitude h B dividig h ito equl prts, N, usig ples prllel to the bses, we obti + circles C, C,,C o S O C we cosider equidistt poits The geertors correspodig to these poits meet the other circles i poits deoted with similr idices Now, from ech circle C k we delete the poits with eve idices if k is odd, d the poits with odd idices if k is eve Ech pir of remiig successive poits o the sme circle d the closest poit of eighborig circle determie trigle Δ Evlute the re (Δ, show tht the sum of ll these res teds to whe, d epli wh this sum does ot pproimte (S Hit There re u u = u 4 such trigles of res (Δ = (si ( cos h r r k for some k > The epltio cosists i mkig evidet the differet directios of the orml vectors to Δ d to S (see lso [NS] vol II 98

VIII FIRST TYPE SURFACE INTEGRALS Similrl to the lie itegrls of the first tpe, the surfce itegrl of the first tpe refers to sclr fuctios defied o domis, which coti the surfce The re useful i evlutig the mss of lmi, its grvit ceter, ierti momets, or forces of iterctio The costructio of the itegrl sums Let S be regulr surfce of prmeteritio φ: R, where is mcd i R Let lso U : R be bouded sclr fuctio, where is domi i R which cotis S (it is sometimes sufficiet to sk U : S R, s for emple whe U is the desit of the mteril surfce S If δ = {,, } is prtitio of, the we cosider the subsequet prtitio Σ δ = {S,, S } of S, d sstem S = {M k S k : k =,, } of "itermedite" poits, ectl s for evlutig the re of S The the sums σ U (δ, S = k 99 U(M k (S k represet the itegrl sums of the first tpe of U o S efiitio We s tht U is itegrble o S iff there eists (fiite I = σ U (δ, S, lim idepedetl of the sequece of divisios for which δ, d idepedetl of the sstems of itermedite poits More ectl, for ε >, there eists η > such tht σ U (δ, S I < ε holds wheever δ < η, d for rbitrr S I this cse we s tht I is the surfce itegrl of U (of the first tpe, d we ote or ltertivel I = S I = UdS S U (,, ds, S Ud, etc Oe of the fudmetl problems is to specif clsses of itegrble fuctios, d methods of evlutig the itegrls Theorem Let S be regulr surfce, d U : S R be cotiuous fuctio The, U is itegrble o S, ie there eists the surfce itegrl of the first tpe of U o S, d S UdS U ( u, v A ( u, v B ( u, v C ( u, v dudv ( ( where φ : R is prmeteritio of S

Chpter VIII Surfce itegrls Proof Becuse is mcd d V = (U φ A B C is cotiuous, there eists the double itegrl i the right side of the climed reltio Let δ = {,, } be divisio of Usig the me-vlue theorem for double itegrls we obti (S k = k A B C dudv A( u~, ~ ( ~, ~ ( ~, ~ k vk B uk vk C uk vk ( k where ( u ~ k, v ~ k Cosequetl, cosiderig rbitrr sstem of k itermedite poits S = { φ(u k, v k : k =,, }, the itegrl sums tke the form = k σ U (δ, S = k U(φ(u k, v k (S k = (U φ(u k, v k A( u~, v~ B( u ~, v~ C ( u ~, v~ ( k k O the other hd, sice V is itegrble o, for ever ε > there eists η > such tht for δ < η, d for rbitrr S, we hve Now, we c evlute k Vdudv ( U ( uk, vk ( k k Vdudv σ U (δ, S ( U ( u k, v k ( k k Vdudv ( U ( u k, v k k k k ( U ( u A B k k, v k C k ( k k ( u ~, v ~ The lst modulus is less th, sice U φ is bouded o, d A B C is uiforml cotiuous o } Whe defiig the itegrl sums, the vlues of U o S, d the res of the surfces S k do ot deped o prmeteritios, hece the surfce itegrl is uiquel defied b S d U I fct: 4 Corollr The surfce itegrl of the first tpe does ot deped o prmeteritio Proof Let ψ : H R be other prmeteritio of S i the bove theorem, d let T : H be the diffeomorphism for which ψ = φ T Chgig the vribles (u, v = T(, b i the double itegrl (, we obti k k

U T (, b A B S H VIII First tpe surfce itegrls ~ ~ ~ UdS ( C (, b ddb Becuse A ~ = AΔ, B ~ = BΔ, C ~ = CΔ, where Δ = et J T (, b, we obti ~ ~ ~ UdS [( U A B C ](, b ddb, S H ie differet equivlet prmeteritios of the surfce give the sme vlue of the surfce itegrl of the first tpe } 5 Corollr Usig the ottios i VIII, the surfce itegrl c be epressed b the formul U,, f (, p S UdS ( qdd, ( where = f (, is the equtio of S, f : R, p = f /, q = f / Other forms of the sme itegrl re UdS [ U r r ]( u, v dudv, d ( u S U EG S v UdS [( F ]( u, v dudv (4 I fct, ccordig to VIII, where we hve epressed the elemet of re i severl forms, we hve see tht p q A B C ru rv EG F 6 Remrk So fr, we hve used the projectio to stud the surfces d the surfce itegrls of the first tpe Similr results m be obtied for, or projectios I prctice, we c divide the give surfce ito fiite umber of surfces, which dmit such projectios This decompositio is frequetl ecessr if the equtio of the surfce is implicit The geerl properties of the first tpe surfce itegrls re commo to other tpes of itegrls, mel 7 Propositio The surfce itegrl of the first tpe is: (i lier reltive to the fuctio, ie ( U V ds UdS VdS ; s S S (ii dditive reltive to the surfce, ie UdS UdS UdS, S S S S where S, S re regulr surfces without commo iterior poits The proof is simple reductio to the similr properties of the double itegrl, d will be omitted

Chpter VIII Surfce itegrls PROBLEMS VIII Evlute the surfce itegrl I = ( ds S where S is the surfce of the cube,, Hit The itegrl o the fces = d =, where ds = dd reduces to ( dd ( dd ( dd Similrl, we tret the other pirs of fces, so tht I = 9 Evlute the itegrl S ( ds, where S is tht prt of the coe, cut out b the surfce Aswer 64 5 4 Fid the mss of mteril surfce S of equtio which hs the locl desit Aswer ( 6 5 (,,,, 4 Evlute the momet of ierti of sphericl surfce of rdius r d of costt desit ρ, reltive to dimeter Hit I = 8 ( ds r 4 The sphericl coordites re S dvisble, sice A B C = r si θ, d I = πr 4 si d 5 Clculte the momet of ierti, reltive to the O ple, of tht prt of the coic surfce, for which, if the locl desit is (,, Hit B defiitio, ( I ds O S = ( ( d d, where {(, R : } The result is I O

VIII First tpe surfce itegrls 6 Fid the ttrctio force eerted b uiform sphericl surfce o poit-wise mss m locted i the iterior (eterior of the sphere r Hit F = km ρ ds r ], where (,, r is the loctio [ ( / S of m, ρ is the desit of S, d k is depedig o uits Usig the sphericl coordites = R si θ cosφ, = R si θ si φ, = R cosθ, where R is the rdius of the sphere, we obti F = πkm ρ[r I rr J], d cossi r I = d (b prts, d [ R rr cos r] / R( R r si J = d [ R rrcos r ] / R( R r Cosequetl, F = Becuse of smmetr, we hve F = F = too 7 Fid the potetil creted t (,, b electric chge of desit ρ(,, =, distributed o coicl surfce of equtio Hit The potetil is Φ = k =, > S (,, ds, where k depeds o uits I prticulr, ds = dd The surfce itegrl c be reduced to double itegrl o = Pr (S, which c be esil evluted i polr coordites The serched potetil is Φ =

VIII SECON TYPE SURFACE INTEGRALS I this sectio we stud the surfce itegrl of vector fuctio, which defied o the surfce I order to epli the meig of this itegrl, we strt out with emple: Emple (The flu of icompressible liquid through surfce Let us cosider tht the domi R is full of liquid, which is i sttior movemet To describe this movemet we use the so-clled vector field of speeds, V : R, which defies the velocit V (,, = (V (,,, V (,,, V (,, t ech poit (,, (ot depedig o time sice the movemet is sttior Now let S be (regulr surfce, for which we eed to determie the qutit of liquid, which is pssig over the surfce i the uit of time (lso clled flu Obviousl, evlutig this qutit supposes sese of the orml vector t ech poit of the surfce, such tht specifig wht "comes i" d wht "goes out" to be possible (see the orietted surfces below V (M k (M k (T k cos k (T k cos k T k M k k Fig VIII If we refer to smll prt S k S, or to its correspodig pproimtio T k of the tget ple π k t M k S k, the seek qutit is cotied i the volume v of prllelepiped of bsis T k d side V (M k k (T k cos k 4

5 VIII Secod tpe surfce itegrls More ectl (see Fig VIII, sice (M k π k, we hve v k = <V (M k, (M k > re(t k = ( V (M k (T k If S k is elemet of the prtitio δ = {S,,S } of S, d M k S k is itermedite poit of the sstem S = {M,, M }, the v(δ, S = vk ( V ( M k ( Tk k k represets pproimtio of the sought volume Further o, if = cos α i + cos β j + cos γ k is the uit orml, the we c eplicit the sclr product (V, d we obti v(δ, S = ( V cos V cos V cos ( M k k ( T k = [ V ( M k (Pr ( Tk V( M k (Pr( Tk V( M k (Pr ( Tk ] k Becuse geerll spekig, better pproimtios correspod to fier prtitios of the surfce, it is turl to defie the flu of V through S s v = v(δ, S lim This emple shows tht before defiig the geerl otio of surfce itegrl of secod tpe, we must clrif the meig of oriettio o surfce (compre to the oriettio of curve i VI Orietted surfces As usull, eplicit writig of the bove formuls supposes some prmeteritio φ : R of S, whe R is mesurble compct domi of the ple Accordig to the defiitios i VIII, S is regulr mes tht φ is :, of clss C, d rk J φ = o More ectl, S is defied b clss of such equivlet prmeteritios, where φ ~ ψ deotes the eistece of diffeomorphism T betwee the domis of φ d ψ such tht et J T Cosequetl, either et J T >, or et J T <, ie the clss of ll prmeteritios c be split ito two subclsses, ech of them cosistig of those prmeteritios which re relted b "positive" diffeomorphism (et J T > To oriette the surfce S mes to chose oe of these subclsses of prmeteritios s defiig the positive sese of the orml t ech poit of S These cosidertios re bsed o the followig: Propositio Let φ : R d ψ : E R be prmeteritios of the regulr surfce S, d let T : E be diffeomorphism for which ψ = φ T If φ(m d ψ(m represet the uit orml vectors t M S, correspodig to these prmeteritios, the we hve: (i φ(m = ψ(m if et J T > (T is positive (ii φ(m = ψ(m if et J T < (T is egtive

Chpter VIII Surfce itegrls Proof Let A φ (M, B φ (M, C φ (M d A ψ (M, B ψ (M, C ψ (M be the differetil coefficiets correspodig to the prmeteritios φ d ψ, t the curret poit M S Cosequetl, the uit orml vectors, which correspod to these prmeteritios, re A i B j C k φ(m = ( M A B C Ai B j Ck ψ(m = ( M A B C Similrl to theorem i (chpter VIII, from ψ = φ T we deduce A ψ = A φ Δ, B ψ = B φ Δ, d C ψ = C φ Δ, where Δ = et J T } 4 Emples If S dmits prmeteritio = f(, o the projectio =Pr (S, the usull, the positive sese of the orml is tht for which the gle betwee o (ie k d is i the itervl [, ] If S is closed, the it divides the spce ito two prts, mel the iterior d the eterior of S The positive sese of the orml is usull chose outwrds (However, the ect meig of oriettio d closeess is obtied i much deeper theories, eg see [SL], [CI], etc Whe referred to the vectors r u d r v, is orietted ccordig to the right-hd screw rule: b rottig the hd of the screw from r u to r v, the screw is drive i the positive sese of I this w the oriettio of is crried to S 4 The oriettio o S c be referred to the prticulr sese, which is defied o the frotier of S I this cse we ppl the sme right-hd screw rule 5 As emple of o-orietted surfce we metio the fmous Möbius strip It is obtied from ple rectgle of sides l d L, where l<<l, b gluig the smller sides cross-wide (s sketched i Fig VIII A l C A A C B B L B b Fig VIII The resultig surfce llows o : prmeteritio The coordites of poit deped o the "fce" o which the poit is lig, eve though we 6

VIII Secod tpe surfce itegrls c pss from oe fce to other without touchig the boudr Therefore we cot specif positive sese of the orml t poit of the surfce, ie the Möbius strip is ot orietted The surfce itegrl of the secod tpe is defied b log to the bove otio of flu through orietted surfce: 5 The itegrl sums Let V : R be vector fuctio o the domi R, d let S be regulr orietted surfce of prmeteritio φ : R, where R is mesurble compct domi (mcd If δ = {,, } is prtitio of, the correspodig divisio Σ δ = {S,, S } of S cosists of prts S k = φ( k S Choosig M k S k o ech S k, k =,,, we obti sstem of itermedite poits S = {M k : k=,, } Let π k be the tget ple t M k to S, d let T k be the projectio of S k o π k, k =,, The sum ( δ, S = V,S = [ V ( M k (Pr ( Tk V( M k (Pr( Tk V( M k (Pr ( Tk ] k is clled surfce itegrl sum of secod tpe of V o S 6 efiitio We s tht V is itegrble o S iff the bove surfce itegrl sums of secod tpe hve limit I = lim V,S ( δ, S, which is idepedet of the sequece of divisios with δ, d of the choice of itermedite poits I this cse we ote I = V dd Vdd Vdd, S d we s tht I is the surfce itegrl of the secod tpe of V o S, lso clled the flu of V through S The followig theorem idictes clss of itegrble fuctios 7 Theorem Let S be regulr orietted surfce, d let V : R be vector fuctio If V is cotiuous o S, the: (i V is itegrble o S, d (ii its surfce itegrl (of the secod tpe reduces to surfce itegrl of the first tpe ccordig to the formul dd V dd V dd V ds V S = Proof Becuse V d re cotiuous o S, theorem i VIII ssures the eistece of the surfce itegrl of the first tpe I = V ds S S 7

Chpter VIII Surfce itegrls Cosequetl, the problem reduces to show tht for ε > there eists η > such tht for prtitio δ d S, for which δ < η, we hve ( δ, S I < ε V, S I fct, becuse V is cotiuous there eists λ = dditio, sice S is mesurble, there eists η > such tht k ( T ( k S k sup V I S holds for divisio δ = {,, } for which δ < η Cosequetl, for such divisios we hve V, S ( δ, S V ( δ, S, ( V ( M k ( Tk ( Sk (* k O the other hd, sice V is itegrble o S, there eists η > such tht for δ < η, we hve (V, S ( δ, S I < (** Combiig (* d (**, for δ < η = mi{ η, η }, we obti ( δ, S I < ε, V, S with ccomplishes the proof } 8 Corollr The surfce itegrl of the secod tpe does ot deped o prmeteritio (s log s we preset the oriettio Proof Accordig to corollr 4 i VIII, the surfce itegrl of first tpe is idepedet of prmeteritio Restrictig to positive diffeomorphisms of the orietted surfce S, is lso ivrit of the surfce, hece the itegrls i 7(ii from bove do ot deped o prmeteritio } 9 Propositio The surfce itegrl of the secod order hs the properties: (i ( V W ds V ds W ds (lierit (ii S S S S S V ds = S V ds + S V ds (dditivit, wheever S d S hve t most frotier commo poits (iii V ds = V ds (oriettio where S is the cotrr S S orietted surfce (of orml Proof (i d (ii re cosequeces of propositio 7, Propert (iii simpl follows from V ( = V, d (i } 8

VIII Secod tpe surfce itegrls Remrk Usig prmeteritios, we m reduce formul (ii from theorem 7 to severl double itegrls s follows: V ds V A V B V C dudv V ( r r dudv= S = ( = ( p V q Pr ( S V V dd Obviousl, these formuls correspod to differet forms of the epressio of the orml, d tht of the elemetr re ds Emple Let us evlute the itegrl I = dd dd ( dd, S where S deotes the upwrds orietted surfce of equtio, restricted to the coditio We m strt b writig the orml vector, for emple i the form ( i j k 4 4 Cosequetl, we m reduce the problem to surfce itegrl of the first tpe, ie I = ( ds S 4 4 Further o, this itegrl reduces to double oe b replcig ds, eg I = ( ( dd, where {(, : } Usig polr coordites, we esil obti the result I = 6 u v 9

Chpter VIII Surfce itegrls Evlute I = dd dd S PROBLEMS VIII dd, whe S is the eterl side of the tetrhedro bouded b the ples of equtios =, =, =, d + + = > Hit The itegrl o the side = reduces to dd, etc For the side S of equtio + + = we hve ( i j k, hece the itegrl c be epressed s itegrl of the first tpe ( ds S Fid the flu of the vector fuctio V (,, through the sphere ( + ( b + ( c = R Hit Φ = V dd Vdd Vdd I prticulr, S (, b, c, R hece Φ reduces to surfce itegrl of the first tpe Φ = R S [ ( ( b ( c] ds Usig sphericl coordites is dvisble, sice ds = R siθ dθdφ, d Φ = 8 R ( b c Evlute I = dd, where S is the eterl side of the ellipsoid S, d iterpret the result b c Hit Usig the prmetric equtios of the ellipsoid = si θ co sφ, = b si θ si φ, = c cos θ, we obti = (b c si θ co sφ, csi θ si φ, b si θ cos θ, hece

VIII Secod tpe surfce itegrls I = c cos θ bsi θ cos θdφd θ = πbc cos 4 θsi θd θ = bc 4 Evlute I = S dd dd dd, where S is the eterior side of the ellipsoid b c Hit I is ppretl improper sice,, c be ero o S, but if we itroduce the prmetric equtio of the ellipsoid (s bove, it becomes bc c b defiite double itegrl; I = 4π b c 5 Let S be closed regulr surfce, which bouds mesurble domi, such tht ech prllel to o, o, o is meets S t most two times Show tht the volume of is give b v( = dd dd dd Hit v( = S S dd, sice S = S S, where S = {(,, : = f (,, (, } S = {(,, : = f (,, (, }, d = Pr (S Supposig f > f, d tkig ito cosidertio the oriettio, dd f dd S dd = f Similrl, we tret the other projectios (see lso problem

VIII4 INTEGRAL FORMULAS Our purpose i this sectio is to estblish reltios betwee lie, surfce, d multiple itegrls i R A similr reltio betwee lie d double itegrl i R we lred hve discussed i theorem,, chpter VII, where we hve proved the Gree itegrl formul Q P Pd Qd dd A chrcteristic of these formuls cosists i some specific restrictios o the cosidered domi d its frotier, which will be icluded i the followig defiitio: 4 efiitio We s tht the domi R is regulr iff it stisfies the coditios: (i is mesurble compct domi (mcd (ii is fiite uio of simple sub-domis reltive to ll es (ie lie prllel to o, o or o meets the frotier S of t most two times, without commo iterior poits (iii S = Fr( is regulr, closed d orietted surfce For regulr domis the triple itegrl m be epressed b surfce itegrl of the secod tpe s follows: 4 Theorem (Guss-Ostrogrdski formul If R is regulr domi of frotier S, d V C R ( is vector fuctio of compoets V, V, V, the V V V ddd Vdd Vdd Vdd S Proof If =, s bove, it is sufficiet to prove the formul for k, k =,, More ectl, we c show ol tht V ddd Vdd, Fr( k becuse ddig the similr formuls for V d V o ll k, k =,,, we obti the climed formul I fct, sice k is simple reltive to o es, there eist f k, g k : Pr ( k R such tht k = {(,, R : f k (, g k (,, (, Pr ( k } B itertig the triple itegrl o k we obti k

gk (, V V Pr ( k f k (, [ V (,, gk (, V(,, fk (, ] dd ddd ddd k = Pr ( k = dd S V V dd, k, k, S VIII4 Itegrl formuls where S k, = {(,, R : = f k (,, (, Pr ( k } d S k, = {(,, R : = g k (,, (, Pr ( k } The sig "" t S k, shows tht the positive sese of the orml is opposite to the usul oe (i ccordce with the sese of the o es Usig the oriettio of the surfce itegrl of the secod tpe reltive to the orml, we m remrk tht S k = S k, S k, is the frotier surfce of k, hece we hve k V ddd S k V dd Similrl we tret the other itegrls } 4 Remrk (i Epressig the surfce itegrl i Guss-Ostrogrdski formul b surfce itegrl of the first tpe we obti V V V ddd V ds, where the lst itegrl represets the flu of V through S The triple itegrl c lso be simplified if we defie the divergece of V s div V = V V V I this cse the Guss-Ostrogrdski formul tkes the form div V ddd V ds, lso clled the flu-divergece formul It is ver useful i field theor b its remrkble cosequeces (see the et chpter (ii The other importt itegrl formul reltes lie d surfce itegrls ivolvig the otio of rottio Therefore we recll (see defiitio,, chpter VI tht the rottio of V = (V, V, V C R V V V V V rot V = i S S j ( is defied b V k

Chpter VIII Surfce itegrls = i V j V V There re lso ecessr some regulrit coditios for the surfce 44 efiitio We s tht surfce S is eplicit reltive to iff there eists ope set O R, fuctio f C R (O, d mcd O such tht S = {(,, R : = f (,, (, } Similrl, we defie the eplicit surfce reltive to or If S is eplicit reltive to,, or, the we simpl s tht S is eplicit Surfce S is clled elemetr iff it cosists of fiite umber of regulr d eplicit surfces 45 Remrk Ech eplicit surfce is orietted ccordig to the covetio i VIII I fct, if S is regulr surfce eplicit reltive to, the the curve γ = Fr hs turl positive sese, mel the couterclockwise oe, which iduces the positive sese o Γ = {(,, R : = f(,, (, γ } Usull, Γ is clled the orietted border of S This oriettio is comptible with tht of S i the sese of the right-hd screw rule Whe the elemetr surfce S is decomposed i regulr d eplicit sub-surfces, b coveiece we cosider tht these sub-surfces hve ol border poits i commo More ectl, ech prt of the border of sub-surfce c belog to t most two sub-surfces, cse i which it is trced i both opposite seses The uio of ll prts of the borders which belog to sigle sub-surfce form the border of S, deoted Γ = Bd(S 46 Theorem (Stokes formul Let V C R ( be vector fuctio of compoets V, V, V o the domi R If S is elemetr surfce of border Γ, the V dr ( rotv ds S Proof It is sufficiet to prove the formul for sigle sub-surfce of S which is regulr d eplicit reltive to (for emple, becuse fill we c dd such reltios to obti the climed oe I other terms, we will prove the formul supposig tht S reduces to sigle regulr surfce, which is eplicit reltive to Let φ : [, b] R be prmeteritio of Γ = Bd(S If we eplicit φ(t = ((t, (t, (t for ll t [, b], the d r V d V d V d = b 4 k V = [( V ( t '( t ( V ( t '( t ( V ( t '( t] dt

VIII4 Itegrl formuls 5 Becuse Γ = B d(s is prt of S, we hve (t = f((t, (t o Γ, hece '(t = f ((t, (t'(t + f ((t, (t'(t Cosequetl, b dt t t f V V t t f V V V d r ( ' ]( [( ( ' ]( [( = d f f V f V ], (, (,, (, (,, ( [ + d f f V f V ], (, (,, (, (,, ( [ Usig the Gree formul for γ d i R, we obti dd f V V f V V V d r ( ( The problem reduces to evlutig the squre brcket uder this double itegrl I fct, sice f C R (, its mied prtil derivtives of the secod order re equl, hece ( ( f V V f V V = f V f f V V f V V f V f f V V f V V = V V V V f V V f = = (rotv f f, where k j f i f f f / is the uit orml to S ( = Fill, S ds rotv dd f f rotv r V d ( (, which proves the Stokes formul }

Chpter VIII Surfce itegrls 47 Remrks (i The Gree formul (which hs bee used i the proof is prticulr form of the Stokes formul I fct, if V =, d Γ = γ is ple curve borderig the domi S = R, the = (,,, hece we hve (rot V = V V, while d r V d V d V (iithe lie itegrl V d r is lso clled the curl or circultio of V o Γ Usig this term the Stokes formul ss tht: "the flu of the rottio of V through S is equl to the curl of V log the border Γ of S" 48 Corollr Uder the coditios of theorem 46, if S d S re elemetr surfces hvig the sme border Γ, the the flues of rot V through S d S re equl Proof Accordig to the Stokes formul both flues re equl to the curl of V o Γ Obviousl, the oriettio of S d S re supposed to be comptible to the positive sese o Γ } 49 Remrk Usig Stokes formul we c improve theorem,, chpter VI, i the sese tht the coditio for the domi to be sttior c be removed from the hpothesis I fct, if the field V is coservtive, ie rot V =, the the curl o closed curve is ull, hece the lie itegrl of the secod tpe does ot deped o the curve, but ol o its edpoits I other terms, ech irrottiol field is o-circultor (or circultio free We metio tht besides their theoreticl importce (obviousl i field theor, the bove formuls re frequetl useful i order to evlute surfce d lie itegrls 6

VIII4 Itegrl formuls PROBLEMS VIII4 Evlute I = dd dd S regulr domi R Geerlitio dd, where S is the boudr of Hit V (,, = (,,, hece div V =, d I = ccordig to the Guss-Ostrogrdski formul More geerll, we obti ull itegrl if V (,, = (f(,, g(,, h(, Evlute I = dd dd S dd, where S is the eterl surfce of sphere of rdius r (d rbitrr ceter Hit I the Guss-Ostrogrdski formul div V =, d ddd is the volume of the sphere Fid I = dd dd S dd if S is the eterl totl surfce of the coe h h, where h > Hit The Guss-Ostrogrdski formul reduces I to triple itegrl; the h result is I = 4 Show tht if V derives from hrmoic potetil i the regulr domi, the the flu V ds, where S = Fr ( S Hit B hpothesis, V U U U =,,, hece div V = ΔU = becuse U is hrmoic Use the Guss-Ostrogrdski formul 5 Prove tht if S is closed surfce, which bouds regulr domi, d cos(, l ds, where is the outer l is fied directio, the I = S orml to S Hit Cosider = (cos α, cos β, cos γ d l = (cos α, cos β, cos γ, such tht I = cos dd cos dd cos dd O the other hd, S cos(, l, l, s i the Guss-Ostrogrdski formul 7

Chpter VIII Surfce itegrls 6 Evlute I = d d d, where Γ is the cotour of the trigle of vertices A(,,, B(,,, C(,, Hit V = (,, hs rot V = (,,, hece usig Stokes formul, I = ( ds, where S is the surfce of the trigle S ABC Sice + + = o S, d ds is the re of Δ ABC, we obti S I = 7 Applig Stokes formul, fid I = ( d ( d ( d, where Γ is the ellipse of equtios + =, + = Verif the result b direct clcultio Hit V =(,, hs rot V = ( i j k, d the ple of the ellipse hs = (,, I = 4π A prmeteritio of Γ is = cos t, = si t, = cos t, t [, π] 8 Evlute the lie itegrl I = d ( d ( d, where Γ hs the prmeteritio = cos t, = si t, = (cos t + si t, t [, π], usig Stokes' formul, d directl Hit rot V = i j k, d Γ is ellipse o the ple = + 9 Fid the curl of V = i j k log the circle of equtios + =, = trced oce i the positive sese reltive to ois Hit rot V =, hece ppl the Stokes' formul 8

CHAPTER IX ELEMENTS OF FIEL THEORY I essece, ll the importt ottios of the field theor were lred itroduced d studied i the previous chpter for both sclr d vector fields Therefore this chpter will be sthesis o the differetil d itegrl clculus, epressed i more ituitive lguge, specific to pplictios For these prcticl purposes, i IX, we will put forwrd the most sigifict tpes of prticulr fields IX IFFERENTIAL OPERATORS For the begiig, we hve to clrif the otio of field, which so fr ws reduced to sclr fuctio φ : R, whe we were spekig bout sclr fields, or to vector fuctio V : R, i the cse of vector field Usull, is domi i R, but similr topic is vlid whe R Some problems rise whe opertig with φ d V, sice the vlues of φ re cosidered s belogig to the field of rel umbers, over which the vector spce R is defied, d the spce R of the vlues of V is idetified with the iitil vector spce R, which cotis I other terms, s log s R is set of pits (,,, or positio vectors r = i + j + k, the defiitio of V (,, i the sme spce, s i Fig IX, mkes o rigorous meig i spite of its prcticl use (eg the work of force, the flu, etc This situtio is clrified b cosiderig the otio of "tget" spce: V r Fig IX 9

Chpter IX Elemets of field theor efiitio Let A = (,, be fied poit i R For other B R, the pir (A, B is clled tget vector t A to R The set of ll tget vectors t A is clled tget spce t A, d is deoted s T A The poit A is clled origi (or pplictio poit, d B is clled verte of the tget vector (A, B The umber B A is the legth, d B A is the vector prt of (A, B It is es to orgie T A s lier spce: Propositio T A edowed with the opertios d defied b (A, B (A, C = (A, B + C A λ(a, B = (A, A + λ(b A is lier spce isometric to R The proof is routie Remrk The tget spce reproduces the geometr of R t A sice we c defie the sclr product of two tget vectors usig the sclr product i R of their vector prts, ie <(A, B, (A, C> = (B A(C A Usig this otio we c itroduce the otios of orm, distce, gle orthogolit, etc, d we c see tht the correspodece T A (A, B B A R is isometric isomorphism The tget vectors i A = (A, (,, A, j A = (A, (,, A, d k A = (A, (,, +A represet the coicl bsis of T A Usig the compoets of the tget vectors we c lso costruct the vector product, the mied product, etc Betwee tget vectors of differet origis we hve the reltio of prllelism defied b (A, B (A, B, B A d B A re collier Now we c formulte the correct otio of vector fields, which is lso pplicble to geerl (o-flt mifolds: 4 efiitio The set T = T A is clled tget budle of R A AR vector field i the domi R is fuctio V : T for which V (A = V A T A for ll A If V d W re vector fields o, their sum is defied b (V + W (A = V A W A t A Similrl, if V is vector field o d f : R is sclr field, their product is defied b (f V (A = f (A V A t A

IX ifferetil opertors Similrl, we c defie the sclr product, the vector product, etc, of vector fields usig, t ech A, the correspodig opertios i T A, ie b "locl" costructios I order to justif the previous use of the term "vector field" for fuctios V : R, where R, we metio tht i the cse of R (which is lier, "flt" mifold we hve: 5 Propositio If V is the spce of ll vector fields o R d F is the set of ll vector fuctios o, the V d F re isometricll isomorphic Proof Ech vector fuctio F : R is defied b three compoets, ie F = (f, f, f, which re sclr fuctios o It is es to see tht ech vector field is lso defied b three compoets, ie V = V I + V J + V K, where V, V, V : R I fct, if I, J, K represet the fudmetl fields, defied t A b I (A = i A = (A, (,, + A J (A = K (A = j A A = (A, (,, + A k = (A, (,, + A the V = <V, I >, V = <V, J >, V = <V, K > The climed isomorphism is obtied b idetifig the correspodig compoets V, V, V d f, f, f } 6 Remrks (i The stud of the sclr d vector fields is relied b three differetil opertors: grdiet, divergece d rottio, which c be uitril treted usig the followig Hmilto's "bl" (or "del" opertor (The Greek νάβλα is the me of ciet musicl istrumet of trigulr shpe: = I + J + K The costt fields I, J, K re metioed here i order to emphsie the locl chrcter of bl, but ccordig to the bove propositio we c simpl ote = i + j + k (ii For prcticl uses the smbol c tke two meigs, mel tht of vector, d tht of opertor As opertor, which cotis the prtil derivtives, it mifests lso two chrcteristics, mel: - lier opertor reltive to the lgebric opertios; - differetil opertor ctig o the compoets of the field

Chpter IX Elemets of field theor These properties of determie the rules of opertig with it d lso the sigificce of its ctio (iii If U : R is sclr field, the U = U i + U j + U k = grd U If V : T is vector field of compoets V, V, V, the V V = V + V + = div V, d V V = ( V i V + ( V V j + ( V k = rot V We metio tht occurs i the otio of derivtive of sclr field U log the uit vector l, ie l = l, i the sese tht (l U = l (U = l U grd U = l The derivtive of vector field V i directio l, which is defied b V V ( tl V ( ( = lim, l t t m be similrl epressed s: (l V = l l l V V = i V + V j + k l l l I such formuls l cts s sclr differetil opertor The Lplce secod order differetil opertor o sclr fields U U U ΔU = + + is frequetl cosidered s Δ =, i the sese tht ΔU = ( U = (U = div (grd U The vectoril behvior of is visible i the followig: 7 Propositio For sclr field U d vector field V we hve: (i (U = R ; (ii ( V = ; (iii ( V =( V ( V Proof (i The vector product of collier vectors is ull; i this cse it mes tht rot (grd U = (ii The mied product, i which two of the vectors re collier, is ull I other words div (rot V =

IX ifferetil opertors (iii (b c = ( c b ( b c is geerll vlid for the double vector product of three vectors, hece lso for = b = d c = V I prticulr, this formul shows tht rot (rot V = grd (div V ΔV, where ΔV = ΔV i + ΔV V V V j + ΔV k = + + We remember tht the strtig formul follows from (b c = λb + μc We multipl b to obti λ( b + μ( c =, hece λ = k( c d μ = k( b If we tke = b = c =, d =b c, the we obti k= The lier chrcter of is essetil i properties s: 8 Propositio Let T, U be sclr fields o R, V, W be vector fields o, d λ R The the followig formuls hold: (i (U + T = U + T; (λu = λu (ii (V +W = V + W ; (λv = λ V (iii (V +W = V + W ; (λv = λ V Proof These formuls epress the lierit of grd, div d rot } The propert of of beig differetil opertor is especill visible wheever it cts o product 9 Propositio If U, T re sclr fields, d V, W re vector fields, the: (i U = if d ol if U = costt (ii V = if V is costt (iii V = if V is costt (iv (UT = T U + UT (v (U V = V (U + U( V (vi (U V = U( V V (U (vii (V W = W ( V V ( W Proof (i (v re obvious The sig "" i (vi is due to the order depedece i V (U = (U V Formul (vii follows b developig the smbolic mied product W W W V V V V V V = W W W V V V W W W (eve though such formul is ot vlid for vectors }

Chpter IX Elemets of field theor Whe we hdle with s differetil opertor it is dvisble to respect the followig: Rule (Step pplied to product gives two terms, i which it cts o sigle fctor We usull mrk this ctio b rrow " ", s for emple i the bove (iv: (UT = ( U T + (U T (Step Relie the ctio of, s idicted b rrows, eg ( U T + (U T = ( U T + U( T (Step Let fter sigle letter, which distiguishes the field o which it cts, such tht the rrows re ot ecessr more For emple: ( U T = T(U Other importt formuls ivolvig re formulted i the problems t the ed of this prgrph Here we metio ol the form of the mi itegrl formuls (estblished i 4, chpter VIII Corollr Uder the coditios stted i theorem, VIII4, the Guss-Ostrogrdski formul tkes the form ( V ddd ( V ds Corollr If the hpothesis of theorem 6, VIII4, is stisfied, the the Stokes formul holds i the form ( V ds V dr S These formuls re useful just for better uderstdig of the divergece d rottio of vector field: Remrk I the cse of sclr field we hve two possibilities of defiig the grdiet of U t A, mel U grd U = (Ai U + (A U j + (A k, s i defiitio, IV, d ccordig to corollr 4, IV, U grd U =, where is the uit orml t the level surfce pssig through A Obviousl, the first defiitio is preferble i clcultios, but it seems to deped o sstem coordites Ol the secod defiitio shows tht the grdiet of sclr field is itrisic chrcteristic of the field Similrl, for vector fields, so fr we hve used ol the coordite depedet epressios of div d rot, so there is problem whether the deped or ot o the sstem coordites The swer is tht the do't d this propert follows from: S 4

IX ifferetil opertors 4 Theorem Let, S d V : R be s i Guss-Ostrogrdski 's theorem Let us fi A, d cosider sequece of sub-domis ( m mn of, cotiig A, d stisfig, together with their frotiers S m, the sme coditios s d S If v m = μ ( m is the volume of m, d d m = dimeter( m = sup{ :, m } ted to ero whe m, the (div V (A = lim V ds m v Proof The Guss-Ostrogrdski formul is vlid for ech m, ie ( V ddd ( V ds m S m m Applig the me vlue theorem to the triple itegrl, we c fid some poits A m m, such tht (V ds (div V (A m v m = There remis to use the cotiuit of div V, which gives (div V (A = lim (div V (A m, m d relie the sme limit i the Guss-Ostrogrdski formul } 5 Theorem Let V C R ( be vector fuctio d let us fi poit A d uit vector T A I the ple of orml, pssig through A, we cosider sequece (S m m N of elemetr surfces of borders m If m = μ(s m re the res of S m, d d m = dimeter(s m whe m, the the compoet of the rottio of V t A i the directio of is (rot V (A = lim V dr S m S m m m Proof Accordig to Stokes' formul for S m d m, m N, we hve ( rotv ds V dr, S d usig the me theorem fo the bove double itegrl, we obti m [(rot V (A m ] = V dr, where A m S m Sice V is of clss C, rot V is cotiuous, hece (rot V (A = lim (rot V (A m m Fill, we tke the sme limit i the Stokes formul } m m m 5

Chpter IX Elemets of field theor 6 Corollr (div V (A d (rot V (A re idepedet of the sstem coordites (ie the re itrisic elemets of the field t A Proof The elemets which pper i the right side of the reltios estblished i the bove theorems 4 d 5, s well s the volumes, res, lie itegrls d surfce itegrls, re ll idepedet of the sstem coordites } I prticulr, we obti the compoets of rot V o the coicl bsis, if we cosider tht re obtied; is successivel equl to i, j d k, ie (rot V (A i V = V (A, etc 6

IX ifferetil opertors PROBLEMS IX Usig rule, prove the formuls: (i rot (U V = U rot V V grd U (ii div(v W = W rot V V rot W (iii rot (V W = V W + V div W W div V W V (iv grd (V W = V rot W + W rot V W V + V W ( UV (v = V (l V grd U + U l l Hit (i (U V = ( U V + (U V = ( U V + U ( V = = U ( V V (U (ii (V W = ( V W + (V W = W ( V V ( W, where the lst equlit epresses the rule of iterchgig the fctors i mied product (compre with (vi d (vii i propositio 9 (iv (V W = ( V W + (V W, where use (b c = ( c b ( b c, we obti ( V W = ( W V ( V W V = W div V, etc W (iv The reltio (V W = ( V W + (V W is ot to be cotiued b W div V + V div W, sice the sclr product would be eglected Usig gi the idetit (b c = ( c b ( b c, i the sese tht we obti V ( W = =(V W (V W, (V W = V (rot W W + V (v (l (U V = (l ( U V + (l (U V = V U l V + U l Let R be fied uit vector ( =, d r = i + j + k Show tht: 7

Chpter IX Elemets of field theor r (i = (ii [ grd (V rot (V ] = div V (write it lso for V = r (iii div [ r ( r ] = Hit (ii Epress rot (V ccordig to problem (iii, d multipl b (iii Combie propositio 9, (v d problem (ii Show tht if R is regulr domi, d u, v C R(, the the followig Gree's formul holds: ( uv vu d u v v u ds, where is the uit orml to S t its curret poit Hit Write the Guss-Ostrogrdski formul for V = u grd v d W = v grd u, d subtrct the forthcomig reltios We strt with div (u grd v = grd u grd v + u Δv, the we itroduce 4 Let R be regulr domi of frotier S, d V C R ( Show tht rotv d V ds, S where is the uit orml t the curret poit of S, d the itegrls of the vector fuctios re uderstood o compoets Hit Appl the Guss-Ostrogrdski formul to V W, where W is rbitrr costt field Sice W =, (V W = W ( V, it follows tht div( VW d W ( rotv ddd ( VW ds W ( V ds S S Cosequetl for rbitrr W we hve rotv d W W S S V ds 5 Evlute the flu of the field V = r + ( r through closed surfce S, where r is the positio vector of the curret poit, d is costt uit vector Hit Verif tht div V =, d ppl the Guss - Ostrogrdski formul The flu is equl to the volume bouded b S 8

IX ifferetil opertors 6 Show tht for, C R (, the field V = grd + grd is orthogol to rot V Hit Estblish tht rot V = grd grd usig problem, (i 7 Let = (,, d b = (b, b, b be costt vectors We ote = grd r, d = b grd r, where r is the positio vector of the curret poit Show tht: (i b grd + r 6 8 grd = ( r (b r r (ii b grd div ( r + grd div (r 6 r = (iii div ( + r ( + = 6r ( r + r 5 (b r Hit (i Estblish the eplicit epressios = r ( + + = r r, = r 5 ( b + b + b = r b r 5, the evlute b grd = r r b r b, d r 5 grd = b r r b 5 7 r r (ii From div ( r = 9 r r it follows tht b grd [div ( r 9 ] = r r b 9 r b r Similrl, sice div (r 6 r = 9 r b r, we hve grd div (r 6 r = 9 r r b r b r (iii Strt with + = r r r b r 5, d deduce div [( + r ] = ( + + 6 r r + r b 5 r 9

IX CURVILINEAR COORINATES Eve though the differetil opertors defie itrisic elemets of the fields, i prctice it is sometimes importt to epress these opertors i other th Crtesi coordites, eg sphericl or clidricl efiitio Let E R be domi, d T : E R be vector fuctio We s tht T is coordites chge (trsformtio iff it is : diffeomorphism betwee E d = T (E, such tht et J T > t (u, v, w E The surfce S = {(u, v, w R : u = u } u is clled coordite surfce of tpe ucostt; similrl re defied the coordite surfces v costt, d wcostt The curve γ u = S v is clled coordite curve of prmeter u; similrl re defied the coordite curves of prmeters v d w The uit orml vectors to the surfces S u, Sv d Sw will be deoted, respectivel The uit tget vectors to the curves γ u, γ v, γ w re deoted b l, l, l Remrks The coordites u, v, w re usull clled curvilier becuse the coordite curves re ot stright lies s i the cse of the Crtesi coordites The chge of coordites c lso be epressed b the correspodece betwee curvilier d Crtesi coordites E (u, v, w T (,,, which is eplicitl writte usig the compoets f, g, h of T, ie f ( u, v, w g( u, v, w h( u, v, w These formuls furish the Crtesi equtios of the coordites surfces d coordites curves Emples (i The sphericl coordites (ρ, φ, θ re itroduced b si cos si si cos hece is sphere, is coe d is hlf-ple S S Cosequetl γ ρ is hlf-lie, γ θ is hlf-circle, d γ φ is circle (see Fig IX, S w S

IX Curvilier coordites The vectors, d re orthogol to ech other, d k for ll k =,, = l k t M S S M S r i S t S r r S Fig IX (ii The clidricl coordites (r, t, re defied b r cost r si t, hece is clider, is hlf-ple d is ple, respectivel Sr b St γ r is hlf-lie, γ t is circle d γ is stright lie (Fig IX, b Similrl, {,, } is sstem of orthogol vectors, d k = l k for ll k =,, (iii Geerll spekig {,, } d { l, l, l } re ot orthogol sstems of vectors, d k l k, for some k =,, For emple we c S cosider the coordites (u, v, w defied b sh u ch v ch u sh v w However, there re strog reltios betwee these vectors: f g h 4 Propositio (i Notig r u i j k u u u r v l d r w l (ii If, reversig T, we ote, etc; we hve r u l,

Chpter IX Elemets of field theor u (,, v (,, w (,, the grd φ, grd ψ, d grd χ (iii {grd φ, grd ψ, grd χ } d { r u, r v, r w } re reciprocl sstems of vectors, ie grd φ r u =, grd φ r v =, grd φ r w =, etc Proof (i d (ii re direct cosequeces of the defiitios of grdiet d of tget to curve The reltios (iii epress the fct tht T T = ι, hece their Jcobi mtrices verif J T J T = I, where I is the uit mtri Sice T is o-degeerte, {,, } d { l, l, l } re lierl idepedet } 5 efiitio If T is chge of prmeters, the the umbers r u = L, r v = L d r w = L re clled Lmé prmeters d grd φ = H, grd ψ = H d grd χ = H re clled differetil coefficiets of the first order Usig the previous propositio we deduce: 6 Propositio (i r u = L l, r v = L l, r w = L l ; (ii grd φ = H, grd ψ = H, grd χ = H ; (iii L k H k = for ll k =,,, wheever the sstem of coordites u, v, w is orthogol (ie { l, l, l } is orthogol bsis i R Proof (i d (ii is bsed o the fct tht l k = k = for ll k =,, Reltios (iii re cosequeces of grd φ r u =, etc } 7 Emples (i I sphericl coordites (ρ, φ, θ we hve L = H = = r =, L = = H r = ρ si θ, d L = r = ρ H (ii I clidricl coordites (r, t, we hve L = H = r r =, L = H = r t = r, d L = H = r = (iii For (u, v, w i the bove emple, (iii, we hve L = r u = = ch u, L = r v = ch v, d r w = = L = H, but ch v ch u H = grd φ = = d H = grd ψ = ch(u - v ch(u - v Cosequetl, L H L H eve though grd φ r u = d grd ψ r v =

IX Curvilier coordites We metio tht grd φ d grd ψ re esil obtied without mkig φ d ψ eplicit, but clcultig,,, from sstems of the form ch u sh v sh u ch v We obtied them b derivig the iitil reltios reltive to d From ow o, we will cosider ol orthogol coordites I order for us to use the Lmé prmeters i writig the differetil opertors of field, we eed the epressios of the legth, re d volume i orthogol curvilier coordites 8 Lemm Let (u, v, w be orthogol sstem of coordites, d let L, L, L be the correspodig Lmé prmeters (i If γ is curve of prmeteritio u = u(t, v = v(t, w = w(t, where t [, b], the the legth of γ is dr b ( L u' ( Lv' ( Lw' (ii If is mesurble domi i the surfce ds L Ldudv E, S w dt ;, the the re of is where T(E = ; (iii If is mesurble domi i R, d = T (E, the the volume of is d L LLdudvdw E Proof (i We hve d r = r u du + r v dv + r w dw = [(L u' l + (L v' l + +(L w' l ]dt, d becuse { l, l, l } is orthogol bsis, it follows the correspodig formul of d r (ii Chgig the vribles (,, (u, v, w i the surfce itegrl o, we replce ds = r u r v du dv, but sice r u r v, we hve r u r v = L L (iii Chgig the vribles (,, (u, v, w i the triple itegrl o, we replce dω = et J T du dv dw, where et J T = r u ( r v r w } The epressios of the differetil opertors i (orthogol curvilier coordites will be obtied strtig out with their ivrit defiitios:

Chpter IX Elemets of field theor 9 Theorem Let U : R be sclr field, T : E be chge of coordites d let U ~ = U T be the sme field i the coordites (u, v, w of E If L, L, L re the Lmé prmeters, the grd U ~ ~ ~ ~ U U U = l l l L u L v L w Proof Accordig to the ivrit defiitio of grdiet, the derivtive ito directio l is the projectio of the grdiet o this directio, hece i prticulr U ~ = (grd U ~ l k for ll k =,, l k O the other hd, if M = (u, v, w is fied i E, lso b defiitio ~ ~ ~ U U ( u u, v, w U ( u, v, w ( M lim, l s s where Δs = r u (M Δu is the distce betwee M d (u +Δu, v, w Cosequetl, ~ ~ U U l L u d similrl, ~ ~ ~ ~ U U U U, l L v l L w From the compoets of grd U ~, reltive to the bsis { l, l, l }, we immeditel deduce the ouced form of the vector grd U ~ } Theorem Let be regulr domi i R, V : T vector field, d let W = V T, where T : E is chge of vribles which hs the Lmé prmeters L, L, L If W (u, v, w, W (u, v, w, d W (u, v, w re the compoets of W i the locl (orthogol bsis { l, l, l }, the div W ( W = LL ( L W L L L W ( L L L u v w Proof Let us fi M = (u, v, w i E, d let us cosider curvilier prlleloid of boudr S d volume Ω (s i Fig IX, hvig the sides log the coordite curves Accordig to theorem 4, IX, the ivrit form of divergece is div W (M = lim W ds S 4

IX Curvilier coordites w e Q C P v e M N A B e u Fig IX To evlute the flu of W through S we clculte it for pirs of fces, eg Φ = W ds W ds ABC 5 MNPQ O the fce MNPQ we hve = l, hece W = W (u, v, w, d (pproimtel the sme o the fce ABC, ie W = W (u + Δu, v, w O both fces ds = L L dvdw, so tht v v w Φ = v w w [( W LL ( u u, v, w ( WL L( u, v, w] dvdw Usig the Lgrge's theorem for the icremet of W L L, d the me-vlue theorem for the bove double itegrl, we obti v v w Φ = v w w ( W LL ( WL L ( M' udvdw ( M * u u, where M' d M * re coveiet poits of the prllelepiped Similrl, there eist M * d M * i the prllelepiped, such tht W ds S uvw W L L L W L L L W ( ( ( ( M * ( M * ( M * uvw u v w O the other hd, Ω = L L L ΔuΔvΔw, hece it remis to use the cotiuit of L, L, L d W, W, W } Corollr Uder the coditios of the bove theorems (9 d, the Lplce opertor of sclr field U ~, i orthogol curvilier coordites (u, v, w hs the epressio

Chpter IX Elemets of field theor ΔU ~ ~ ~ ~ L = L U L L U L L U L LL u L u v L v w L w Proof The compoets of W = grd U ~ U ~ U ~ re W =, W = L u L v U ~ d W =, hece L w L L U ~ W L L = L L U ~, L W L = L L U ~ d L L W = L u L v L w Fill, ΔU ~ = div W } Theorem Usig the bove ottios, uder the coditios of Stokes' theorem, i orthogol coordites (u, v, w, we hve: L l L l L l rot W = L L L u L W v L W w L W Proof Aimig to fid the compoet ito directio l t M= (u, v, w E, we cosider the surfce S i S coordite surfce, bouded b the curvilier rectgle MNPQ = Γ (see Fig IX u w e Q P C v e M N e u Fig IX Accordig to theorem 5, IX, this compoet of rot W is rot W l = lim W dr, where is the re of S = (Γ I order to evlute the curl o Γ, we evlute the lie itegrl o ech side of Γ, eg 6

IX Curvilier coordites 7 MN v v v dv w v u L W W dr,, ( ( sice o MN, we hve d r = L dv l, etc Cosequetl, w w w v v v dw w v u L W w v v u L W dv w w v u L W w v u L W W d r ],, ( (,, ( [( ],, ( (,, ( [( Epressig the icremets b the Lgrge formul, d usig the me theorem for the bove itegrls it follows tht: v v v w w w vdw M v W L wdv M w L W W d r ' ( ( ' ( ( w v M w L W M v W L ( ( ( ( * * Tkig ito ccout tht = L L ΔvΔw, d tht ll the ivolved fuctios re cotiuous, we obti rot W l (M = ( ( ( M w L W v W L L L Similrl, we fid the other compoets of rot W, hece we hve rot W = ( ( l w L W v W L L L ( ( ( ( l v W L u L W L L l u W L w W L L L which formll c be writte s the bove determit } The sphericl d clidricl coordites re frequetl most used ifferetil opertors i sphericl coordites If, i prticulr, we tke (u, v, w = (ρ, φ, θ, the L =, L = ρ si θ, L = ρ, d: grd ~ ~ si ~ ~ l U l U l U U, div W = si ( ( si si W W W si cos si W W W W W

Chpter IX Elemets of field theor 8 rotw = si si si W W W l l l = = si ( si l W W ( si ( si l W W l W W, U U U U ~ si si ~ si ~ ~ 4 ifferetil opertors i clidricl coordites If (u, v, w = (r, t, re clidricl coordites, the L =, L = r, L =, d grd ~ ~ ~ ~ l U l t U r l r U U, div W = W r t W r rw r (, rot W = l W r t W r W rw W t r l rl l r ( l t W r rw r l r W W, ~ ~ ~ ~ U t U r r U r r r U The proof reduces to direct substitutio, d it is left s eercise 5 Remrk The formlism bsed o smbol like is ot possible more I fct, the epressio of grd u might suggest to cosider ~ l w L l v L l u L, but obviousl div W ~ W, d rot W ~ W, etc Cosequetl, it is dvisble to use ol to epress differetil opertors i Crtesi coordites

IX Curvilier coordites PROBLEMS IX The geerlied sphericl d clidricl coordites re defied b si cos r cost bsi si d br sit ccos c where, b, c R * + (i Idetif the coordite surfces d the coordite curves (ii Fid the vectors, d ; d l, l, l, d check their orthogolit (iii Evlute L, L, L, d H, H, H Hit Compre with emples, 7 (i d (ii of this sectio Cosider the (o-orthogol sstem of coordites (u, v, w defied b T : R R, where (,, = T(u, v, w mes: sh u ch v ch u sh v w (i If I is the stright lie segmet of ed poits (u, v, = (,, d (u, v, w = (,,, fid the legth of γ = T(I usig the Lmé prmeters (ii If Q is the squre of digol I, fid the re of S = T(Q (iii If K is the cube of bse Q, fid the volume of = T(K Hit L = ch u, L = ch v, L = We hve = iff u = v, hece γ is the stright lie segmet of edpoits (e, e, d (e, e, ; prmeteritio of γ is = e t, = e t, =, t [, ] Becuse the coordite curves re ot orthogol, we hve to ppl the formuls d r d r = the legth of γ, Q I ru rv dudv = the re of S, d K r ( r r dudvdw = the volume of u v v Let U : R be sclr field, T : E be chge of coordites, d let T be epressed b 9

Chpter IX Elemets of field theor u (,, v (,, w (,,, where (,, Show tht ~ ~ ~ ~ U U U grd U grd grd grd, u v w d use it i order to obti the epressio of grd U ~ i orthogol curvilier coordites Hit grd φ = H = l, etc L 4 Estblish the formuls: (i rot lk grd( Lk lk, k =,, Lk (ii rot W = grd( LkWk lk k,, Hit (i Sice grd φ = l, it follows tht rot ( l = O the other L L hd rot ( l = l + grd L L + L rot l, where grd L = L grd L (ii rot W = k,, rot ( W k l k, where rot(w l = l grd W + W rot l = = [grd W + W L (grd L ] l = L grd (L W l, etc 5 Estblish the formuls div l = L L (grd L L l, etc d use them i order to obti div W i orthogol coordites Hit l = l l, hece div l = l rot l + l rot l, (see problem (ii, i IX, where we c use the previous problem, d the properties of mied product Further, div (W l = (grd W L L l ( W L L =, etc L L L L L u 4

IX Curvilier coordites 6 Let us cosider the sstem of coordites (u, v, w, defied b ch u cos v T : sh u si v w Show tht it is orthogol, determie the Lmé coefficiets, d write the Lplce equtio i these coordites Hit r u, r v d r w form orthogol sstem of vectors t ech poit L = L = ch u cos ~ ~ U U ~ v, L = + (ch u cos U v = u v w 7 Evlute div W d rot W if, i sphericl coordites (ρ, φ, θ, the field is defied b W = ρ θl + ρ l Solutio div W = ρθ + ρ ctg θ, rot W = 8 Fid the potetil from which derives W = ρ θl + ρ l i sphericl coordites (W is o-rottiol ccordig to the previous problem 7 Hit U ~ (,, (ρ, φ, θ = W d r, where (,, d r = r dρ + r dφ + r dθ = dρl + ρ si θdφl + ρdθl, hece W d r = ρ θdρ + ρ dθ Usig prticulr lie, U ~ (ρ, φ, θ = d d = ρ θ + c 9 Fid the potetils of the followig fields i clidricl coordites: V = l + r l (rot V = b W = l + t l (rot W = r Hit I clidricl coordites d r = l dr + rl dt + l d I prticulr, V d r = d(r, hece U ~ = r + cost Similrl, W d r = d(t implies U ~ = t + cost I the locl bsis {l, l, l } of the curvilier coordites (u, v, w defied b 4

Chpter IX Elemets of field theor ( u v uv w we cosider the field W = ul + v l + w l Show tht rot W =, d fid the potetil which geertes W Hit L = L = u v, L =, hece d r = u v l du + u v l dv + l dw The potetil ( u, v, w U ~ (u, v, w = u u v du v u v dv wdw ( u, v, w c be obtied usig the formul u P( t, v, w dt Q( u, t, w dt u ( u, v, w ( u, v, w Pdu Qdv Rdw v v w w R( u, v, t dt ie evlutig the circultio o prticulr broke lie up to costt The result is U ~ (u, v, w = (u + v / + w 4

IX PARTICULAR FIELS So fr we hve bee studig ol the o-rottiol fields s prticulr tpe of vector fields (see lie itegrls o-depedig o the curve, fidig fuctio whe the prtil derivtives re kow, etc The cetrl result o o-rottiol fields refers to the fct tht the derive from potetils d these potetils, c be epressed s lie itegrls of the secod tpe (circultios The most represettive emple of o-rottiol field is the Newtoi oe V = k r, r where k depeds o the uits, d r is the positio vector of the curret poit (V = grd U, where U = r k It is es to see tht div V = too, so we re led to le similrl other tpes of fields efiitio The field V : T is sid to be soleoidl iff div V = holds t ech poit of R Becuse of the prcticl meig of div V, epressed i terms of flu, the soleoidl fields re lso clled fields without sources Propositio V : T is soleoidl field if d ol if the flu of V through closed surfce S is ull (i the coditios of the Guss- Ostrogrdski theorem Proof If div V =, the V ds div Vd for domi Ω S with Fr Ω = S Cosequetl, we c use the ivrit defiitio of div V, specified i theorem 4, IX } Theorem The field V : T is soleoidl if d ol if for ech M = (,, there eists eighborhood N, of M, d there eists vector field W : N T such tht V = rot W o N Proof If V = rot W the div V = ( W = Coversel, let us choose M = (,, for which N = S(M, r for some r > (which eists becuse is ope B hpothesis div V V = V V o, which is vlid o N too The problem is to costruct the field W, such tht V = rot W o N We show tht there eists such field i the prticulr form W = W i + W j, ie the followig reltio is possible: 4

Chpter IX Elemets of field theor 44 V = W W k j i I fct, the problem reduces to solvig the sstem: V W W V W V W (* o N The first equtio gives W (,, = dt V, (,, (, where φ is rbitrr rel fuctio of clss C o N Similrl, from the secod equtio we deduce W (,, = dt V, (,, ( Replcig W d W i the third equtio we obti V dt t V V,, (, (, (,, ( or, usig the hpothesis tht div V =, V dt t V,, (, (, (,, ( Applig the Leibi-Newto formul to the bove itegrl, it follows,, (, (, ( V Obviousl, there re fuctios φ d ψ stisfig this coditio, hece W d W re completel (but ot uiquel determied } 4 Remrk The costructio of W b solvig (* lso represets the prcticl method of solvig problems i which W is sked Usull, the method furishes W o the whole, eve though the proof is restricted to some eighborhoods of the poits t If the costructio of W must be

45 IX Prticulr fields relied repetedl t differet poits, there rises the problem of comprig the fields o the commo prts of the correspodig eighborhoods This problem is solved b: 5 Propositio A ecessr d sufficiet coditio for the fields W d Z to verif the reltios rot W = V = rot Z o the ope d str-like domi is tht W Z = grd U for some sclr field U o, d rot W = V Proof If rot W = V = rot Z, the rot (W Z =, hece W Z derive from potetil U Coversel, if V = rot W d Z = W + grd U, the rot Z = V } B log to the cse of the o-rottiol fields, which re sid to "derive" from sclr potetil, similr termiolog c be used for soleoidl fields i order to epress theorem from bove 6 efiitio The field W, for which rot W = V, is clled vector potetil of V If so, we lso s tht V derives form vector potetil W Usig these terms, the bove results tke the forms: 7 Corollr (i V : T is soleoidl field iff it locll derives from vector potetil (ii Two vector potetils of the sme field, o str-like d ope domi, differ b grdiet (iii If V derives from the vector potetil W, d S is surfce of border s i Stokes' theorem, the S V ds W d r, ie the flu of V through S reduces to the circultio of W log Proof All these ssertios represet reformultios of some previousl estblished properties, mel theorem, propositio 5, d respectivel the Stokes' theorem } As geerlitio of the potetil fields we cosider ow other tpe of fields, which re geerted b two sclr fields 8 efiitio We s tht the vector field V : T is bi-sclr iff there eist two sclr fields φ, ψ : R such tht V = φ grd ψ The bi-sclr fields c be chrcteried i terms of rottio 9 Theorem V : T is bi-sclr field iff V rot V = Proof If V is bi-sclr field, the V rot V = sice rot V = grd φ grd ψ Coversel, if V rot V =, this is sufficiet for the equtio

Chpter IX Elemets of field theor V d + V d +V d = (** to hve solutio, where V, V d V re the compoets of V I fct, this equtio is equivlet to the sstem A(,, B(,, V where A = V d B =, which is itegrble iff V V A A B B B A, ie V rot V = This coditios ssures the possibilit of itegrtig successivel the equtios of the sstem, ie i = f(, + C( obtied b itegrtig the secod equtio we c determie C such tht the first equtio to be stisfied too We s tht U be solutio of (**, if U(,, = represets the implicit form of the solutio = (, I this cse there eists itegrd fctor μ such tht μv d + μv d + μv d = du, or, equivletl, μv = grd U I other ottio, mel φ =, d ψ = U, this mes V =φ grd ψ } Aother chrcteritio of the bi-sclr fields is formulted i the more geometricl terms ivolvig the field lies: efiitio The curve L is clled field lie of V : T iff V (M is tget to L t ech M L Accordig to this defiitio, the field lies re solutios of the sstem d d d, V V V where two of the vribles,, re serched s fuctios depedig o the third oe Theorem V : T is bi-sclr field iff there eists fmil of surfces i, which re orthogol to the field lies of V Proof If V is bi-sclr, the V grd ψ Therefore V is orthogol to the surfce ψ = ψ(m, where M L, d V is tget to L t M Coversel, if the field lies re orthogol to the fmil of surfces ψ(,, = cost, the V grd ψ, hece V = φ grd ψ } The followig propositio itroduces some of the most remrkble properties relted to bi-sclr fields, lso kow s Gree formuls 46

IX Prticulr fields Propositio Let V = φ grd ψ be bi-sclr field i R, d let Ω be regulr domi of frotier S (s i the Guss-Ostrogrdski theorem If is the uit orml to S t the curret poit, the we hve: (i ds [ Δψ grd grd ] d S (ii ds [ ] d S (iii ds d S Proof (i If we ppl the Guss-Ostrogrdski theorem to V = φ grd ψ, the we obti V = d div V =φδψ + grd φ grd ψ (ii We write (i for V = φ grd ψ d W = ψ grd φ, d subtrct the correspodig formuls (iii Tke ψ = i (ii, such tht =, d Δψ = } Remrk Coditio V rot V = is useful i prctice i order to recogie the bi-sclr fields The problem of writig bi-sclr field i the form φ grd ψ m be solved usig theorem I fct, solvig the equtio V d r =, we fid the fmil of surfces ψ = cost, which re orthogol to V, hece V grd ψ Fill, we idetif φ such tht V = φ grd ψ The lst tpe of fields, which will be cosidered here, is tht of the "hrmoic" fields Eve though the re more prticulr th the previous oes, these fields re the object of wide prt of mthemtics, clled hrmoic lsis 4 efiitio Let be domi i R, d V ( We s tht V is hrmoic field iff it is simulteousl soleoidl d irrottiol i The sclr field U : R is sid to be hrmoic iff ΔU = (ltertivel we c s tht U is hrmoic fuctio A sigifict emple of hrmoic field is V r = k r 5 Theorem Let be ope d str-like set i R, d let V : T be field of clss C o The V is hrmoic o iff there eists sclr field U : R, of clss C o, such tht ΔU = d V = grd U i C R 47

Chpter IX Elemets of field theor Proof V is irrottiol iff V = grd U O the other hd, V = grd U is soleoidl iff ΔU = } The followig theorem shows tht the hrmoic fields re determied b their vlues o the frotier of the cosidered domi 6 Theorem Let R be ope d str-like, d let Ω be regulr compct domi, bouded b S (i If the hrmoic fuctios U d U re equl o S, the the re equl o Ω (ii If the hrmoic vector fields V d V hve equl compoets log the orml to S (t ech poit, the V d V re equl o Ω Proof (i For U = U U, we hve ΔU = o Ω, d U S = If we ote V = U grd U, the div V = grd U + UΔU = grd U, d V U = U = o S Cosequetl, ccordig to the Guss-Ostrogrdski theorem, grdu d, hece grd U = So we deduce tht U = costt, d more ectl, U = o sice U S = I coclusio, U = U o Ω (ii Let us ote V = V V Sice V = V o S, we deduce tht V = o S Becuse V is hrmoic, there eists U : R such tht V = grd U If we cosider W = U V, it follows tht div W = UΔU + grd U = grd U o, d W = UV = o S Usig gi the Guss-Ostrogrdski theorem, we obti tht grd U d, hece grd U = Cosequetl, V = o, ie V = 48 V } 7 Remrk The problem of fidig the (uique hrmoic field, which is specified o the frotier, is specific to the theor of differetil equtios with prtil derivtives of the secod order Without other detils of this theor, we metio tht lot of properties of the hrmoic fields re cosequeces of the previousl estblished results cocerig other prticulr fields For emple, from propositio, it follows tht ds ds, d S S

S ds IX Prticulr fields wheever φ d ψ re hrmoic fuctios A problem which leds to the solutio of the Poisso equtio Δφ = λ is tht of determiig field of give rottio d divergece: 8 Propositio The field V : T for which rot V =, div V = b, where is give vector field, d b re give sclr fields, is determied up to the grdiet of hrmoic field Proof We serch for solutio of the form V = V + V, where rotv rotv d Becuse V divv b divv is irrottiol, there eists sclr field φ o such tht V = grd φ The secod coditio o V gives Δφ = b If φ deotes prticulr solutio of this equtio, ie Δφ = b, the φ = φ + Φ, where ΔΦ= Cosequetl, V = grd φ + grd Φ (* Now, bout V, we remrk tht div = div rot V =, hece is soleoidl, d V is vector potetil of As usull, this potetil is determied up to grdiet, ie V = V + grd ψ, where V is prticulr vector potetil of O the other hd, becuse div V =, we obti Δψ = div V, which is other Poisso equtio If ψ is prticulr solutio of this equtio, the m write ψ = ψ + Ψ, where ΔΨ = Cosequetl, V = V + grd ψ + grd Ψ (** Usig (* d (** we obti V = V + grd (φ +ψ + grd Ξ where Ξ = Φ + Ψ is rbitrr hrmoic fuctio } 9 Remrk The w of provig the bove propositio furishes prcticl method of fidig vector field for which we kow the rottio d the divergece The cocrete determitio of the left fuctio Ξ is depedet o the form of, d of the vlues imposed o the frotier of Solvig the Lplce equtio o uder give coditios o Fr is specific problem i the theor of differetil equtios with prtil derivtives of the secod order (see pproprite bibliogrph 49

Chpter IX Elemets of field theor PROBLEMS IX Verif tht the followig fields re irrottiol d fid their potetils: (i V = ( + i + ( + j + ( + k i Crtesi coordites; (ii F = l i sphericl coordites; (iii W = r si t l + r cos t l + r si t l i clidricl coordites Hit Evlute the rottio i the correspodig coordites d evlute the lie itegrls of V dr o prticulr broke curves (s i the previous prgrphs Show tht the field V = r ( r is soleoidl d fid oe of its vector potetils, where is costt (fied vector, r = i + j + k is the positio vector of curret positio i = R, d r = r Hit div V = (r( r + r ( r = + r r ( r ( r =, where, hece V is soleoidl Sice this propert is itrisic, we c choose the referece sstem such tht sts log the is of Crtesi sstem Becuse r hs ivrit form i + j + k, the problem reduces to fid the vector potetil of the field i j k V = r = r (i + j Becuse lookig for the vector potetil of the form W = (W, W, s i theorem, leds to icoveiet itegrls, we m tr other forms, eg W = (W,, W I this cse we hve to itegrte the sstem W r W W r W 5

IX Prticulr fields We fid W = r + φ(,, W = ψ(, d ccordig to the secod equtio, I prticulr, we c choose φ = ψ =, hece vector potetil is W = r k = r Show tht the followig fields re soleoidl, but ot irrottiol, d determie vector potetil for ech oe: (i V = i j + k (ii V = i + j k Hit (i Followig theorem, we obti W = + φ(,, W = + ψ(,, where ; eg φ =, ψ = (ii Similrl, we fid W = + φ(,, W = + ψ(,, where, s for emple φ = φ ( d ψ = ψ( 4 Show tht for ever irrottiol d soleoidl field V, we hve grd (r V + rot (r V + V =, where r is the positio vector of the curret poit Hit grd (r V = r rot V + V rot r + V r, d r V rot (r V = V r + r div V V div r V r 5 Evlute the divergece d the rottio of the fields; (i r f(r grd r + r (ii f grd g g grd f (iii r ( r 6 We ote u = r, where is costt vector d r is the positio vector (s usull, u deotes the orm of u Fid the coditios o the rel fuctios F d G of rel vrible, such tht: (i u F(u is irrottiol (ii G(u is hrmoic 5

Chpter IX Elemets of field theor Hit (i rot [u df F(u ] = (grd u + u + F(urot u, where du ( r grd u =, d rot u = r df Fill we obti u + F = du dg d G (ii ΔG = div grd u + (grd u, where div grd u = du du (grd u = dg d G Cosequetl, ΔG = implies + = u du du u, d 7 Verif tht the followig fields re bi-sclr, d write them i the stdrd form φ grd ψ: (i V = ( i + ( j + k (ii V = grd f + f grd g (iii V = r ( r Hit (i V rot V = We write the equtio of the surfces, which re orthogol to the field lies ( d + ( d + d =, i the form d d d, we deduce = costt From the reltio V = φ grd we idetif φ = ( (ii rot V = grd f grd g, hece V rot V = The surfces orthogol to the field lies hve the equtios fe g = costt, hece V = φ grd (fe g, where φ = e g (iii Write V = r ( r r ; the orthogol surfces re coes of verte d is, of equtio r = Cr, where C = costt Like before, V r = φ grd, where φ = r r 8 Let { e, e, cos e } be the locl bse i sphericl coordites (ρ, φ, θ, d let V = e cosθ e be vector field Show tht V is bi-sclr si d fid the sclr fields f d g for which V = f grd g 5

Hit rot V = cos si d r = e, hece V rot V = Usig the formul e dρ + ρsiθe dφ + ρe dθ, IX Prticulr fields the equtio V d r =, of the surfces orthogol to V, becomes d si d, cos This equtio hs solutios of the form ρ cos θ = C, ie g =ρ cos θ From V = f grd (ρ cos θ, where grd g = ρ(cos θe siθcosθe, we deduce f = si 9 Show tht the field V ( r = r, where is costt vector, 5 r r d r is the positio vector, is hrmoic, d fid sclr potetil of V Hit rot V =, div V = ; V r = grd U, where U = r + cost etermie the hrmoic fuctios (sclr fields which deped ol o oe of the sphericl coordites ρ, φ or θ Hit If U depeds ol of, the the Lplce equtio, ΔU =, reduces U to If so, it follows tht U(ρ = c + c Similrl, U(φ = c φ + c, d U(θ= c l tg + c etermie the field V : R T, for which rot V = ( i + ( j + ( k d div V = Hit ecompose V = V + V, where rot V =, div V =, d rot V = ( i + ( j + ( k, div V = It follows tht V = grd φ, where Δφ = Tkig φ = 6, 5

Chpter IX Elemets of field theor we obti φ = φ + Φ, where ΔΦ = Cosequetl, V = i k + grd Φ O the other hd V = V + grd ψ, where V = W i + W j + k is prticulr vector potetil of ( i + ( j + ( k I prticulr, f g W = + g(, d W = + f(,, where = Tkig for emple f =, g =, we obti V = ( + i + ( + j + grd ψ Coditio div V = leds to Δψ =, which is verified b ψ =, hece ψ = + Ψ, where ΔΨ = Sice grd ψ = k, we obti V = ( + i + ( + j +k + grd ψ The solutio of the problem is V = V + V = = ( + + i + ( + where Ξ = Φ + Ψ is rbitrr hrmoic fuctio j + ( k + grd Ξ, 54

CHAPTER X COMPLEX INTEGRALS X ELEMENTS OF CAUCHY THEORY B its costructio, the comple itegrl is similr to the rel lie itegrl of the secod tpe Ad et, the properties of the comple itegrls of the derivble fuctios re so importt from both theoreticl d prcticl poit of view tht the re frequetl qulified s ucleus of the Clssicl Mthemticl Alsis As emples of remrkble results we metio the fudmetl theorem of Algebr ( Alembert, the uifictio of the itegrl d differetil clculus (Cuch theorems llowig the evlutio of some itegrls b derivtios, d the pplictios to the rel itegrl clculus (icludig some improper itegrls This prt of the Comple Alsis is kow s Cuch Theor We begi b itroducig the comple itegrl i its most geerl sese, ie for rbitrr fuctios: The costructio of the comple itegrl Let f : C be comple fuctio of comple vrible ( C, d let be simple piecewise smooth curve (the mtter bout ple curves i VI remi vlid sice R ~ C We ote b : I, where I = [, b] R, d (I =, comple prmeteritio of ; more ectl, (t = (t + i (t, where ( t, t [, b] ( t represets the rel prmeteritio of (refresh I for more detils A prtitio of is defied s fiite set of poits o, which is oted { k ( tk : k,, t t t b}, where A = ( d B = (b re the edpoits of The umber ( m{ :, k, } k k is clled orm of the prtitio With ech prtitio we ssocite sstems of itermedite poits, which re sets of the form (see Fig X S = { k ( k : k,, tk k tk} Fill, the umbers f, (, f ( k ( k k k S re clled Riem itegrl sums of the fuctio f o the curve, ttched to the divisio d to the sstem S of itermedite poits 55 k

Chpter X Comple itegrls B = ( b R k+ b = t tk+ k k tk k t I t = t A = ( efiitio We s tht f is itegrble o the curve if there eists the limit of the geerlied sequece (et of itegrl sums lim f, (, S C ( Altertivel, if we work with usul sequeces of itegrl sums, the we sk the uiqueess of this limit for ll sequeces of prtitios ( p with (, d ll sstems of itermedite poits p If this limit eists, the we cll it itegrl of f o, d we ote it f ( d The ottio f ( d is sometimes greed, sice the comple itegrl is defied o curves, b log to the rel lie itegrl The first turl questio bout comple itegrl cocers its eistece d evlutio This is solved b the followig: Theorem The cotiuous fuctios re itegrble o piece-wise smooth curves, d their itegrls reduce to rel lie itegrls of the secod tpe More ectl, if f = P + i Q, the f ( d Pd Qd i Qd Pd Proof If Similrl, if we ote k k ik = ( k, where k,, the the vlues of f re f P(, iq(, ( k k k k k i for ll k,, the k k k k ( k k ( k k, k hece ( equls the orm of s prtitio of the rel curve i R If we itroduce these elemets i the comple itegrl sums of f o, the we c seprte the rel d imgir prts of these sums, d we obti: 56

X Elemets of Cuch theor f, (, [ P( k, k ( k k Q( k, k ( k k ] k S + + i [ Q( k, k ( k k P( k, k ( k k ] k It is es to see tht these sums coverge to the rel lie itegrls Pd Qd Qd Pd lim ( lim ( k k [ P(, ( k k k k Q( k, k ( k k ] [ Q(, ( k k k k P( k, k ( k k ] These limits eist ccordig to theorem } 4Corollr If f : C is cotiuous fuctio, d is piecewise smooth curve of prmeteritio : [, b]c, the f ( d = b ( f ( t ( t dt Proof The comple prmeteritio (t = (t + i (t, t [, b], comes from the rel prmeteritio ( t, t [, b], ( t hece the hpothesis tht is piece-wise smooth mes tht the fuctios d (d cosequetl hve cotiuous derivtives o fiite umber of subitervls of [, b] The rel lie itegrls from the previous theorem become defiite Riem itegrls o [, b], ie b / / Pd Qd P( ( t, ( t ( t Q( ( t, ( t ( t dt, b / / Qd Pd Q ( ( t, ( t ( t P( ( t, ( t ( t dt To ccomplish the proof, we replce f = P + i Q, d = + i i these itegrls, d restri the result i comple form } 5 Emple The fuctio f : C \ { }C, of vlues f ( (, is itegrble o the circle C(, r, cetered t, of rdius r >, d d i C(, r I fct, the itegrl eists becuse f is cotiuous o = C \ { }, hece lso o C(, r, d the circle (trced oce is simple smooth curve i Usig the comple prmeteritio of this circle i t ( t r e, t [, ], / 57

Chpter X Comple itegrls / i t we obti ( t ri e, d ( f ( t Cosequetl, ccordig to i t r e the formul from Corollr 4, the vlue of the itegrl is d i dt i C(, r The geerl properties of the comple itegrls correspod to the similr properties of the rel lie itegrls of the secod tpe: 6 Theorem The followig reltios hold for cotiuous fuctios o piece-wise smooth curves: (i ( f g( d f ( d g( d,, C, f, g C ( C (kow s lierit reltive to the fuctio; (ii f ( d = f ( d + f ( d, f C ( C (clled dditivit reltive to the coctetio of the curves; (iii f ( d f ( d, f C C (, where d re cotrril trced (med oriettio reltive to the sese o Proof Without goig ito detils, we recogie here the similr properties of the rel lie itegrls of the secod tpe, hece it is eough to recll the coectio estblished i Theorem } The followig propert of boudedess remids of rel lie itegrls of the first tpe, sice it ivolves the legth of curve 7 Theorem (Boudedess of the comple itegrl Let f d be s i the costructio If M sup f (, d L is the legth of, the f ( d M L Proof Becuse I = [, b] is compct set i R, d the prmeteritio is cotiuous, it follows tht = (I is compct set i C The cotiuit of f ssures the eistece of M, such tht f ( M t ll Sice the smooth curves re rectifible (ie the hve legth, there eists L def sup { : } k k k Cosequetl, for the modulus of the itegrl sums, we obti f, k k (, S f ( M ML, k k k k k k where it is eough to tke the limit ( } 58

X Elemets of Cuch theor 8 Corollr Let be domi i C, d let be piece-wise smooth curve For ech N, we defie fuctio f : C, which is cotiuous o If the sequece ( f is uiforml coverget o, the lim f ( d lim f ( d Proof B hpothesis, u f lim ( N, such tht > ( implies M ot sup f f mes tht for ech > there eists ( f ( < L, where L stds for the legth of Accordig to theorem 7, we obti f ( d f ( d, which is reformultio of the climed equlit } 9 Corollr Let fuctio f be lticll defied b f ( ( If is piece-wise smooth curve i the disk of covergece of this power series, the we m itegrte term b term, ie ( d f ( d Proof The prtil sums of the give power series c pl the role of f i the previous Corollr } Remrk A importt propert of the rel lie itegrls of the secod tpe cocers the idepedece o curve I VI, we hve see tht this is the cse of coservtive fields, which derive from potetil Simple emples (see the problems t the ed, s well s I i Emple 5, etc show tht the comple itegrl geerll depeds o the curve of itegrtio However, if the itegrted fuctio is C-derivble, the its itegrl does ot deped o curve, but ol of its edpoits The followig theorem sttes coditios for this cse, which will be ssumed i the etire forthcomig theor: Cuch s Fudmetl Theorem Let f : C, where C, be C-derivble fuctio, d let be simple, closed, piece-wise smooth curve If the iterior of is icluded i, ie (, the f ( d = 59

Chpter X Comple itegrls Proof (bsed o the dditiol hpothesis tht P Re f, Q Imf CR ( The itegrbilit of f o is ssured b Theorem The dditiol hpothesis llows us to use the Gree s formul from VII, which gives Q P Pd Qd = dd ( P Q Qd Pd = dd ( Becuse f is derivble o, hece lso o (, it follows tht the Cuch- Riem coditios hold, hece the double itegrls from bove vish It remis to use Theorem } Remrk The ssertio of the bove theorem is correct without dditiol hpotheses, but the proof becomes much more complicted (see for emple [CG], [G-S], [H-M-N], [MI], etc Before discussig more cosistet cosequeces of the bove Theorem, we metio severl immedite corollries, which re lso sigifict for the reltio betwee comple d rel itegrls These properties hold o domis of prticulr form: efiitio We s tht domi C is simpl coected if the iterior of ever closed curve from is lso i, ie ( I the cotrr cse, whe there eist curves for which (, we s tht is multipl coected (w, is coected, sice domi mes ope d coected Here we void further cosidertios o the order of multiplicit (bsed o homotopic curves, d other properties of the domis, but the iterested reder m cosult [BN], [G-S], [LS], etc 4 Emples ( The followig sets re simpl coected: C, C, { C : Re }, d other hlf-ples; isks (ie iterior of circles, d iterior of simple closed curves; Arbitrr itersectios of simpl coected sets (b Most frequetl, the multipl coected sets hve the form: C \ { }, C \ F, where F C is fiite, C \ N, etc (, r \ { }, ie disks without ceter, (, r \ F, where F is fiite (or eve ifiite set of missig poits; Coected sets with missig poits or missig sub-domis 5 Corollr If C is simpl coected domi, d f : C is derivble fuctio, the: (i The itegrl of f does ot deped o the curve; (ii f hs primitives o ; (iii The Leibi-Newto formul holds for the itegrl of f 6

X Elemets of Cuch theor Proof (i Let d be two curves i, which hve the sme edpoits, s A d B The curve =, obtied b coctetio, is closed, d sice is simpl coected, we hve ( Accordig to Theorem, it follows tht f ( d =, hece b virtue of properties (ii-iii from Theorem 6, we obti f ( d f ( d = (ii We fi, d we prove tht the fuctio F : C, of vlues F( f ( d is primitive of f, ie F is derivble o d F / = f I fct, for ech, there eists (, smll eough to ssure the implictio Becuse ( d holds t ll d i C, we m write ot F( F( E(, f ( = d = f ( d f ( d f ( f ( Usig the idepedece o curve of the lst itegrl, we m evlute it o the stright-lie segmet [, + ] For this itegrl, the Propert 7, of boudedess, holds with L, hece E(, M ot m { f ( f ( : [, ]} The derivbilit of f implies its cotiuit, hece for ech > there is ( >, such tht mi{ (, ( } implies f ( f ( I this situtio, fortiori f ( f (, hece F( F( f ( lim (iii We hve to show tht the formul ot f ( d G( G( F / ( holds for rbitrr,, d for rbitrr primitive G of f I fct, the previous propert (ii poits out prticulr primitive of f, mel F( f ( d Becuse the differece of two primitives of the sme fuctio is costt, ie (F G / = implies the eistece of CC, we hve F( G( = C t ll I prticulr, we tke here = d = } 6

Chpter X Comple itegrls We strt the series of mjor cosequeces of the fudmetl theorem b the cse of multipl coected domi: 6 Theorem Let C be domi, d let,,, be pir-wise disjoit closed (d, s usull, simple d piece-wise smooth curves i If f : C is derivble fuctio o, d ( \ ( k, the k f d ( = k k f ( d Proof The ide is to ppl the fudmetl theorem to f d some closed curve, for which ( To mke it possible, we tke d Bk k for ech k,, eg the closest poits betwee d k, d we coect them b stright-lie segmets (s i Fig X The sought for curve results b the followig coctetio: A A ] [ B, A ] A A [ B, A [ A, B ] A k = Im B A A B B A = Re Fig X Cosequetl, ccordig to 6 (ii, we c decompose the itegrl o i sum of itegrls o the costituet rcs Becuse the segmets [A k, B k ] d [B k, A k ] re opposite i order, we hve f ( d f ( d = I dditio A A [ A k, B k ] [ B k, A, hece A A A A A A f ( d A A f ( d k ] A A f ( d To complete the proof, we ppl the propert 6 (iii o d the fudmetl theorem o, ie f ( d k, k,, f ( d = } 6

X Elemets of Cuch theor 7 Remrks The mi pplictios of the bove theorem cosist i reducig the itegrl o to itegrls o k, which i geerl re simpler I prticulr, if =, the f ( d f ( d, wheever \ ( ( This propert is frequetl formulted i terms of cotiuous deformtio of to, relied iside For emple, usig 5, we obti tht d i holds for rbitrr curve uder the coditio ( To complete the list of vlues of this itegrl, we metio tht it vishes if (, ie is eterior to, d the cse is udecided (see the et sectio The et theorem, b Cuch too, sttes remrkble reltio betwee itegrls d derivtives: 8 Theorem (Cuch formuls for derivble fuctios Let f : C be derivble fuctio o the domi C If is closed (simple d piece-wise smooth curve i, such tht (, the the formul (! f ( f ( d i ( holds t ever (, d for rbitrr N Proof Cse = We hve to show tht f ( d f ( i Usig the result i Emple 5, d Remrk 7, this reltio becomes f ( d d f ( Becuse f is derivble t, there eists M > such tht the iequlit f ( f ( M holds t ll i eighborhood of If we replce = C(, r, the ccordig to Theorem 7, we hve f ( f ( d M r C(, r It remis to tke r The remiig cses ivolve mthemticl iductio d will be omitted (the iterested reder m cosult [CG], [G-S], etc We metio tht importt prt of the proof cocers the implicit ssertio of the theorem, mel the eistece of ll higher order derivtives t ech poit of the domi where f is oce derivble } 6

Chpter X Comple itegrls 9 Corollr If the fuctio f : C is derivble o the domi C, ( ( the it is ifiitel derivble o, ie there eist f t ech, d for ll N Remrks We m use the Cuch formuls for derivble fuctios to evlute comple itegrls b the simpler w of derivtio The cse i 5 is immeditel recovered from Theorem 8, pplied to the ideticll costt fuctio f :C{}; o derivtio is ecessr The sme theorem, pplied to the sme fuctio, leds to d C(, r ( t ll C, d for ll r >, N * A lot of comple itegrls llow the form of the Cuch formuls, hece we c clculte them b derivtios For emple, C( i, ( d ( i C( i, ( i follows for = i, =, d ( i d i ( i f The Cuch formuls hve importt theoreticl cosequeces: Theorem (boudig the derivtives Let f : C be derivble fuctio o the domi C, d let If = C(, r, d r is smll eough to esure the iclusio (, the (! M (, r f (, r where M (, r sup { f ( : C(, r}, d N Proof If we ppl Theorem 7 to the Cuch formul for f (, the we get (! f ( f ( L sup : C(, r, where L = r is the legth of } Theorem (Liouville If fuctio is derivble o the etire comple ple, d bouded, the it is ecessril costt Proof Usig the hpothesis of boudedess, we m ote M sup f (, / i C such tht the iequlit M (, r M holds for rbitrr C, d r > Accordig to the previous theorem, writte for =, we hve / M f ( r r Cosequetl f / = o C, hece f reduces to costt } 64

X Elemets of Cuch theor Fill, remrkble cosequece of the Cuch Theor is the followig propert of C of beig lgebricll closed, which is cosidered to be the fudmetl theorem of the Algebr: Theorem ( Alembert Ever polomil P, of degree, with coefficiets from C, hs t lest oe root i C Proof I the cotrr cse, whe P ever vishes o C, we c defie fuctio f : CC, which tkes the vlues f ( P ( Accordig to the lgebric properties of the derivble fuctios (discussed i Chpter IV, it follows tht f is derivble o C I dditio, f is bouded I fct, becuse lim (, we hve lim f (, hece there eists some r >, such P tht f ( wheever r The boudedess of cotiuous fuctio o compct sets gurtees the eistece of umber M r >, such tht r f ( Sice f is derivble d bouded o C, the Liouville s Theorem ss tht f is costt, which is ot the cse if } The list of cosequeces of the Cuch s fudmetl theorem cotiues with m other remrkble results, icludig those from the et sectio Without goig ito detils, we eouce here etesio of the Corollries 8 d 9, which shows tht the Cuch Formuls resist to limitig process: 4 Theorem (Weierstrss Let C be domi If: (i is closed (simple d piece-wise smooth curve i, d ( ; (ii f :C re derivble o, N; (iii there eists F u the there eists lim u ( lim f ( : ( C is derivble; u ( k ( ( k f ;, such tht (b lim f, k N; d M r ( k k! F( (c ( d, (, dk N i k ( I prticulr, we m ppl this theorem to series 65

Chpter X Comple itegrls PROBLEMS X Evlute the itegrls o [, ] of the followig comple fuctios: t i i t i t ( f ( t, (b g( t e, (c h( t si( i t, (d sig ( i t t i i t Hit Idetif the rel d imgir prts of the give fuctios, d itegrte them seprtel I the emple (, from we obti t t f ( t i, t t t t f ( t dt dt i dt = t rctgt i l( t t t Evlute the comple itegrl I ( d log the followig curves: ( Stright-lie segmet = [ i, i] ; (b Left-hd hlf-circle cetered t, of rdius ; (c Broke lie [ i,] [, i] Hit The itegrl refers to the rel fuctio, but the vrible is comple Replce d d from the prmeteritio of the curve, ccordig to the formuls i d 4 Stud whether ( i d depeds o or ot, where is curve of edpoits = d = + i Hit Accordig to theorem, sice the fuctio i is ot derivble, there re chces the itegrl to deped o the curve To poit out this depedece, evlute the itegrl o the stright-lie segmet [, + i], o the broke lie [, ] [, i], d o rcs of prbol, circle, etc 4 Let be closed (simple d piece-wise smooth curve i C Show tht the re of ( hs the epressio A d i Hit If we ote = + i, the Theorem leds to rel lie itegrls d d d i d d The rel prt of this epressio vishes s itegrl of totl differetil d d d( Accordig to Propositio VI5, the imgir prt equls A 66

X Elemets of Cuch theor 5 Some of the followig comple itegrls c be clculted b Leibi- Newto formuls Idetif them, d fid their vlues I = e i d ; I = e si d ; I = i d cos d ; I 4 = ; I5 = d i C(, Hit The method is workig for I (s i R, I (b prts, d I (chgig I I 4, the domi C \ {} is ot simpl coected, d i I 5 we hve multi-vlued fuctio However, there eist coveiet cuts d 6 Evlute I ( r, where r C(, r Hit If r (,, the I(r = ccordig to the fudmetl theorem If r >, the usig theorem 6 with coveiet r, r >, we obti d d I ( r C (, (, i r C i r Tkig ito ccout the emple 5, d the decompositio, i i i i we fid I(r = gi 7 Usig the Cuch formuls, evlute the itegrls: d I = ; I 5 = I = i C(, si d ; I 4 = C(, C(, e d ; I 5 = si 5 ( d si ; d Hit I I, use the Horer schem to fid the roots of the deomitor I I, the Cuch formul holds with =, d = I I, put forwrd the fuctio si I I e si 4, we hve lim, d i I 5, lim si 8 Evlute F( ( d, where ( k F( k k Hit The power series of F is coverget i the uit disk, where we c / use the Weierstrss theorem Tke ito ccout tht F ( k 67

X RESIUES This sectio is further developmet of the ide of reducig the itegrls to derivtives vi specil (mel Luret power series The theoretic bsis follows from the Cuch theor, which hs bee sketched i the previous sectio From prcticl poit of view, the iterest is to gi ew powerful tools for the clculus of comple s well s rel itegrls Theorem (Luret Let C be domi, d let C be poit such tht (, r, R for some r, RR +, r < R, where ot (, r, R { C : r R} represets the crow cetered t, of rdiuses r d R (s i FigX If f : C is derivble o, the, t ech (, r, R, its vlue is p ( f ( ( p ( Proof We epress f ( b the Cuch formul, usig the closed curve C R r [ A, B] C [ B, A] Im R C R r B C r O A Re FigX I fct, ( (, r, R, hece Theorem X8 gives f ( f ( d i Becuse r C [ A, B] C [ B, A] R d, we obti [ A, B] [ B, A] f ( f ( f ( d C R d i i C r I the lst itegrls, hs differet positios reltive to d, mel: 68

X Residues 69 Cse I R C, hece Cosequetl, i R C we hve ( = Becuse, the lst frctio is the sum of geometric series, ie holds for ll R C, i the sese of the uiform covergece If we put ( ( ( i R C, the we m itegrte term b term, d so we obti C R d f i ( = (, where R C d f i ( ( We clim tht this series (of positive powers is u coverget reltive to,, ( R r I fct, the remider of order equls R ( k k k def = ( ( ( k C k k R d f i Usig the u covergece of the geometric series o,, ( R r, we fid R = d f i R C k k ( ( ( = = d f i R C ( If we ote, the we hve R Accordig to Theorem X7, the followig iequlit holds R R R R M, where } : ( { R C f M sup From < R we deduce limr =

Chpter X Comple itegrls Cse II Cr, hece ( I this cse, i C r = we write Cosidertios similr to those from Cse I led to the developmet f ( d = i C r p, p p ( where p p f d i C ( ( r A similr evlutio of the remider def r p k k k p ( leds to the coclusio tht limr p = Combiig the two cses, we obti the proof of the climed equlit f ( = p + p ( p ( i the sese of the u covergece of ech series o (, r, R } Remrks We usull refer to the series p ( ( p ( itroduced b Theorem, s Luret series This series hs two etries, d its covergece mes cocomitt covergece of the two series, (, clled regulr prt (Tlor, of positive powers, etc, d p, clled pricipl prt (of egtive powers, etc p p ( The formuls for d p re similr, vi the correspodece! p The itegrl i looks like the Cuch formul for f ( (, but geerll spekig Theorem X8 cot be pplied, sice C R The coefficiet hs direct coectio with the itegrl of f, mel f ( d i C r Its specil utilit i evlutig itegrls justifies its distiguishig me: 7

7 X Residues efiitio Let us cosider,, (, r, R, d f, s i the bove Theorem If the hpotheses of this theorem re stisfied for rbitrril smll rdiuses r >, r < R, the we s tht the Luret series is developed roud I this cse, the coefficiet of the correspodig Luret series is clled residue of f t, d we ote = Re (f, = Re f (, etc 4 Emple Fuctio f ( ( is defied o C \ {, } To obti its Luret series i (,,, we decompose it i simple frctios f ( Becuse, these frctios represet sums of the geometric series, ( ( Cosequetl, the Luret series of f i the crow (,, gives f ( 4 8 The coefficiet = from this Luret series does NOT represet the residue of f t =, sice the developmet is ot vlid roud To get this residue, we write the geometric series for (which implies, ( 4 8 The resultig Luret series (roud hs ol positive powers, mel 7 f ( 4 8 hece Re (f, = To obti Re (f,, we cosider i the geometric series ( ( ( d we write the Luret series of f roud, which is f ( ( ( Cosequetl, Re (f, = Similrl, we fid Re (f, = +

Chpter X Comple itegrls 5 Remrk We c develop fuctio f : C roud C i the followig cses ol:, is pole, or is essetil sigulr poit I the first cse, is regulr poit, d the Luret series cotis ol positive powers; more ectl, it reduces to Tlor series / f ( ( ( ( f f f ( (!! I fct, ccordig to the fudmetl theorem X, the coefficiets of the Luret series (see Theorem hve the vlues p p f d i C ( ( =, p N * r d ccordig to the Cuch formuls X8, f ( f ( C d i R ( =, N! Ecept Cse, c be uivlet isolted sigulr poit The pricipl prt of the Luret series roud cot vish more, d the sigle differece we c mke refers to the umber of terms If the pricipl prt of the Luret series roud hs fiite umber of terms, ie the // ( p f ( ( ( p ( p lim ( f ( p is fiite, hece is pole (Cse The remiig possibilit for the pricipl prt of the Luret series roud is to coti ifiitel m terms Here we recogie Cse, p whe lim ( f ( does ot eist, p N * The evlutio of the residues t poles reduces to derivtios: 6 Propositio If f : C hs pole of order p t, the p d p Re (f, = ( f ( ( p! d p Proof B hpothesis, fuctio f hs the developmet p f ( ( ( p ( p The resultig developmet of the fuctio ( ( f ( hs Tlor ( p coefficiets, hece ( ( p! } 7

7 X Residues 7 Corollr The residue of meromorphic fuctio, s A f, where B A d B re derivble o, t simple pole, is A( Re (f, = / B ( Proof It is es to see tht f hs pole of order p t iff F f hs ero of the sme order t this poit I our cse, this mes tht A (, B (, d B ( To complete the proof, we tke p = i the previous propositio } 8 Emples (i Fuctio f ( hs simple pole t = (divide si power series, if ot coviced The bove corollr gives Re (f, = (ii The sme poit = is double pole of the fuctio g( For si p =, the formul from the Propositio 6 leds to d Re (g, = si cos = lim = d si si I this cse, the method of opertig with series seems to be simpler th derivig I the Luret series we m look ol for = Re (g, (iii To fid the residue of fuctio t essetil sigulrit, we hve to develop i Luret series, d idetif the coefficiet For emple, k / k h( = ( si k (k! hs essetil sigulrit t =, d = Re (h, = 9 Remrk As we hve lred metioed i IV5, the clssifictio of the sigulr poits refers to the poit t ifiit too I prticulr, c be uivlet isolted sigulr poit of fuctio f : C, which mes tht there eists some r > such tht ot { ( C (, r { C : r} This coditio shows tht is the ol sigulr poit i eighborhood V (, r def { ( C (, r {} I the spirit of Theorem, we m iterpret { ( C (, r s crow (, r,, d the Luret series s developmet roud If this series cotis ol egtive powers of, ie it hs the form p f ( p,

Chpter X Comple itegrls the we cosider tht is regulr poit (sice f ( mkes sese I the cotrr cse, whe positive powers do eist, we cosider tht is sigulr poit, d we mke the distictio betwee pole d essetil sigulrit b the umber (fiite or ifiite of positive powers For emple, i the crow (,,, the fuctio f ( ( from 4, hs the developmet f ( = ( ( Cosequetl, is regulr poit of f O the other hd, is simple pole of the fuctio (, d essetil sigulr poit of e Recllig the fil prt of the Remrk, we m spek of residue t, which should be turll relted to the itegrl o C r Becuse the border of (, r, is (, r, this residue should ssure the reltio Re (f, = f ( d i C (, r More ectl, the epressio of the Luret coefficiets i Theorem, re suggestig the followig: efiitio Let be uivlet isolted sigulr poit of the fuctio f : C, d let r > be umber for which (, r, If is the coefficiet of / i the Luret series of f i (, r,, the is sid to be the residue of f t, d we ote C def Re (f, Altertivel, if we ote g( = f (/, the the chge leds to g( f ( d = C (, r d, C(, r hece we m defie the residue of f t b the coefficiet b i the Luret series of g i the crow (,, r There is o formul similr to tht i Propositio 6 for the evlutio of the Re (f,, so we hve lws to relie Luret series roud Emples (i For the bove fuctio to 6 d 9 we hve Re (f, = f ( ( (compre (ii The vlue Re ( (, = 4 results from the developmet 4 8 ( = which holds i (,, The vrit of replcig leds to the sme result, but it is bsed o the developmet of the fuctio ( i the crow (,,, mel 74

X Residues 4 8 ( (iii is essetil sigulr poit of e, d Re ( e, = comes out from the ver defiitio of the fuctio ep, mel, which!! mkes sese i the etire C = (,, {} The clculus of the itegrls b residues is bsed o the followig: Residues Theorem (Cuch Let C be domi, o which the fuctio f : C is derivble, d let be closed (simple d piece-wise smooth curve If,,,, re the ol uivlet isolted sigulr poits from (, ie ( \ {,,, }, the Proof Let k = C( k, r k, tht ( \ ( k k f ( d i Re ( f, k, k k, be disjoit circles (s i FigX, such Accordig to Theorem X6, we hve f d ( = k k f ( d = Im = Re Fig X B Theorem, the coefficiets roud k, k,, hve the vlues k i It is eough to replce k k k f ( d 75 i the Luret developmets of f def Re ( f, k f ( d i f ( d }

Chpter X Comple itegrls Emple Let us evlute I 76 f ( d, where C(,, N *, d f ( ep The first step is to fid out the sigulr poits of f, d the secod oe is to estblish wht sigulrities re i ( ; fill, we ppl the Residues Theorem, d we mke clculus Fuctio f is ot defible t the poits =, =, =, d The ture of these poits is: is essetil sigulrit, d re simple poles, d is regulr poit Obviousl,,, } (, d ( { {,, } ( for ll, sice f is derivble i { ( C (, Theorem furishes the followig vlues of the itegrls: I = i Re ( f,, I = i [Re ( f, + Re ( f, ], I = i [Re ( f, + Re ( f, + Re ( f, ] I dditio, we hve I = I for ll, d ltertivel, I = i Re ( f, Now, we evlute the residues From the developmet i (,,, 7 f ( 4 8,!! it follows tht Re ( f, =! e Usig the formuls for simple poles (see 6 d 7, we obti ep( ep( Re ( f, = e, d Re ( f, = e Fill, becuse i (,, we hve f ( =,!! it follows tht Re ( f, = To coclude, we metio the vlues i e e if I i e if if This emple obes the specific restrictio i Theorem, which sks the sigulr poits of f be either i the iterior or i the eterior prt of the (closed curve from the itegrl A turl problem is to fid the vlues of the itegrls J f ( d, where C(,, or L f ( d, where { i : }, etc To solve such problems, we dd couple of improvemets to Theorem :

X Residues 4 Semi-Residues Theorem Let the objects, f,, d,,, stisf the hpotheses of Theorem, d i dditio, let be simple pole of f If is gulr poit of, where the tget jumps b rdis i the positive sese of, the f ( d i Re ( f, + ( i Re ( f, k I prticulr, if belogs to smooth sub-rc of, the k k f ( d i Re ( f, + i Re ( f, Proof The Luret series of f roud hs the form f ( = ( k, ( where is derivble i eighborhood of Let r be rc of circle i such eighborhood, which isoltes s i FigX Obviousl, r is closed (simple, d piece-wise smooth curve, o which we hve r AB f ( d i Re( f, k r k r r A r A r B ( B (b FigX O the other hd, we m decompose this itegrl i two prts f ( d = f ( d + r f ( d AB r For the first itegrl we hve lim f ( d = f ( d The secod r oe c be decomposed i two itegrls ccordig to the form of f, mel d f ( d = r r + ( d r AB 77

Chpter X Comple itegrls Usig the prmeteritio = + r e, t [, ], of r, we obti d = r i d = ( i = ( i Becuse r is compct set, d fuctio is cotiuous o r, there eists M = sup { ( : r } < I dditio, the legth of r hs the form L = ( r, hece the propert of boudedess (X7 gives If we tke r r it ( d M ( r r i ll the itegrls from bove, the we obti i Re( f, = f ( d ( i, k where we hve to replce k = Re ( f, I prticulr, if belogs to smooth sub-rc of, the there is o jump of the tget, ie = } The vlues of J d L from re give i Problem 7 t the ed I the fil prt of this sectio we will ppl the residues theor to the rel itegrl clculus The mi difficult rises from the curves o which we itegrte: the Residues Theorem holds o closed curves, while the rel itegrls re defied o prts of R, which re o-closed curves Therefore we re iterested i costructig closed curves b ddig etr curves The problem is to cotrol the itegrls o these dditiol curves, which usull mes tht we m eglect these itegrls i limitig process: 5 Jord s Lemm # Let C be domi, o which the fuctio f : C is derivble, d let C be fied With verte we cosider gle of vlue (rdis, i which r represets the rc of circle of rdius r, cetered t We suppose tht holds for ll r i eighborhood of r, where r R defies limit positio of r, (s i FigX4 We clim tht if lim m ( f ( = rr the we m eglect the itegrl o r, ie lim f ( d = Proof Let us remrk tht rr r r r ot r r M m ( f (, which ppers i the hpothesis, mkes sese s fiite umber, sice r is compct set d f is cotiuous fuctio We recll tht the legth of r is L r = r Accordig to Theorem X7, we hve 78

X Residues r ( f ( f ( d = r d M r L r = M r r Im r r r r O Re FigX4 Usig the hpothesis tht ( such tht r r ( we obti f ( d } r 6 Emple Jord s Lemm # is useful i fidig the itegrls d I, N, As usull, the first step is to idetif the sigulr poits I this cse, fuctio f ( ( hs simple poles, mel k k ep i, k, Let [ O, A] [ B, O] be closed curve like i FigX5 If we tke r r M r r > d rg, the will be the ol pole i ( r Usig Im i B ( = r e r O A ( = r FigX5 Re 79

Chpter X Comple itegrls the Residues Theorem, we obti i i f ( d i Re( f, = e r O the other hd, this itegrl equls sum of itegrls, mel r d f ( d + f d r ( + ] f ( d r [ B, O Followig the defiitio of improper itegrl, the first term gives r d lim I r Accordig to the bove lemm, pplied to f, t =, d r, we m eglect the secod itegrl, ie lim implies lim f ( d = r r Fill, usig the prmeteritio e i, [r,], we m reduce the third itegrl to the first oe: i d i r d f ( d = e [ B, O] = e r i e To coclude, the limit process r leds to I si 7 Jord s Lemm # Let C be domi, for which { C :Im }, d let the fuctio f : C be derivble o, ecept fiite umber of uivlet isolted sigulr poits If lim f (, the lim r C r i e f ( d =,, where C r is the upper hlf-circle of rdius r, cetered t i t Proof Usig the prmeteritio r e, t [, ], we obti ot i r(costisit i I r e f ( d = C r e f ( r e r i e dt If r is gret eough to iclude ll sigulrities uder C r, the we m ote ot M r m { f ( : C r }, sice f is cotiuous o the compct set C r The propert of boudedess for itegrls o itervls of R, gives I rsit r r e M dt r Becuse fuctio si is smmetric reltive to, this iequlit becomes i t i t Ir rm rsit r e dt 8

Tkig ito ccout tht Ir B hpothesis, rm r r t e dt = M r r M e r si t t t ech t [, ], we obti X Residues r implies M } r 8 Emple Let us show tht the Heviside s fuctio if t ( t if t llows itegrl represettio, epressed b the followig formul e ( t d, i R i To prove this equlit, we first remrk tht the itegrl preserves its vlue if we replce R i b (, ] [, The defiitio of this i t Im - r - R _ i O r r Re - r O - Im _ r r` Re b FigX6 improper itegrl ivolves the curve ( r, ] [, r, sice ot i t e e I d = R i i e d = i t lim d r r We costruct the closed curve, which we eed i the residues theorem, i differet ws, depedig o sig t (compre FigX6 d b Cse t > If we ote r r e i t d = Cse b t < The iterior of r` r r r t, the we obti i t e ire, = i r `r cotis o sigulrit, hece e i t ` d = r Usig the Jord s Lemm #, we eglect the itegrls o `r d r 8

Chpter X Comple itegrls 9 Clsses of rel itegrls To be clculble b residues, the rel itegrls shll hve certi prticulr forms I prctice we hve to idetif the clss the give itegrl belogs to, d to ppl the specific techique A Improper itegrls of rtiol fuctios Let us le the itegrl I P( d, Q( where P d Q re polomils with rel coefficiets The eistece of this itegrl (see V is ssured if Q hs o rel roots, d grd Q grd P + We suppose tht these coditios re stisfied, d we pursue the techique of clcultig its vlue The strtig poit is the ver defiitio of improper itegrl, which (i this cse llows the form I = lim I, where r r r I r r P( d Q( To ppl the Residues Theorem, we costruct the closed curve r [ r, r] r, where r is hlf-circle of rdius r, cetered t (s i FigX7 Im r - r q O r Re FigX7 Let,,, q be the roots of Q, where grd Q = q, i the upper hlfple { C : Im } These poits re poles of the comple fuctio f, of P( vlues f ( If r is gret eough, ll these poles re i ( r, hece Q( r f ( d i Re ( f,, q k where the right-hd member does ot deped o r Wht remis is to decompose this itegrl o the sub-rcs [ r, r] d r, d to tke r, sice the Jord s Lemm # opertes o r k 8

B Itegrls of rtiol fuctios i si d cos Let us evlute J R (sit,cost dt, where R is (rel rtiol fuctio Bsed o the Euler s reltios i t i t 8 X Residues e e e e sit, cost, i it we m chge the vrible t e, t [, ] Becuse this chge of vribles represets prmeteritio of the uit circle, we obti P( J d, C(, Q( where P d Q re polomils with rel coefficiets The vlue of J comes out b the Residues Theorem for the poles i the iterior prt of C (, C Improper Itegrls of rtiol fuctio times cos (or times si Let us cosider itegrls of the form K c i t i t R( cos d, K s R( si d, where R = P/Q is rtiol fuctio for which grd P < grd Q, d > I dditio we suppose tht Q hs o roots i R, d R is eve fuctio i K c, respectivel odd i K s I prctice, if K c is give, possibl o [,, the K s = becuse of prit, d vice vers Therefore we m combie K c d K s i comple itegrl of the form K R e i R( d def lim r r r i e R( d To obti closed curve, we costruct r = [ r, r] Cr, where C r is the upper hlf-circle of rdius r, cetered t (s i Lemm 7 Let us ote b,,, q the roots of Q i the upper hlf-ple (compre to clss A These poits re poles of the itegrted fuctio, which re cotied i the iterior of r wheever r > m { k : k, q } Theorem gives r e i q i R( d = i Re ( e R(, k, k where the right-hd term is idepedet of r If we decompose r e i r R( d = e R( d r i + C r i e R( d, the the limit r voids the itegrl (see Lemm #, d we obti K = q k i i Re ( e R(, k

Chpter X Comple itegrls The reder m fid emples of such itegrls t the ed of the sectio There re m other tpes of itegrls (eg with multi-vlued fuctios, for which we recommed lrger bibliogrph To be more covicig bout the dvtges of usig the Cuch theor, we coclude b severl remrkble rel itegrls, which re esil obtied b residues techiques Remrkble itegrls ( The Poisso s itegrl is defied b sit P dt t Usig the prit of (si t / t, d chgig the vrible, we obti si t si t P P( dt = t dt R t Accordig to the Euler s formul for si, the problem reduces to the Heviside s itegrl, d the result is P = / (b There is remrkble itegrl of clss C i 9, mel cost L = dt, >, t which is kow s Lplce s itegrl Combiig with the correspodig itegrl of si, we obti L = R t e i t Becuse i is the ol (simple pole i the upper hlf-ple, the stud of the bove metioed clss C leds to the vlue i dt e L = ire (, i = (c Kowig the Guss itegrl G = e t dt e, we c evlute F = cos d = si d =, which re clled Fresel s itegrls I fct, if r is the cotour from 6, for = / 4, the r e i d = Itegrtig b prts i the correspodig rel itegrl, it follows (without Lemms # or # tht e d r The itegrl o [O, A] teds to G, d chge of vribles i the itegrl o [B, O] leds to G too r i 84

X Residues PROBLEMS X Write Luret series for the followig fuctios i the specified crows: ( e, (,, ; (, (,, ; ( si, (,, Hit ( Compre to 4 ( Multipl the series of ep b geometric oe ( ecompose si, the write the series of si d cos evelop the followig fuctios i Luret series roud : Hit (i si roud 99 (i ; (ii ; (iii si ( ; (ii ; (iii Trsform the series of ( ( Fid the residues of the followig fuctios t the specified poits: ( Re (, i ; (b Re ( si si, ; (c Re ( 4, ; (d Re ( si, ; (e Re ( si, ; (f Re ( ep (, ; si cos (g Re ( si, ; (h Re (, 5 ; (i Re (, cosh si Hit irectl use 6, 7 i ( d (b (c Prolog to derivble fuctio For (d d (e, use the defiitio of si (f I the series of ep, ech term is the sum of series; idetif the coefficiet of / (g Similrl to (f, use the series of si d idetif the coefficiet of / i si si cos 4 (h Use the developmet = (i ivide the 5 45 series of cos b tht of cosh 4 Stud whether the residues lws vish t regulr poits Estblish the ture of, d (if fid the correspodig residue of the fuctios: ( b c ; / ; e ; si ; e ; si (/ ; 85

Chpter X Comple itegrls Hit Accordig to Remrk 5, Re ( f, = wheever, is regulr poit t fiite distce Otherwise, Re ( f, is ot ecessril ull, eg Re (/, = Re (si, d Re (, mke o sese 5 Use the Residues Theorem to evlute the comple itegrls: e d C(,4 ; d ; si C(, e d ( C(, ; (, cosh d C Hit Fid the poles d the essetil sigulrities i the iterior prt of the metioed circles, d evlute the correspodig residues 6 Let f : C be derivble fuctio, which hs ol fiite umber of uivlet isolted sigulr poits, s,,, Show tht Re ( f, + Re ( f, k = k d Use this fct to evlute I =, where C(, r, d r = Hit If R > m { k : k, }, the C(, R = C \ {,,, }, so the Residues Theorem gives C(, R f ( d i Re( f, k Compre to the defiitio of the residue t, which shows tht ire( f, f ( d Fuctio f ( C (, R, from I, hs te simple poles, mel the roots k k of order of, k cos isi, k, 9 Ecept =, the other ie poles lie i the iterior of C(, Isted of evlutig the ie residues t these poles, it is esier to write I = Re ( f, + Re ( f, 7 Evlute the itegrls J d L from Emple, N Hit The circles C(, from J re smooth curves, hece we use 4 with = The curves i L re squres, hece we tke = / The clculus idicted b Theorems d 4 leds to the vlues i e e if i e e if J i e if d L i e if if if k 86

8 Evlute the improper itegrls I = 4 d ; J = d (, > X Residues Hit Use the method described i 9 A To fid I =, evlute the residues t the simple poles i I the upper hlf-ple, the fuctio from J hs double pole t i 9 Fid the vlues of the followig rel itegrls b mes of residues: I = dt ; J = cost d cos, (, Hit Appl the scheme from 9 B For I, the ol (simple pole i the uit circle is For J, the correspodig pole is Evlute the itegrls cos I = ( d ; J = si d b,, br Hit The itegrls belog to clss 9 C I the cse of I there is double pole t i J vishes for =, d it J reduces to the Poisso s itegrl for b =, hece we m restrict the problem to, b > 87

Ide A bsolutel coverget - improper itegrl i R ti-derivtive 49 re - of domi 56, 6 - of surfce 94 B Beroulli 5 biomil itegrl 9 C coic - bsis - form of curve 5 Crtesi - coordites 74,, 8 - decompositio 66 - equtios Cuch test - for improper itegrls o R - formuls 6 - itegrls with prmeter - theor 55, 65, 7 chgig the vribles - i improper itegrls - i multiple itegrls 7 - o surfce 9 coefficiets - differetil - Lmé compriso test - for itegrls i R 4-7 - itegrls with prmeter coctetio 6 cotiuous -curve - surfce 9 covergece - boudedess criterio 84 - of multiple itegrl 8 - uiform coordites - Crtesi - chge of - curve - curvilier - clidricl - polr 7 - sphericl 7 - surfce Alembert 55, 65 rbou 6 defiite itegrl - reltive to prmeter 5 - with prmeter divergece divisio - of curve, 9 - of domi 6 domi - closed, compct 6 - mcd 6, 94 - regulr - simple 68 - simpl coected 6 - str-like 5, 45 double itegrl 64 E elemetr bod 56 Euler fuctios 7 Euler-Poisso itegrl 9 ehustig domi 8 88

F field - bi-sclr 45 - coservtive 49 - fudmetl - hrmoic 47 - irrottiol 49, 4 - soleoidl 4 - vector 4, - without sources 4 flu-divergece formul Fresel 84 fuctio - gmm (Euler 7 - bet (Euler 7 G Guss 95, 84 Guss-Ostrogrdski, 4 Gree 75 Gree-Riem 5 H Hmilto Heviside 8 I implicit equtio - of surfce 9 improper itegrl - i R 9 - multiple 8 - rtiol fuctios 8 - reltive to prmeter 6 - with prmeter itegrtio - b prts - comple 56 itrisic - propert of curve 5 - propert of surfce 9 itertio 69 Jcobi 7, 9 Jord 56, 78 Lmé Lplce, 8, 7, 5, 84 Luret 68 Lebesgue 56 Leibi-Newto formul - for improper itegrls o R legth - of curve, 58 - of vector 9 Liouville 64 Lipschit - curve - surfce 9 M me-vlue 65 mesurble - compct domi (mcd 6 - i Jord s sese 57 - surfce 94 mesure - iterior / eterior 57 - Jord 56 Möbius 6 multiple itegrls 6 J L N bl egligible set 58 Newto 54, 4 o-circultor 6 o-compct sets - itegrl o o-sigulr surfce 9 orm - of divisio 6, 55 orml vector 9, 89

O opertor - del - differetil 9, 7 - bl orietted - border 4 - curves 5, 58 - surfces 5 P prllelepiped 56 prlleloid 56 prmeteritio - comple 55 - equivlet 4, 9 - of curve - of surfce 9 prt - iterior / eterior 6 - positive / egtive 85 - regulr / pricipl 7 - vector poit - gulr 77 - criticl 7 - itermedite 9 - sigulr 7 Poisso, 84 pole 7 potetil - sclr 47 - vector 45 Pricipl vlue 8, 87 projectio 68 p-volume 56 R rectifible curve regulr - surfce 9 - domi residue 7, 74 - semi-residues 77 Riem itegrl 4, 6, 55 rottio - of vector field 49, S Schwrt 98 sectio 68 series - reductio to 7 smooth - curve - piece-wise - surfce 9 Stokes 4 sum - lie itegrl 4, 44 - multiple itegrl 6 - surfce itegrl 95, 99, 7 T tget budle tget ple - to surfce 9 tget spce 9 tget vector - to curve, - to the spce totl differetil 47 triple itegrl 64 vritio - bouded Weierstrss 65 V W 9

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