ome Clculus bckgroud ou should be fmilir wih, or review, for Mh 404 I will be, for he mos pr, ssumed ou hve our figerips he bsics of (mulivrible) fucios, clculus, d elemer differeil equios If here hs bee oo much of gp sice ou ook hose courses, ou mus sped ime reviewig h meril or ou will o be successful i lerig pril differeil equio echiques s give i Mh 404 Below is brief guide o some eeded clculus meril, bu i is b o mes complee represeio of l relev meril Iegrio d iffereiio Leibiz Theorem: A hd resul we will ppl over d over, les i resricive cses o he seme here cocers he iegrl b( I ( = f (, d Theorem: if f(, d f / re coiuous o he recgle [A,B] [c,d], where [A,B] cois he uio of ll he iervls [(,b(], d if ( d b( re differeible o [c,d], he I b( = ) b( f f (, d = f ( b(, b'( f ( (, '( + ( ) (, d ( Eercises: Le f ( = si( s) ds irs, use he Leibiz heorem o compue df/d ecod, iegre he iegrl direcl, he ke he derivive o obi he sme resul efie he wo-vrible fucio = + u (, g( d for rbirr iegrble fucio g(, d show h u(, is soluio o he pril differeil equio u u = If f = f (,, z) is sclr fucio, d = (,, 3 ) is vecor fucio, he he oio for he grdie of f is give b f = grd( f ) = ( f, f, f z ), where f = f /, ec IN Cresi coordies his is someimes wrie s f = f iˆ f ˆ + j + f zk ˆ, where iˆ, ˆ, j kˆ re he ui vecors i he,, d z direcios, 3 respecivel Here f hs domi i 3-spce (h is, i R ), bu we hve he logous formule i he ple or i -spce The direciol derivive of f he (vecor) poi f ( + v) f ( ) i he direcio of he vecor v is lim 0 = v f ( ) I follows h he re of chge of qui f() see b movig pricle ( is ( d / d f ( ) = f ( d / d (
The divergece of he vecor fucio =,, ) is give b div = = + Therefore, he Lplci of u is 3 + ( 3 u u u Δ u(,, z) = u = div( grd( u)) = + + Also, u = grd( u) = ( u ) + ( u ) + ( u z ) The curl of he vecor fucio =,, ) is give b 3 curl = = (, ( 3 3, ) = Theorem: Le, G be wo vecor fucios such h curl(), curl(g), div(),div(g) eis, d le f be sclr fucio such h grd(f) eiss The ) curl( + G) = curl() + curl(g) b) div( + G) = div() + div(g) c) curl( f ) = f curl() + grd(f) d) div( f ) = f div() + grd(f) e) curl(grd(f)) = 0 f) div(curl()) = 0 g) curl(curl()) = grd(div()) - iˆ ˆj kˆ 3 If equio ivolves soluio eedig pril derivives of secod order, he we re ieresed i coiuous fucio o domi h hs coiuous pril derivives up o order wo, d we s he fucio is of clss C () If he fucio ol eeds coiuous pril derivives of firs order, he he fucio is of clss C () or emple, if we re ieresed i solvig he oe-dimesiol he equio for u o he domi < < b, 0 < < T, he u should be of clss C ((,b)) i, C ((0,T)) i (The equio holds i he ope domi, ie he regio o cosisig of is boudr becuse we c o ssume he derivives of he fucio hold o he domi s boudr) Now le be ope, bouded, simpl-coeced se i he ple (or i 3-spce) wih smooh boudr (B simpl-coeced we me hs o holes B smooh boudr, which is bouded here, we me here is ui orml vecor ech boudr poi, ie, ech poi o he boudr, here is vecor h is perpediculr o he ge ple he poi, of ui legh) Cll he ui (ouwrd poiig) orml vecor ˆ ivergece Theorem: Le g be coiuous sclr fucio o d is boudr, d coiuousl differeible o, d le be coiuousl differeible vecor fucio o The
{ gdiv( ) + grd( g)} d = ( g) ds ˆ Eercises: Use he ivergece heorem, wih leig = u = grd(u) o obi ( g u + u g) d = g( u ˆ ) ds This epressio is ofe clled Gree s irs Idei Tke he epressio i pr, ierchge g d u, d subrc he wo ( u g g u) d = ( u g g u) ds ˆ This is epressios o obi Gree s ecod Idei I hve wrie he divergece heorem s if i were plr heorem, bu i fc i holds i higher dimesios A wo-dimesio heorem o keep i mid is Gree s Theorem: Le be bouded plr domi wih piecewise C boudr curve (someimes is deoed ) (Piecewise C me coiuousl differeible ecep fiie umber of pois) Cosider prmeerized such h i is rversed oce wih o he lef (rversed couerclockwise) Le p(, d q(, be C fucios defied o he closure of ( +, ie he uio of he wo ses, ie cl()) The ( q p ) dd = pd qd + A compleel equivle formulio of Gree s heorem is obied b subsiuig p = -g d q = +f If = (f,g) is C vecor field i cl(), he ( f + g ) dd = ( gd + fd If ˆ is he ui ouwrd-poiig orml vecor o, he ˆ = (+d/ds, -d/ds) Hece, Gree s heorem kes he form dd = ds, where = div( ) = f + g is he divergece of Trigoomeric d Hperbolic ucios Addiio formuls (rig): si( A ± B) = si Acos B ± cos Asi B cos( A ± B) = cos Acos B m si Asi B o, for isce, cos() = cos () si () = si () = cos () ; hus, for emple, si () = ( cos())/ Eercises: usig he ddiio formuls, show he followig iegrls re rue: or rbirr posiive iegers d m, 0 m si( mπ )si( π) d = 0 / m = cos( mπ )si( π) d = 0 for ll d m 0
kech grph of () for > 0 d superimpose o he grph he grph of he fucio / Numericll pproime he firs 5 soluios o he rscedel equio () = / 3 how h si( π )cos(0π) = { si(π ) + si( π) } Hperbolic fucios: sih() = (e e - )/, cosh() = (e + e - )/, h() = sih()/cosh(),, ec d d sih( ) = cosh( ), cosh( ) = sih( ), ec d d sih( ± = sih( )cosh( ± cosh( )sih( cosh( ± = cosh( )cosh( ± sih( )sih( Eercises how h sih(), h() re odd fucios d skech grph of ech The show cosh() d sech() re eve fucios d skech grph of ech how h sih() d cosh() form fudmel se of soluios for he d u differeil equio u = 0 d equeces d eries of ucios We re goig o be delig wih series of fucios (ourier series), so ou should recll few higs bou sequeces d series = efiiio: covergece of series of (rel) umbers: = + + L coverges if he il ed c be mde rbirril smll; ie give olerce ε > 0, here is M > such h for m > M, < ε = m efiiio: bsolue covergece of series: coverges coverges bsoluel if Remrk: he Compriso Tes: If b for ll, d if b = coverges, he = coverges bsoluel The corposiive ecessril follows: If = diverges, so does b = The limi compriso es ses h if 0, b 0, if lim / b = L, where 0 L <, d if b = coverges, he so does =
Remrk: he rio es: he series + = coverges bsoluel if ρ < for some cos ρ, d for N (We do cre if he iequli is o me for he firs N erms) Emples: or ( ), so = + = /, hece = = ρ, so series is bsoluel coverge or, + = + Hece, here is o upper boud less h, so he rio es fils, ie give o iformio This series cull diverges The rio es lso fils for he series, bu he series coverges o π 6 / I fc, he p- series p coverges for p >, d diverges (is ifiie) for p efiiio: uiform covergece of sequece of fucios: ssume he sequece {f ()} =,, of fucios is defied o iervl I of he rel umbers The {f } coverges uiforml o I o f() if for olerce ε > 0, here is M such h for m > M, f m () f() < ε for ll i I efiiio: uiform covergece of series of fucios: wih f s defied o iervl I, f () coverges uiforml o I o f() if he sequece of pril sums {s N } N=,,, s N = = Compriso Tes: If coss, d if [,b], s well s bsoluel N f ( ), coverges uiforml o f() o I f ( ) c for ll d for ll b, where he c s re c coverges, he f () coverges uiforml i he iervl Covergece Theorem: If f () coverges uiforml o f() i [,b] d if ll he fucios f () re coiuous i [,b], he he sum f() is lso coiuous i [,b] d b f ( ) d = f ( ) d b The ls seme is clled erm-b-erm iegrio Covergece of erivives: If ll he fucios f () re differeible i [,b] d if he series f (c) coverges for some c, d if he series of derivives f ' ( ) coverges uiforml i [,b], he f () coverges uiforml o fucio f() d f ' ( ) = f '( )