Analisi II - Appunti del corso

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Edie polito@ itcode goneMarco: ..→ Lun 15 appointment16 con-. .ArgotetestTobaccoCannatai .derivates) Limit integrals ,,,2) Serie di funtime Potenzadinumerical oh Fourier., , , .E testIsame cartacei t Iesi aperto- .⑨- - -)2h( 3×7 punt punt1/10/18;DDifference Analisitea II eAnalisi DEIR JRLI →→ : ) YERfcxy XEIB= R'Analisi BDEII fRIB→ : →\↳ "Z fcx HERC) ZEBX.- y,B'DE Rf →: fcx.biz ?) ( ERW WER-27= x. y .g c-nssrE→* ( )fg Xu XmXz, ., . ,-RnDEBI →: iii.:*. .FIX% ' 'B) YERYERExon oI oh'e in-eguazione curveunafoznra parametric .EST151=49.394 ) escorts: it I,C xI:}:¥E escort ,)I±In& IXzFsRZBre ( ) EXo •. - ... .. . .,: yITER '>r o }{ .ME/JTixitxice) x±Belko C= - ]µEtraseatonostro Caso 's't( )) Hz UXr Xss Xor--{ R2Brcxosbo 't(E) 'sb) }C- LyxdLx gotx- :, - .y n RTIyo - -- --- -- -I :! Y XXoBe delsuperficies Orchis B= yy. ^ ⇐Belko ) (Br ←yo )YoXo Zo #o, , , IBe dellasuperficies sgera ×RnA c-pt.es#rwasIiia* T, . internspt .A- E frontiersA corneaA. di Atutti imteruii pit .XIa-a" I vectorsDm " xf unB: " E Vellorey unER⇒ usersYERXERProdolto sealane→ →y YatX XzyztXn=. .. .txnb.nl/Ill=I.x-,&gxi.xi-=xitx7t...tXnIXz - -- .. - -- - . I:" I }IKE1115)fcx ?Brcxoiso b) ER) -= ,)={ ?Brett }llxI' IKEER Xo-LimiterDEF - l=L)eimcfcxi ERanalisi I xo→•a• n ,-7×0x elsexolef Ifcxi820 -4IxHE I> O : --in ¥ETE - .e -.. ..e E ✓-- :,ii Ty. II Xo tfSXo Xo- limoIn amah 1-'sBB - → IT. "ERf⇒Hf%¥wlkEHEFillsHE SooI> o .Fix E -) incontinua seXoTERM. eimcfcxh-fcx.inI→nel pianoeimlfcx.us#Iyn(x.y)-iCXo.yoS-i )( YoXo ,y - - - ----{ 'cosetx= re iMtsink )reis y-- xx^His )see traseato y: e.{ trustxox. is . . .b. resin tgot= I X× Xo)Ling Got )roost datmob indifferentyotrsimt in,Eseurpio {y= roostf xC =g)x. = 'tresin't 'yx IIIst.IE?II)=e=.dicwittsinat(ixAk(em ),Whosto 't tsimt)( ( )X. →y 0,0 -I))ya( rycostttsimot O== TxYaCurveILYCurve forms parametricin Sono :{ roost×= .Bf C-;D resist4- -✓ x[D= roostegjmo.Suisham't 'syX re)( nescientFc osteal)t =0,219GBA①z deltat - parametere-Y deeea curvey .Ea circumference- funzioneE dellaimagineed deltaE sostegnsy × della curvay ,SW' 't yX describe in foundLe essenecurve possess :BZPARAMETRICI )'L (DERA RJ→: oroost )eseurpio TEE( ]: guy cost0,2T re ., rsimt8LIMPLICIT2 A ) eseurpiocostX. ×2ty2=ry - :- . )y=hCx73 ESPLICITA X=oppure^ Y,-eselnpio Vex )-M: the i'y#y xx = - xx [Vr VriesX=ye y --a.5/10/18-)Titiastsbscts.fgi.tt#)aetebES-eurpio. , Xo Cost- Xero( III.Fit )sent often) rosineb= =I x¥1Eiti .Esemp coffees:→ I661=(30%51) to'OSS=↳o(→→ E it- )= 8664 Curva *→ "8lb ) antis*b)8cal ¥ .issue imieltiva "imielhua "ahimsa%ff-FBfngen.amOsservatoreng.gg/Fttoth#)=eniyg(gu.)--8lb )x . . 6cal )zctoth )facto-CtothQ Eto&) )' Xx -Xo #goF )as@)F' Tgfcxo ) =)Ctoth -8 I81 )Is(vector tangent g. ,}- bh bq=tg8,g- a⇒) )th factact ]-.e 'm =- ' .act .tn acts- OF ]#incurve to ))T=⑧?gto¥ Eanchetangent nel tangentveltreieuetlore topunt ,81¥ I;( ) %)±cent gets. . /It e-Se cegoeare) curvauna flea nelpttTangVettori curvaOSSERVAZIONE ren- fcx )fcx ) -y - - - - -- I y×Ein 6¥) I Cha )ca. , - ,Fumzioni pin variabieidi oppurefiDGR3-i.IRDEBT( )Befi →DSR JRfi → tserrrpio Z=xZty2)=L iC2- yx ,- geometricaltopografocac interpretationsuperficies Emitsi Canoeingassaultvalonesostituisce a )( =L!foxy Go€ Edefumito )continuity em yoin ) : ,yo, ( C) )Xo yox Y -0 ,,2- n Y- .CC/Hhishl-fcx#y28557 ein=> ×2fgj.us#=heiyg(fcxisthL-fcxy2fEserupio-xztyztxy-fcx.es TX01-29) Jy=Z = f2 27=2×1-0+5CI uellotangentdue fcxsy ' puntsnel) Ealla superficies 2- =ix. si sons i )( 11--1ItSia unversoree 2%32¥ )fth fab'%( fathers ))"he 2day21=228%-0- e, Icon -=. , -- ( 0,37( Venstre)Versace o,CinGRADI ENTE- che ha rateie vetere 'EedencomponentgradientChianaSi comeparziale ' sie sonic i (stasis )f=(III aiadf-HI.gg) e )EYEo-Cim , ' ))pi RdieiveeeoCurve Afcx ) costC C=y -_, ¥. TY( )describe in implicitformscurveeserup.io i Z=×2ty2=C I xMatrices Jacobian → mTIER "fixy→ YER)> IT 5% ()( )biXn,XzX Yz Ym= , ,. .., , ., . . ,II÷=(3E÷ i3f)3¥ iscon ,z. - - - in, , . . .. mxn3¥2¥3¥ - - . .÷÷÷:÷÷÷.. )3¥J 3ft 3¥- . . --. Js --(I÷ " "÷IDi'DERI →-se . "f BSe DEB Camra-7 ): :# iii.sis .'TDER BSef ITJf=(2¥: 2¥ gta) f 'd I= -, )(Taverna catenadellaRegladi derivationRegla olifunriohe compost"DERI " BERMA ER JooJL mxdol Xhmatinee matricesBEIT I e=g→oF=CFIxD=cJs=( )delle matricpzodolto→→ f) Jg(J If Jacobi=go . ane. I⇒ Dnalisi: inIttgcscxsD-IT.de#EsemDi-glqcsimkatancnD=Iy( It ) )C¥×D=I) bossatomssing -.¥+2)( atanxcos= ' REMA1LCASI ARE TEOUSIMPORTANTDUE IN WIIF③ R' CLCIIBf )E:D → goof =ITSIR:D3 → icyz→=g→Cy )fix )y= f=(Fg¥Y( II. )'LL 2¥D= )JCB .Jf= , =,La 'sCyl3fx3¥gits ) )8 ...rs#aicss3fx:c .② fog→=fcg→Cy )) .FI/9gi!iI,)=3f.9'stF.9'zI.Jcscsicssh-ts.us -13¥BY= .→deeInterpretation Tgeometric z ya 7'DSRf Rfcx.is )2- := )( ¥T¥ LexisTf dieseled J) Curva=L= xx-, [.ohuppohiarns ' scenecons )'E equation parametric a ,factj)-8cg astsb)=c TintI )(Essempio, *I x2ty2= Ifcx b) C prodolto matric= oh ', -offoff )=( ) ft) ) =LI fact Jf)Ct 8.J Ctso %=, . .=DCt )f= . - sealaneprod .E alleIe okortogomoale eivellocurve dellatopografic-Z-fcx.DKSupers .×.fcx dilivello)=L C )W C=yX Z sonsz supcy w, , ,, &orthogonalEJie alle eiveeeosuperficial 'Lexis Che fcxAppeichiarns Serino) )Za y 2- Ocome= =-,--28¥ -oh live.eeSup fcx. di we )y. ,:c 433) ueEo÷g¥sqma' ⇒" ' =. . )=(¥§gfefcEa consuperficies →)Non Zconfondere xOI Yi , ffg))Fcs 4373⇒ anew= o. =ortogonale allaSuperficies =L )CXZ y,⇒Direction Odel YtZ)fcx ay⇐ ,iverson IxLisse.sealaneprod .see- Is asftp. Hillf yacosa11= .XoinsSH'2L →Hacioemassimoha Tfvalve guardsgwansb E→2=0Jp ,↳ Ofimdividua diIe direEa Massimo pendentone a .)( ate partialOSSERVAZIONI derivYz=fcx )y 3¥, - o.¥11 ) 2L 0=Tyx-Teoreema )ftp.s-FL.vfeg?g%4FIe..6' parzialedecorate continueha →EESe f ¥qy' )duehaEE fSe f variabieie o =-Havetiffs Vz22¥ = %Iie =L(piano )tangent ) Cfcx )in) CZXo tXo2- yoyo xoxoya .= , ,,Dims stations i f- g)21 ( X Xo-- 3¥ Hayes)) → ex →? zz,xo +:{ yoso -Est --- .-. wL C )Yo,)FIL(tllfcxayo ))fcxo So→ e- ,OSSERVAZLONE iolerivabieita ' sufficientparziale continentalgarante- - EaLa anon re,'LCF ) delle derivates directionaldi esisteuzaee .Differentiable 't fixdella funzioue )fcx.yiz-A.CIIt ( tf )Hx) ) Call Xo Zoast Yo- - ,, → )( II TfASe cheapplication eine are ="LEESe differentiablee-- →↳ tangentieE derivable Icontinua pianoed,,Deviate Ordinedi Superiore)( txs''Esenr-pioiz-s.im yx{ (2×93+5×4)2¥ )x2y3tx5(cos .=Deriv I (3×52))zzg=. 'scary txsco . )'t txS7Czy3t2ox3¥Z_×= (2×51-5×4) >coscxzy)( x2y3txs{ sin-Deri It" (( )xDzg⇒y= > Cx2y3txs )Ctrysin + cos,6×29))tcoSC×2y3t×s c2×931-5×4 Gxyz)(x2y3txs )) C}Tyg←= ( 3×25sin-3×554222 )tcoSC×2y3t×s C 6×4212×931-5×4 )(X2y3txS )) C( 3×25sinFy = -€2222=D 2×27=2924 9/10/18Terrene 273¥21' it jSe SEE → 2Xj2X ;DIFFERENTIAE "R definiteJRAsealane f e-C-funzcure -0:. I -in tXo I A )↳se matrices reigeruns Cxn . .differentiableHIIIIIt LIED (A.f )=L xoxoto 11C per- 'nel di pianoERS e-caso 'e di uneq . )' CVX-xoitcy-y.ITJR fcxo C )%)Lex As AzC ) Y toC-× t) t YoXoxy = --,,( )per C )Yoy Xox →, ,OSSERVAZIONE =ejm→×?"¥£5 RI JRg. →c :C ItDuaeisiin gc× few )tactgicx, )) XoCx, + Xo T--, , reltatawgentexoxoperPer oliffeeenziabieitadi ta 'pin variable derivableLunzioui -. e-diverseSono i It 2¥Eederivabieita tutte derivatesI parzialein→ = ,iCon =] h. -,fixoliffereenziabilita fix ALFIE→ )tech) ) xoll, t x= .XoXperA =@fcIEDtEseI.TT.fEIfssecxisi.co.o ,D=C )o Copse x.AT L fix fchepoi 07=0 ( =De )o y= , ,¥y31-6,07=0 0,01=0Cfcx.yj-fcosotf.ly/tohlx2tyT)fosse differentiableSe → sealaneP .¥yyXZ ¥④y )obvreebbesi limavere : . =( L)X 0,07y →,{ .sk#xci=vzTl-*t-oconx=s¥= D- 00 Tf)Teoreema haf differentiable AESe si→ ei =IoTfcx II I'llIIfcxo fcxoiso ) C +10111 ))Eg ) tdee → = .. ~tangentpiano per sealP are.Teozeura differentiable f continuaf e-Se I →: 'E Eallora differentiablefSe E: 'LEE derivates continueprimeparzialeHaOSI e→:Curve-RIASSVNTO- !&¥? )(③ Ict ) Piinastso Curva= 8nCt7 "E' Bimmagine e-oh E indicator8 edincontenwta-supports di 8come g ,{ roast IIct , ,, ,yaxxa>reosimt • Xasi icostsCon 2TYa>Cireconfereenza descrilta parameter .caV1eqancon . .' doi Orient amentsL parameq un- .. IX,. ;yEST 5(tous .ox→= !- T Isenza1st 2I-52 orientaments- -② leg diimplicit data-una comecurva essenepuoaanche. ohlive.ecoh ' fun -.Curva oneuna -21BZII fcxe- costy=fc )→se y =, .4×2+42EI Lex ^) Z-2y Z=×2ty2con=, -- -re - -zz c= - -- ( ToCiccone oh ) raggioCentro. e0,0. - -- y( -E implicitIn guest 'yCaso ×.x re - →y•( )Bo③ especitataeealmeutefcx.yl-cp.wssolo esseneyXZty2=EI ] Xn -- XVI y= xz yyet N y - -- x(