*

\EKHEEIE OYEIKHE\ . . .1

\ "

TIAYAO> IAANNOY XPH>TOX TPIKAAINOXEnixougog Ka0nyntriE Avon)'lgrotrlE KcOnUtnE

flavenlotqplou A0lvtbv

AIKHIEIX OYIIKHTMqXavJ?4,i

A, EKAO>HAr,oq0op6v4 - 2upnl4gtop6vrl

AOHNA 2006

ISBN; 960-90045-0-4

O TIAY O' IOANNOY XPH'TOE TPIKA INOETPIOYAAIAE 42 - 757 72 ANO IAI'IA IIAATONOE 17 - 757 77 EAAHNIKOTE .: 210 7777339 THh; 2lO 9613922

Go* fff:ft1?i1t1,|t-1til'!E rs rrszrar - e-n,ir, eorena€acsmi.eroo)rcorcxEroo€oio . Hi€Krpovrxri l€il6onolnon . oril . Mo\rdz . Eninoon . Br8R|o6tdid

Ilegrelripeva

XrpBotruopoi

N Auvcpr,x{

VII Xrqoqoqp{

VIII Auvcpr,xri )uorrlpatog )<rrpcr 16 ltr.rv

)riwopo Mo0qpcrrx6

Afiovrnos[S Aoxtloetov

10

l f f

167

271

?RJ

II96),oyogH euoLxi odv erlmipn €ivdt yla rov tQevqri pn 6nQxi5 ttB6 'non.

Eyet va awqletanioet npat6p,apa rQof iparu (plxgd i peyd a), q I'tonrav onoiav nt0cv6v va 66oet xar,votipr,a 6eqon mnv avgpdnml yttito4.Ilptner. Iotn6v, yptyotponoui;wag rn ouooaQ,tuptvq e1meqfu, va Bqer', ptoaan6 ng no]'uaetqlteg 6uvarorqreE, exeivoug toug 6q6pouE nou 0a rcv o6q-yioouv otn Ltotl ctutrbv ttov npop)'rlptittttv.

Ettol.ttvaE, ax61t4 xat oto stine6o qg ewwi6euo4g, n @wui oe xapldreBintaoq 6ev eiva6 ourc pnoBel, va dvat, 11 anopvqp6veuoq xdnonv od'['-6tov, tuttuv, i owo6d)v t6pav- H acpopofunn rav Bdoeotv qS emotipnsnQoijtto ttt, a)'],a xat owemyerdl u1v mav6nlto eqaAqoyqE mtv v6pavrnE otnv etti).uo4 x6motav (re.pel.Idrtltd)v i 9eaprytxtbv) npoB)'qpatov.Erct n 6t6aoxatria arord.ei pn otffietn 6r,a6r,xaoia, nou nept)'appdver' t4vava),uoq rav 0epello66v v6pav rq; erwtiq.nE xat tryv tqaQpoyi touE.

H )'tion qeaerytxdv amqoeav anorc],ei ottorarmd m avd|t6ofiaoro xo-

ltctr tou pa9fp,atoE xat Baom6 xetriEo rnE Lxc:drnxds rou oltou\aoxi vaerpapp6(et \rlptouqymct ttg ywboer,g rou. E[,vo.L xdurdxQovd xr 6va ouvaQfid-onxo <natyvt6t> )'oytxiE, 6vct ave$dwlurlto fie,io duroeLtryou, aorql,Qcwtln4y4 S1ltt out2ytxLbv ouyxw4oruv.

OLd dwd fuBan otrawovv oiyoupa. ta),two, a),)"d xtipn eivat ngo|6vpaxpoXpovrlE xt minov4g 6ou)'uiE.

n Po\oro2

To Bt$),io nou o avay,vtbotrlg tyet oru ytpn rcu eivat xo.Btt6E qE noLu-Xeovqg €WeLeiag, nou anoxni9ryte ond tn 6r,6aoxa),ia tou ctwlmotyov pa-?ipdrog xtela oro A ' ttoE rov <Dwtxot TqiparoE, d,Ld xat oe dhla Tpi-pard rqE ZyoArig @etmbv Entmqpdv tou Ilavenmqpiou A9nvdv. IoL]\tgaJt6 rLE aoxioelE nou ncp#yovtat e5d iTow ),tfre(, xard tr1 \dpxeta tunnaqa66oeav, evb aLleg irdy 0epdrd eEerdoEav.

Idnheea paE npo$),npdxLoe q 6opi rou, n an6gnon yl(t rtry oxroid ttctE t-qeee avrlpixattow pe pld oelpd reo$),ipam, rou pE rn pta i r1v alt),qltoeEi pa; t9e ruv ol ot EoltnttE ttdg.'Htav ;wovoE o fieLQ,aopdE va ex66ooupe tva duuoapu', oav duTd Jtouxatax),tilouv ta pnEta tav l3t$Lr,onolueiov xat npoo@pow oxouE EoLttlrtEn1v <eyyv4ptvq> (xar, \umuxdS noL),tE EoetE aveu9wtD <tltLtwla>.

Zuvtpryre xat ro aw(,9eto. YnieEav orlyltEE ltou mtEtdp,aotav va na-pouotaooupe Luptveg aoxrioer,g elunveE, dpopEeE, dJ,a rtg fieeLoo1ree€gEoptE aprctd 6tioxd,eg otn ),tion rcug. Me 6w ),6yta doxioelE fiou 6€v Jteo-op(,(owat yta rc ptoo Eonqti, a)J,a yn exelvouE, ra ev\taqtBowa xat ot, t-xdvdrnr€E rav oJroiav t[,vdl derlETd fiLo fidvu.

H rc]tmi paE mr,},oyfi xaqoa\inrye &n6 ro oxoltd fiou 06\oupe vd eEufiq-peqoer. rc B$)'to ctur6, rou elfiGoupe 6tt, 0a anorc),toet tya Xeiolpo Poi-9npd yld rouE qotnptg nou QA),ow va xatavoioow xahneQa t4 Mryavtx1xdr vd pd9o1N va tgxtBp6low ttg ywinetg roug yta rn ),toq aoxfioeotv \ta-qopen xoti l3aqpoti 6uoxo),iaE.

Xae[,oape tqv tLtl tttE Mrlxdvlxi;, xtgw yr,o. peqo\o\oylxoiE oxonoug,oe 11 lleEd)"al(r, evb oro tt),og \tvoupe tva ohropo Ma0ry.attx6 Zup '"ri-

Qopa, xtou negttyet ouvonttxa xat TapiE aumqpri rtrytqpiaoq ug anapai-tn6g paqnqdro6E yv^oelg yLd xq l,liotl xov doxioqov.

H oepd. rav xeEd,aiav avttmotTei onl oelQd itou roeow.dEeruL n t nt4E M4Xavutig om neEoodrepa Brf),ia <Duotxrlg. To r€Ieud(a xetpal,"ato ei-val aqleQnDpivo arnv E6oti O€aei.d rnE 2yetm6tqruE, tva plgog tqg onoi-ag 66aoxetat oro QLtoLxd Tpripa pa(i. pe 14 Mrygvr,xri, xat" etefifi ntmaj-oupe 6tt ot daaioErE nou ltcer,tXowat, e6oo e[,vat, npootttE yld rouE qonnftSnou aoyd,oirrat p.e t4 M4Xawxfl.

2r4v apyfi xafu xeqd,atou neqtyetat otw-ropq 9eatpla, nou Bo40a rcv a-vawd)orn, pplg va avatp$pt oe a),),eg nnytg, va Bper rcug l3ctoLxotE tt-ttouE nou 0a rytroBotjoav va eivat ygriot lt"ot, ottt ),ton xov ctoxioeov Jtou (r-xolou9otiv.

2rr1 ouviyeta napouor.d(etat tvag apr,0p6E napaSerypduov ttou e[,var, Lu-ptva avo),wua, hot dnre o dvctw^oxqE vd Woee[ va fiaeaxo].,ou0e[, tn ],"o-ymi, a),La xdL xn mQ,(txtlyLxi mq Lw4 rouE. Ta raBa6eiyltata tyouv eni,e-

IIPoAoroE

w[ pe a.aEoeeuxa xe.rield. Ild,]nd qr' aurd eivat az],a, a d xpqor,pta, t6-oo yLd rn Xenolpoxtorcr.ipevq p60o6o ),tons rouE, 600 xat yta tqv tryvLx4XAiottE rou pa9rytanxot epyak[au. YniQTow napa6el,ypata, l,oag nto no-LutrToxa, fiou 6paE n fiInanE xarav6qori rcuE Boffiet otqv xatav64o4 ap-x€ra ,rjoxo[ay ewoubv, 6nag n.y. rnE 6u'aptl rnE elqng. Ttluog, unagyow,td. paelxd naBa'ciyparu nou enr,l'tyqqxav )udyot tou 6(.unvou rp6rou m6-VrnE, xrou dJruLreircL yLa t4v mi).uo1 touE.

Zro xt\oE ladfu xeqdhdiou tmdppt tvag apr,01t6E dlutav (loxioEav noufiQare[,vowdl yn hio\ otov d.vdryborq. H waoT urlorl p' auttE elvat ana-Qaltrltq nQ]oiht60eon y.a rnv JrLieq ltdrdv6non rqg t ng rou xetpalainu. Oldoxioe.E axoLotfrotiv Paolxd (6XL 6paE anoxlelorlxd) rq otLed x@v xtcQa-\erypd.rtov. Or. anawfioet"E nou uttdpyow oro re og tou BlB)"iau 0a l3on9n-oow tov avayvborn va elty$pt t4v oB06trya qE )aioqE tou.

npoB),npafl,oriwape apxeta yla ro dv 0d ttetfieL va yaqaxr4pinoupt rtgaoxiaELs avdToya pe ro P(r0p6 6wxoL[ag xouE. Kd.rL rttoto op,ag epneB6-XeL aQ er6 lnoxelryv,opd xa4 oe oEoptveE fieLitx66e.E, xd9e aIIo napdl3on9cet. Etor, dEiyoup€ rov avayvbotq va xplvet p6voE tou yn ro l3a0p6Suoxd,taE trlg xd9e doxnong.

@t),nupe ),out6v va ntorcrioupe tutq, to flt$Ib aw6 0a qnvei TprlotpooxouE qoLrnrtE rov @uolxdv Tltrllldrav rov AEI, xa06E ,4dL d)'l'ov Tlnlltd-tav mw 2Xoltbv Osrlx(bv Eltlottlltrbv, nou aoyolnotiwdl pe ro dwLxti4cvor1E Mtycvtx4g ota r)'a(.or.a qE lwtxrlg <Duotxrig. Eln(loulte e.nlaqE, m4ptoa o' aur6 0a ppow xdnorcg 6teg owd6e)lEor, t6oo an6 ra AEI 600 ,4dLast6 rr7 ptoq euwideuon. T6),oE, iooq va pnopeoouv va. ro fiQaoery(,oouv xaLxdnont ar6 rcuE paqrytg rnE f ' Alnebu, ta ev\nqttpawa tav onoiav $e-(Qeliyouv dJrd ta n)'aton rcu oyo)'txot npoyedltpctxoE.

Eivat nd,tj Euotd,oytx6 n fiQdrn tx8oon vd xt€eLtptl Ld9q xar, ex ravnBo#grov Entoip.e ovyytbpry &rd roug avayvinreE yt' cwa- E)n4ovpe ou,or, 6txtg toug napdrtteioelE, etttorlltdvoe r,g xat npoftoetg, 0a Aoryqioouv ornonpawr.xi Ad,rfutoq rcu ftfiAiou oe pe),)towr,xtE ex56oet g.

NAYAO, IOANNOY XPH2TO2 TPIKAAINO,

/ t -

Iup$olrcpoi

{ >to pt$},io Xerprponororjpe roug 6le0veiE ougpolrogoriE, xriguc yr,c trglQ0YOvOp€rQrXfg OWCpnrlOeUg. H cv-ftororylo rolrE l.r€ rouE el]urryDrorig oup-0ofwUoG e lvcl rt odlou0il:

sinq = npq, cose = arttq,tanq= eW' cotq= oq9'

arc = r6Eo,arcsinx= to$qpx, arccosx = tGotmx,arctan = ToEeqx arccotx = toEo(gx,

. 'Ol,c rc oft$olc, rcr onoic elvcr pcriqrc cvc96qowo oe cvfopcrc. An-)'c6ti:

r =? '

" canst = oTa0ee6.

XriwouoI

Ma0qpuu,x6Bori0qpa

1.'Y)THMATA >YNTETAIMENON

2ro e*i:rr6o ouvno(l)E XQrtorpofiorotvrcl 2 ouorripcrc olvrerclpevruv:o) Kaer€onvd. H e6oq evoE onp€lou A (0I. oX. M.l) ngoobrogi[erar 0Jro

buu pryr0q fiou rxuuv broorriorrg, Urixoug. ro \A xol JA.

>xiua M.l zxipa M.2

13) Ilol,rx6. H 06on evo6 olpei,ou A ([3I. o1. M.2) nqooSrcql(erar, and ev<rprlxoE r xcr pra yovi,a 0 (O < 0 < 2n).

MeraE{ no}.Lxdv ror xoQx€orcvrbv ouvr€-Tcypdv(l)v roxfouv or ox6oolE:

x = r c o s q , y = r s i n d .

2ro lrirgo ouvrlorog lqnolptonolorivr<[ 3oxoTl[0 ouvTETdY|rrv(l)v:

o) Kaemorov6. H 06on ry<iE olpelou A([)'.. oX. M.]):rgoobtorrl(rrot c.no tprra ptyr-0q :rou €1ouv brcotaoag gilxotsg, ra x4, ynXIL Zt

[J) Kulrv6eua6. H 06oq evoE o[p€iou Azxittq M.3

2n ,yNToMo MAoHMAI rKo B)HSHMA

(lll. oX. M.4) rreoo8roQiL€rar ofi6 6to prilrt, '(a ZaxaL r = (OA'), 6nou A'r1r[eo0olri ro1j A oxo erfur€8o xy ]4or rn y(0v[o E Q s E < 2n).

,Xipa M.4 zxipa M.5

Mercfri xcqteorcvrirvrd.r xul,rv6erx6v olrws-rc1p,€valv uolriouv or, o16-o€t5:

x = r cosq,Y = r sinq,

1) Eqcrqrx6. H 06oqrou onpolou A (pl. ox.M.5) rleoo6roelEsrur ar6

6vc prixoE, ro r = (OA), rcu. 6uo yrr.rv[eE, xLS (p (0<E<2xt) ̂naL e O<e< fi).Mercfri xrqreor,cvdrv xoL xul.[v6e[x6v orlvrexay$ivov loltiorw or, o1€orlg:

x=rsinqcosq,Y=rsinesin(P,z = z cose.

2. AIANY>MATIKO: AOII'MO>

>Xipa M.6

rt

t{ipa M.7

Xuvrot6oeE crv6op<rtog. To bldwopo 06019(€nl,|3orrrrt cxrivo) ev6E onpsi,ou P (0I. oX. M.6)€lval ro OP = r, ro ofioi,o trjloQe[ €fiiolq va yQa-qe[:

,= r f *y t +z f<= tx , y , z t ,6nou x, y, z ot ouwerclpriveE Tou orlpe[ort P, JTou€b(t) noiLouv xcl ro qoi'.o ru)v ouv(orood)v rou d-uiolrcrog r, xar f, t, f rc povc6n[tr a\riopororou xoQTf ordvo$ ouorrlprurog ouvrmclpivtrlv.

flqrio0eoq c"noprirtov. )qv lrq(rrooq 6uocrruopdrrrrv 11 xcll 12 GI. oX. M.7) ro d0Bolopo.TOUE

r= r t+12- reooE[oe(Eex0[ ar[6 ro |r6reo:

r | = r = ttrl-l7f a 2a7, gos'

ZYNfoMo MASHMATIKo BoHoHMA 273

Zmpa M.8

lror rI 6r€i0uvori Tou, Jrou 6(vercl ur6 r1y(,)v[(t g yr,0 Tnv ofioic roxuel:

6 = 674g1f r * fz cos9' 12 sin9

Tlloi,o dn-ropo oro 1ripo, ro AB (ptr. oX.M.8), W[oeol vo lgaqe(:

A B = r S - r t = r

ttot cvr A = x A i + Y A i + z A l ! ,

A Ar B = x B , + Y B J + Z B r ( .

DGA)B=A(cB)=c (AB) ,6 )A (B+C)=AB+AC

0(r €Xoutr€r=- (xB-x f +On-y ,o) i *Qs-2a1r t = r , f + ry i + r r f t= 1 r * r * r r ) .

Irrp negi,nrooq oeeoloporog rlol],6v owoptorov 0a tolria:

r = I

ri = |

r, i + z 16 j + z r,,i.

Eorotagrx6 { $crOportri lrv6pevo 66o cwogitorv. Aur6 €[va[ [ovo-pexQo p6ye0oE r4ar oelq€ror (l)S €ris:

A' B = AB= (A' B) = AB cosq = Ax Bx + Ay By + A? Bz,

6nor A xcl B rc pdrqc row A xcr B, I q pmclri roug yarvic, Ax, Ay, Az oLowrordosg rou uvUopcroE A xol Bx, By, Bz oL ottvrordo€S To! dv]ioudrogB.

Meer,xes t6rornreg rou €oor€Qtxou l,woufvou:o.) AB = BA,

f) AA = 42,

frc rc povablcic 6rcvtopcrc rol xcqleou.vori oum{pcroE otrv[€Tayuf-vtrlv tOlriel:

f f : i j =f t t = rA A ^ A ^ Ai j = i k = j k = 0 .

lxcvr] xul ovo.yxcic oul€r1x4 1la vq clvor rd0€ro 2 avfopcrc A xar B,

YIC rO OJrOIO AB * 0, Ei.\dL

A B = O .

Elorteqrx6 ri 0rcvuoptrtrx6 lrv6pevo 66o cr"ucpritorv. Aur6 elvcr6r,cwopcrlx6 p6leeoE xcr, oeiEerar (l)E eElE:

274 'Y^IT)Mo MA@HMATIK) BoH6H MA

= n ABsinq,

6]101-l Ax, Ay, Az oL orJvr,ox6o€E Tou c\niopd.Tog A, Bx, By, B, or olvrmrireErou cvioporoE B, I n prlrQ6r€en yolv[c perc[t rov o.vuopdrrlv A xor B (13]..

oX. M.9) (xrlaQx€r xcl q 8', 1rc qv onoict"ay(n 0' = 2n - 0) xct|, povc6ro.i,o 6rd1ru-opc xd.Oeto oro €r[(,fis6o nou oqi(ouv rcA xal B, pe qogi 6nrog n qoaq E€Er6-mqoqou xo1l,ic, o.v orqtqoupe ro JTQdroawoFc (ro A) xard Tn ytovla 0, pflqt vcorwcw{oel ro betreQo (ro B).

MeBlx6E l6dr4reg rou eEtrlr€Qrxo{ y[-vop€vou:

$ (cA) x B = Ax (cB) = c (Ax B),

6 ) A x ( B + C ) = A x B + A x C .

lirilA,B)= A x B = l Ax Ay Az

l " ,"nu,

tx4pa M.9

a ) A x B = - B x A ,

\ ) A x A = 0 ,

flc rc p,ovo.6rci,tr 8rcvriolrarc rou xcgreonvori ororrlptrrog ouvretc.ypi-vov r,oxte[:

f" i =ft , i " t =! , t " f=i .Ixo.vri xcl cvclrci,c owO{xn ncQcl"l,ni"i,ag buo cvuolrcr$\r A xcrL B (AB

>Xnpc M.lO

t 0) ti,vuL rlAxB=O.

Ioxt€r (0I. oX. M.10):

lAxB l=ABs ine=Ah=5 ,6rrou S ro epBcb6v rou nceclln]'oyedp-pou, nou oql(ow r0 cvuopcrc A xcr B.

3. AIAOOPIKO: AOTI)MO>

Ilcprilolog owtipt4oqg. fra xcOe owcqtqo4 y = f(x) n lcgcltoyogrqg oQiLercr ols eE{g:

v ' = f ( ^ ) = l t = d f = f i m a f = l ' m t ( x + a x ) - l ( x ) .

dx dx ^x) oAx a. o Ax

H rcqc'ytrryog pcg 6[veu ro QU0p6 trg percpo],r19 'ntS y, 6rsy percBd]rl,e-rcl. n x, ricr y€opexeix(r €xqEiqel rqv x]'ion rnE equrr6u.eqg orqv xcpru],4y = t(x) oro a\tutLo (x,y).

Or Baorlroi, xov6veg ncBcyr.iryr,o4g e[,val:

o, dlci , l t 'x)) =c t (x). b 4lrtxl t er. l= t G)r c G),dx ' dx '

y1 ! _t7x1g1xl)=f (x,g(x)+t lx) t ( ,ox

^, 4 | r (x) ] _t ' (x) t6)- f (x) t ' (x)" ' a*lgi l) lgr*r l z '

t t Av z=t(y, xcr. y = t(x) L$Xi)t, + =++ =f 0)C G).dx dy 0x

,Yl'rroMo MA@HMATI Ko BoH@H MA 275

Iltrqdltrlol pegr.xdrv owcqrrioeorv

ta) f '(x) Kx) r'(x)

0 a^ a\ lna

x stnx cosx

a x + b a cosx -stnx

nxn-l tanx cor2x

ex ex drcsin(+)'{at-F

Meprzf tcgriloryog" Av fxoulr€ pr,c owdqnp4 nollrrirv percB)"qr<iw u= l(x, y,2,...), 16r€ n p€Qrxn rluQdyoyoE qE owcprloqg curriE oS fieog x o-Qi.Lercl og eErts:

df _du _ tim f I + Ax, y, z, ...) f (x, y, z, ...) .dx Ex a\ ,0 Ax

Avoloyo ogi.(ovrcl or pegn6g r[aQdytrlyor trlE nQoE y, z, x.'r,.L.OrroE goiv€ro.r, lcci cr6 Tov opr"op6, yLc vc pqoupe rn pee|,x{ ircBcyr:-r1o

276 ,Yt'rroMo MASHMATIKo BoH@H MA

ptaE owdQrnonE, I.X. og JreoE x, dQftel vo rroeay(l)Ilooups auni rrt ouvae-rrtort (oE r[eog x osoedvrag 6]'eg g dl],eE percp],rpdE fioose€E. II.1. yr,c u =x3y + 3y3z tyorspe.

du Er' )"-- = 3x2y, --1 = xJ + 9y22, "". = jts.

dx dy dz

Kcrd ra dl,),a loyiow ol t6u6rqreg xcl or. xcv6veg r4g W<oonig ncecyrir-you, q onolc peglx6g 9oq6g ovopi(eror xor ohxri ncgdryoryoE (o€ al'[[e€onpe rq prBuxr).

l r 'Orcv y = qx) toluer: 11 = -4L .

dx ox

Arcqoqrx6. Eoror owtrqrlol y = f(x), xbxl cv perafiilrl,otpe q pertr-fl)'nrri x xcrd ),i,yo, 6n),cbri an6 xl at xz,I perapo)'ri rq5 exqEilercr cno rnblaeoeo ̂x = x2- xb evtil 1 percfu],ri rqg orvdqrqorg y ar[6 rn 8roeoed dy= f(x2) - f(x )

Av ro /x -+ 0 xnl livercl are4roord plxg6 ro oupflotriloupe pe dx xcnro ovopdEoupe 0ueoerx6 tqE Fxc[lqr{E x.

Awioroulc r1 percpo]'{ rr1g y rqv oup$o}.i(oupe pe dy xcr, rlv ovopd(oupe 6rc9oqw6 rqg ouvrtq,r4or6 y. Ioliel:

dv = f' txl rlx = 4 dx.ax

In ouvdgrlol no)'}.riw percB),1r6v, r.X. rnv u = l(x, !, z), xo 6ltrqoQlxdlQroq€Td'r:

. . -a t a t a fo u = o l = - o x + - o y + - o zdx dy dz

xal ovopd(eror olrx6 0rcqoprx6.

Ilcprilolor. cvtot6pcE trile<oE. fla rlE ol.rx€g rraerrydlorlq mnv rleQi-rlr(,ooq ouvderlolE pr,cE percl3]'rprlE y = f(x) i.Xou$t, rr.X. yLc rn b€rtr0Ql tto-qdyo-ryo:

v = !!t-= I Ptl .dx2 dx \dxl

lBriypc nou olpaivel 6rr aexlxa peioxoupe qv rqtbq raqdy<oyo rqg ou-vderrlorlg, rqv onoi,c orq ow6lero rcqcyo'yi(oupe. Tc i6lc pnoqoupe vcroupe Tto rrE rxoQoy6youg avor€QdE roE€og.

ILa xrs p€erxEg rraeoyd)yous, ooov apoQa ornv rexvl,xri fioeaydyrorlg. r-ox{o! / rc (6rc, 6prog n.1. m1v neqf.rrorl owrigrprlE 2 Wxag}*lr.,(in t(x, y)€Xoupe 4 8errrepeE rcqc'ytiyolE:

,yNToMo MA@HMATIKj B)HqHMA 277

atf atf a'f a'fdxdy dydx dx2 dy2

- 2 ^H rloedytllyog I qusiy6l drl nprino. noqralory(loupe <oS rreog y xar

dx dyo.ur6 fiou l3eloxoup€ to ncqoy<o1i(oupe rrrE JleoE x.

:Tq ysvrxrl r€elrxtoorl

a,f * atf

dxdy dy dx

cl,l,c oq Ouorlr{ (xar r,6lcireqc ml levrxi) ouvri0og lltel 1 to6rr;rc.Avriluorl owrigtrp4E oe oerpri Taylor oto pr;06v. Ano6er,xvuercr,

orr FLa ouvdQrlol f(x) Ufioesi vo. lqcgei, oq pogcpri:

i l x t= r io t+ - ! , t u t l , * l r 'u t l xz + . . . + ! ln , r1 l xn + . . . ,t : ) \ = O Z ! t \ = O n ! x _ o

q o:roic. ovoyri(tro.l o€ted Taylor.Me q fu1ftutt qg prroqroripe vo rdQ,oup€ peplx6E lqriolpeg rqooryyiogE.'Ero[ yrc x -+ 0 61oupe:

o ( t x x) '= r x nx, F) , L ,n= t - ( ! nx) ,

( r 1 x ,

I f i ;= t tx , 6 ) e l r = / 1 x ,

E) ln (l ! x) = lx , ctr,) sinx = x,

l) tanx = x, rl) cosx - I _I1 .

. IIXOritrofoS avrlopctog. 'Eortrl

d.wopa A(t) = Ax @ f + Ay @ I +Ar(t)k.T6tt q rcqriytoyog rou cvuoucroE aurou org nqrog r ei,vcr:

A'(t)= 6 = 6^ A(t + At) - n(D -dt alno At

At

e,o i +Ato t j+e, (u f f )- * l

a.-o

278 >YNTOMO MAOHMATIKO BOHOHMA

= *:rle-#4i * A'(t * ̂ '),- Av(t) i + A,(t + at: u* i)=

=44- i *4L1 +44. i .d t f t ' d t

Exr6g ar6 roug yv<oororig xcv6vrg ncpaldrylolg 61oups:

! r tn t=ndL+84L.dt dt dt

d 6 y 9 1 = d A y B + A x d L .dt dt dt

Ef,vtn olpal'nx6 mqv negbrrorq rou elorreqnori ylvop€vol vc p1v a),,-trd(olpe rn o€rQd rtllv A xal B.

Bc0pi6cr. Av 6xoup€ owdqrrpq t(x, y, z), r,6r,e r,o brdwopa

t radt= v t=af i *a t3*a f i .dx dy dz

ovopdlercr, 0c0pi6c (gradient) rnE owdexqolE fxcl elvcr, xdOero orug e:rr-qdv€reg rfou oeiLer, n ox6on

f (x' Y' z) = const'

To orlpBol.o 7 l,6prau teleouiE avdb€ka (i nabla), lqrplponole i,ro.l eu-grircra or4 puorxr{ xcr, 61eu ev6lap6p,ouoeg r,6r,6rpeE. H pe}.6rq rou pyciver,

6lro u:r6 rc 6gr,a rou pr,Blriou curori.

4. OAOKAHPS'TIKO> NOTI>MO>

A6eroro oloxhip<opc. Av f(x) xcll F(x) tivcll 6rio owaexioerg rou x,yro 1rg ofio[€g rcXf€r.:

dF(x) = ftxt.dx

"t&t. tl F(x) ?'€yercl c6qlmo o)'oxlrletrrlr<r rns f(x) r{ar oup0oliEsral:II t(x) dx = F(x) + C,l

6fiou C auea[Q€rq orceeqtr, q ono(c orc nQoplripcra cpuolxrig nQoobr,oei(e-r0r dnd xrE ottvOrix€q.

Ou 0aolx6E l6l6rrpeE ror coquorou oloxl.qq6pcrog:

,Y|ITjMO MA@HMATIKO BOH@HMA 279

AX I o,,rl= r(x) dx.

tt latr,ta,=^lnr*,t , . r

6) J

L f tx ) ts (x l dx = ) f t x ,dx !

l*,,.)=*,'bd

loo*Megrxd otolleub6q oloxl4qrirptrtc

f(x) lto a' ftx) lroa,

XnX n * t + cn + l

1x ln lx l + C

e x + Cd ^ n

lna

slnx - c o s x + C sin4 x -totx + C

COSX sinx + C1

l a ' - x 'arcsin \f;) + C

tanx - /nlcosxl + C 1a 2 + x 2

L 2yss1n (L) + C

flo rw eteeon tov o].ox],,4Bopcrorv lq4olponolorirnor, or.rvri0tog ol eEnE

611o u€0o6or,, or oroitE pas 0oq0orlv v<r cvdloupe 6vc lol'{n},'oxo o}.oxlti-

erllpo o€ 6vd, { peqr,xd orol1el66l:a) Avtuaaaotaoq pcnS7rytfi;, f lc rov unoi'o'yrop6 rou oloxtrnQcbpc-

roq I

r = | t(x)dx

p:roqoupe vc cwtxcroorrlooupe rq peraB?'rprl x pe pr,c tr?,'l,q, r.X. xnv f, XQI-orponor6vrog q o16oq x = EG), x6r,e, rdlQvowog ll' 6rpq 6rl dx = E(t) dt

x0j f(x) = f[(Q(t)] = g(t), ro oloxlnQolro lQdQ€roL:

,YNroMo MA@HMATIK) B1H@HMA

r =[ s{t)vro at.I

'Orcv ro uno),,o.y[ooulr.e, enrorQdeoupe finv oQXDi{ [€ro{3lTtri x XQnoqro-fior6waE licr ndlt q a1xia1|| x = q(t).

D naQeyoyt.ni o),anlaigaoq. Mloqotpr va lqqorpononiooupe rqo?(eoI:

I I II r1x1E7x1ax = | tO a\q61= fd) a(x) - | v(D r(4 dx =J J ]

t:Kx)E$)-l qtsatrd.)

Ogrop6vo oloxltigorptt. Av F(x.) elvor, ro a6Quoro olox]uriQtopo, x\E f(x)ioxur[:

t ' r 'I f(x) dx = F@ - F (a)= F(x)ll , ra

IDroflF€E tov oel,opevov oluoxl,rporpdr<rw:

q | f t x )dx=o,I

[ o t "$ | Kxtdx=- l War ,

J" lo

t lT) | cf(x)dx=cI r(x)dx,t , L

[ o [ o [ bat I lrt i ltetxll* =l rtxt*xl eetd",

t " t "L

t c t D t o

o I r@**l r7)a*=l 4x1dx.l. t" 1"

Ot tgonol ol,ottlrtip,<ooqg eivcr, i6or, zcr, mrprercglppf,vc:

,YNroMo MA@HMATTK' BoH@HMA 281

t) Aw.r4atd6tdoq pctuAl,rrttig flc rov rmol.o.yr,opo rou ol,ox)lqqdlpc-ro5

[ 'I = l"r(edx

'

XQrfoqrofioroups q oxtart x = q(t).'EXouVe ffie t = KxI f[EO] = F(t), dx =q'( )dt. EJ.oVivtl,E, ro o)'oxlrlErrpa. yqaqercr:

1 = ["0' r61,p,61ar.J ",o,'Orcv ro unol.oylooupe 6ev Xperd(ercl vo. snov6],0oup€ ornv aexrx1 p€-

rc0lnri.$) Ilagayowtxti oloxAtip,tootl. loyiset

[ ' l b [ o

I f(x) v0) dx = t(x) E@ | -

| v6) r(x) dx .! " t " ! ,

Etrrcp:rf lrn oloxlrlq<irpr:tc. et r=xf +yi+zft ro brdwopa 0€-onE rov orlu€i(l)v plcq xcg-mil,r6 L Jro! ouv66€r rc oqpe(,cA xcr, B oro XdAo, (01. oX. M.l l) xctr" F = F, (x, y, 4 f +F, (x, I, z) i + F, (x, !, z1 ft. pro.6r<rvuopcrr,xq ouvriBrq-orf rtrlv x, y, z, "c6te to urr,xqrmi)'r,o ol,ox)'riEopc ei,vor,:

[ " t II Fdr= | Fttr =

| (F,rtx + Frdy + F, dz).l a l t l t

fra rov urol,oytop6 rou 6r,onqi.voups 6uo lT€errlrdo€rE:c) H xcp:rri).q L 6ivercr, na.p,cprrerrd on6 rr5 elr.otir,oelE x = x(t), y = y(t),

z = z(t) xcr| oro oqpela A rqr B ovTtoxorxouv or, rr,p6g tA = axo,j te = b, torcnci,qvoupe

I l f .

I Fdr = | lF^r t )e! + Fu( t )YL + F,0oa .l , ) ' t d t ' d t

' d r )

bqlc6{ ro enlxc.l.urri}.lo o}'ox}'riqropc ovdy€Tor oe c0Bor,opc 3 }row6v o}.o-xlqeopdr(t)v.

fl) H xcp:rflq oe[,E€rar afid rr,E efr,otiloeq y = y(x) xol z = zlx) wrL ariuonpe[c A xor B cwiorot1oriv or, rr,p6g xA = d rLCtL xs =f. T6re €1oupr:

2x11!tG M.l I

'YNroMo MA@HMATIKo BoH@HMA

I tn,

l r* = I fr,at * r,atfr * n,at fil a,,l L ) " '

or|trcbi rcu rdlru eloupe vc utolroy{,ooqr dogouopc tquiw xor,vtiw o}rox}"q-eoUdt@v.

Av n xcFnrll'n I elvcu x)'"elorri, 6nlc&i ro oqpetc A xcl B oup,ni.nrow,ev6 ro prlrcE rr;E l. elvcr 8rdgoqo rot prper,6g, lq,rpqonontipe to ot4tFoi.t-ou6

tf Fdr.) L

Aravtrioerg Aoxr{oeow

KEOAAAIO I

x"=Q y" = - ys6s (qD/g - - 0.06 A

cl - I ,3 ,z l\ 1 0 1 0 5 '

r.4 x"=rl t,71pz -r? -h' l ,y"=rl Lry6z -rl -r l ).r.5 a = g y" = nJ(zt- t,{1).

1.6 x" = 0, lc = 3II/5.

1 .7 x "=o , y "=3 (R+H l24 2 R + H

1.t

1.9

Lt2

1.13

1.14

1.15

1.15

r.t7

1.18

1.19

L20

t.2l

P) tw tanq > p, F = gpmtanE / [2(tanE - 0].Irr rang -<p 16ox6E 6ev 0c llvr;Oel 6oq xor ov elvcl q 6tvcpq F.

r.zz h.u= 1(1 - I /{ii\ .

1.23 NA = mgcosa, NB=m$inq, Q=90"-2a,

1.2

1.3

x" = 0, y" =!L sin 7q25.J @

xc = 2L"/3'

v1/v=1q-89119,-rS.

Q = 3h/4.

tnE=(1 -kt) /zkt .

x < 2(M + 2m) p[ l,tM, +6 M + rnr- .

tanE = 3cota, T=mg'fi+kuza, m"/mt=tan2a

x= ! !P- h 1'z\- ̂ tuur-)T = ng/4cosq.

T=mtG+P/ ' {V +-zm,

tanE = 2tand.

a1 F = g4rntanq / [2(p + tnq)]

N=m$/ ' l L '+2LR

AIIANTH2EI' A'KHEEAN

r.24 ti = (1/3) tune.

r .2s L=t t t t f -1

r .26 m = !(

1.28 h=2.6m.

KEOAAAIO II

2 ,1 a) u > o^ l t r r t<-1 s xct t>3s.p ) z < O y r , c - 7 s < r < 3 s .y ) a > o ^ { L a t > 1 s .6 ) a < O T m t < 7 s .e) Bl. oxriF0 A.1.

>Xipq A.1

2.2 u)u=rf t /2, u=f2.p) <u>=x$12

2.3 y = ( t +2)3/ i ,u= ( t+2F, a=2(t+2).

2.4 s = ehk)u|2. t=2\ruo/k.

2.5 x=xo+ln(1 +kuTt)/k, u=u0/(1 + ku|t) .

2.6 x =2t, y=16t2, tddt =2, dy/dt = 32t, &x/dtz =0, Py/ft2 = 32.2.7 x = f/4, ncguBo),r{.

2.E 4x-3y-7=0, etOeia, a=10m/s2.

2.9 a) y=pp1y2, g\ a=kf+2ntt, a=Zni, i tanE = k/ (2nt).

ANAAffHZEI, A2KH2E{2N 285

2.f0 a\ y=1a-nvz1Y,

F l u = * ' [ t + A -m1 'z , a=2kn ,\) x = 1/n.

2.ll a\ u = b (1 - 2ctt), a =-2o.b

F)A t=1 /d , S=bQa.

2.12 a) y = (nBx) x2.$)a=G/dn=[1 + (xn/xy]32 04/n).

2,!3 R = (2u3k) cos9 sin (0 - E)/coPE.2.14 x = uo2/g, y = a - u&/zs.

2.r5 u=3P = tzn/s, a=st,[7iVF/F o 17m/s22.16 tanE = 2s/R.

2.17 d = 2R = z Att+ Sl'z /'[uz- SA'z = 10.8m

2 . 1 8 a = c

2.t9

2.20

2.21

U

, =lsT-,(!91'la.*l d t2 \d t t )

q=@o(1 -e -q t ) /1 r , a

B),6m oxriFa A.2.

U e .

. ^ ^ . ^ - a a + bt @ t \ y - t , -

dD

lzd,9*,I d t d t

= 0)0 e-xt.

>xiua A.2

2.22 tanq = 2tan.

2 .23 uonn=4 e lh + ' l h2 +s2 ) .

,.ro ,o^,=\f ry:l +, *fl,

286 AIIANIHZEI' A'KH'EAN

2.2s cD =[./ ' [x'r

2,26 sin01/ sin02 = n.

2.27 9 qf.etg.

2.2E ur=u/2sinE, ac=u2 cosq / 2lsin3q-

2.2e h-". = n * Y1 *eR: .2s 2#

To orpei;o trQoooroQaqexar o:16 rt ol6oq oosq | =-$, orrou q rtlt"

yovi,a-nou ollpctilel 1 cxrlvo rou oqprlou pe trry xcrcx6Quqq eu0ela.

2.30 lra 2,>---!-' 4t @. + I,U2)

I t c 0 . u ' ,-' 49 1u2 + g[r1

JG-2ro . -*=' E l4ze[. ,*,,Erciq161olpe

2.31

2.12

2,3t

2.t4

p) Rr = P/a.

Q = 6 0 "u =',lta.

q = f& si at + Fcofatl3z I ab.Eltdlrorl 6trrrt x=la, y=0.M6'yr,cn'4 6tav x = 0, Y = x b.

z.ss ,^=- -!L-i .- ',1 L2 -u2 t

2. !6 a=usin2E/h.

2.37 oK = a | [2sin2 @/2)]-

2.3E R=eTtTz/ l? ' , lZ).

2,39 T.a rcos9 > ,[Zgt-sin|

t =(u/e\coto U -'tT-teLtnd@ co,fl-l ,

y,"a tscos? < ,{7gLsin|

t =(2u/t\ cote .

R =!.,F + 2st2 .

ANAN"rH'EI' A'KH'EON 287

KEOAAAIO III

3,1 c) u' = , - u, g) u' ) u + v, y) u'=1[G +vz xcun u'o1npcrlleu pe

rrry u yarvlo E, ttat" <bo'ce tanE = 1 v1v, 6l u' ='[ut - "t

xot n v oxl-pcrlfel pe rqv u lrovi,o 0, &at" thate tanq = u'/?.

! . 2 a ) v= -7 i .3.3 F) Iro1/Il xcr, cvnyxcic ouvS{xq vc elvou ncqrd),]'1lo rc rrcl a eivcr r1

rx a = 0, y) xuxh,xr i p€ trxr ivc l ,6)v=-3tr .3.4 q = u'stan (% / up ).

3.5 u = 10{a n/s .

3 .6 t ) u *=L - t z - t t , P1 uo=L tz+ t r .' ' 2 \ t 2 2 q t z

", ,r-j----1 . 7

v r - t u - - u 'u1 u

3.t q = 47gs63Ut * uz .Ug

3.9 n) s = l6u2sinE/9, 9) s = 2uz (cosE ! sinE) ' 6in2q + cos2q) / Gcos2E).3.10 uu = zat sin(Ez), aB = 2asin (qn). H rQoxrd rou B e{,vor, ru0€ic rrou

o1t1parlEer ytovlo (r - O / 2 [.e ro oeiEdwro €r[ine6o.

3.ll S= Zbuo / 3u.t.l2 a = uu/L.t.l3 ct) E = 39". g) D = 0.62 n/s.

! . r4 t ) un =ral ' [7+{, P) s,=a' [ i i - j /u (6xa, l u<u).

3.15 o.) us= 61p13inE, F) ut = roR (1 - sinq) / sinE.3.L6 us=vfi/(P5inE-r) \ro' Rsinq* r.

!.17 a = 2usin2(q2) / @ + r) .3.18 u2=@t/cosq)-ur

3.re a) s = s70 m, g) ,f = Rc(+ + t). ct = aran(r / 2') .

3 .20 q e=60 . , g \ L=200{ in .

S .Z t t=v t / ( v2 -& ) .

288 AIIANTH'EI' A'KH2EAN

3.22 450 m.

t.2! 2 min, 240 m.

3.24 d) s =ro(ro*\[G *An) te, 0) tncp = 11o /rtG *2th,

y1 tan e =l

3.25 (J") y = gx2 / t27uo + u6fl, pe aqryrl rou ouor{pdroE mn'rercyp,6v<ov oToolp.eio exr6feuolg , b y = gx, / (2rP0), \) y = -gx2 / QGy).

u = uo (uo t - L) t 4 (uo t - LY - h'z

,=u r / ' [ r \ f , e=u2 L2 7 (x ' z+ t )32 , uo=0 .8u

KEOAAAIO IV

t t a=k- L r , u=* t - +F + t )o . x= l t t ' - ! f +u6 t+xe ,mmm2m"2m6m6nots vsxat" xs'\ roxrirnra xdr n 06oq 14 lqovlxrl orlyp{ r = 0.a) Bl.ere opipo A.3.

F) Fy = - myo u2 sinwt. Yy\ a = {u6, y6 rttcosat l, y6

s=6r j=-y6a2sinat j .

F F..u x = t t + u \ o . U y = ; t ,

F , F.. -x = : - L f + U " ^ t - V = L t z .

2m 2m

a ) a = ( m 2 - m ) g / ( m 2 + m i ,

F) (AB) = 4 m2m3 L / [mt @2 +m) + 4m2 m3l,

(BC) = mr (m2 + m) L / [mt (m2 + m) + 4m2m3],

y l N = m ' p + 4 f r 2 f r 3 p .Ifl2 * lDq -

3.26

r.27

4.2

4.3

4.4a

txnpa A.3

l _ \ 1 )

/ lw+\u6 '+zsnJ) - t l

/ :>i

AIANTH'EI2 A'KH'EAN

4.5 'lctr-c6:XQo1va, t = I"/3u.

4.6 T = mr mzg (1 - sift@ / (mt + m2 sin2E).

4 . 7 a ) a B = 2 r = m " g / ( m e + m c ) , T = m B m c g / ( m B + m c ) ,p ) F = m s ( m a + m s + m c ) t / m c , T = m n g .

+. t t = ' [ t lM+5n)/ (2ng.

4.9 a1 =a2=at=Q, a4 = (m3 + m4 - mt - m) I / m4'

4 . L 0 a , = 1 e . a , = € , a , = - B' 2 ' '

2 ' 2

4.ll x = [(m0 + m r ) (m t + m) x t + (m0 + m2 ) m t x2 ] / m t (m0 + m r + m2 ).

4.12 L n in =t o , lna*=lo+me/Gk).

4.r3 F = (tRsinE - u2) m / R, u ='[eRsinE.

u = L /ltD + t\,t7h/e I .L \M ] - )

ut = mu / (M - m), uz = Mu / (M - m)

u ,_m(u+u )+Mu D)= t ) , y ,=m(p - -_u )+Mu- ' M+m M+m

u = 2coscp [u tuz / @t + u)].

u) a=mg / (M +m) T = mMg,/(M + m),g) a = m(ar + 9 / (M + m) T = mM (a j + I / (M + m),

\) a = m e - a) / (M + m), T = mM (8 - a2) / (M + m)

4.2s 7 =1 pmscosq, a =(sinq-]pcoscp)e, u =ltanv.

4.2r c\) Ft =W'r = 38.4 N, n h=uS!r = 64 N,

6ev petcPd),)',ow<rl.

4.22 rp = 1114, a = eQ -\ry ) = O.ae .

4.2t F = (trt + U) (M + mE = 22.5 N.

289

4.16

4.17

4.1E

4.19

a.1a oz =af g,;@-o"ty /t .

Fhn - UC (1 + nM)'4.24 t = Fo= lmg (1 + n/NI) .

290 ANANTH2EI2 AzKH2EAN

4 . 2 5 a ) a = ( 1 - m r / m 2 ) 9 ,

p , . - m 1 T - m 2 ( ! - a 2 )m t + m 2

4.26 f.ct 0 < F < lrr @ + M)g = Ft x0'| xo. 6uo orbpcrc elvcr cx(wpd.fla. Ft < F < (ltr + tt) (M +m)g = Fz xct xc 2 o6pdrd0o xrvotarcr poL[ pe err[-rdpvoq

a = F - u , 2 .m + M ' , -

Iuo F2 < F rc 2 o<ir;r.cra 0a xl-votw0r lre 8ucrpoqerlx€E erlra-

xuvoet5a , " = F - F 2 + u , p -' ', Ma . = U 2 8 .

'Otcv ro orilpc m fi6o€l (rfi6 To Mro 6eriteqo 0c xrv€iTdt p0 fiLTdXUvona ' ' = L - p ' g > a v '

evtb ro m ea €rq3eobtv€rdr F€ erlrrdxworl a'n = - lit g,6nou p3 o ou-vx0ileorrlg rel0lg p€rdEt m ].col, €fi17r66ou. >ro oxlipa A.4 tlaQtordvo-wo.t ra ay(F) xu a-(F).

4.2'l a = (Msinq - pMcosq - 4mE / (M + 16m).

4.2E ct t ao = gy, P, az=G:uaY + 'a 'z .

4.29 d) m=Ml tanq-u l . B t .m> u [ ta t * -u r .\ t r \u r

4.30 o) F > llg g) uo< "l 2aF (M + m) / (nM) .

4.3r xs = lne * t[.' { * 2.sH (i3j4 ] t Atsf/t.

4.t2 t=fttnl ++E). n,*=f;luo-fretn(t. ,*W)]4.$ u=T*(uo-eEr)e-" t ' , '=rygr*(*-T) iQ -e rg.

2 =[s - kuol .- u,^.\ - m l

E -mt m2 (2t - a2)' t - - = , -

r r r t i r r r2

zXWa 4.4

4 .34 AM=2(Ns-M)

AIIANTII'EI' A'KH'EAN 291

4.!5 u6a / u, = t[2 .

436 K=m uo - ut U6U

4.v s =6psq , n,*=$7sinq-tn (1 +sinq)).

r f .3E F= 0V+ 1)m(a+W), et=nia+-{8 .

4.39 u = u/4 rrQoE rc fidv(l).4.40 {Z qogcg.

4.41 t = 2usinq / [cgin2q - pcos2q)].4.412 lL(J tanE>p t =mu (sinE - pcosq) / [p (m + M)g]

lL(I tanE < p f = 0, ro xr$<inro 6€v 0o rlqoei.I a , \ l

4 .43 g t= ! l q :g : -1 ]14 ,' t k \ k 'R ' I

$ tna=lP'? e' - tY L.\ k2 R2 |

+u a = 1['[2 s /R .

4.45 c)N=1t49+ va&, 9) N= Mt+vAB . iJN=Mt+Veg.4,46 tanq = 76ttQ - 211.4 .47 n<3 .4.48 t = 4rekR /($fk-nLP).4.49 s = (m2 - mt) L / (m1 + m2 + M).

n.t u=fi(t -"-* ')**"-h' . *=xo-*f,: ,,(m)aa-(w-u),

, =*,- !n l t -m ( , - . - I ' -

m[ _ ; , ) )+uomtr_"_ ; ' , ; .

4.51 ao=O, sr=29.4.52 to= (mt + m) p&/4, at=d2=At/ (mt+m), dt=W=const . ,

a 2 = ( A t - p m r 9 / m 2 .

292 AnANtHzErz AzKHzEoN

KEOAAAIO V

sr ,=^{@t(F--"'fsinq.

S.Z F=ltu(gt+u).s.3 F=p(tp+e(.).s,a a = sR tr - cos ( !./R)l /e.5.5 F=lu.5.$ n = nOg-kt24u.

5.7 u=uo+uaP.

s.s I Ar = psufi, ii) AF = - psufr, tii) aF = 0.5,9 u=0.99m/s.

5.10 a) aA = gsinq - uA @A ̂ x + Id / mA ,

5 . ! l a = E s i n @ - u a n( M + A t ) '

u = E t s i n @ - u l n M + Q n t .M

5.12 FA= 74m N.

5.13 o) m1= mo g-u/v. ,

fl) mo= mt e utoe , [iI' ax' A'5,

y \ E p . t 8 .' ' I D l

$)u=ys=sons1.

mt

u

2xtpa As5.L4 m=moe-(qt+u)fu.

5.15 Av o dlovcg y e[,vcr, xcrcxdQuqo5 U.e qoaa r[aoE ra rdr<rr xal aex{ rrl060l lou orpelrou oro orolo 0Qlo]€ra[ xo o6Fa yia t = O, sv6 o x oQr-l6wrog, q rgoluri rou o6po.roE 6(v€rot cn6 rr1 o16o11

y = _ (&/au)x,

6llabf elvar €!0r[a.

5.15 y=gt/4, a=44.

5.L7 T = mo@o + uln2)z / 2L.

,-2Ip

5.19 F = &mt/P.

5.20 s= 11.6 flds2.

5.2r u G) = E In --4!-- .

5.22 t = kln2, Hnax = k?g(1-ln2).

,.r, ,=-""*[

s.2s 1, =Ml1 _"- I

5.24 u=uln tu

OtIOtJU=UO+AUt .

uz=uln M , h =0.66 s, w=220m/s.M-mo

. 1_ _-:-- | ,

28 (. )

mo-m

5.25 u1 = 134 m/s, uz = 124 m/s.

6.9 t- 6.5 s.

6.10 u=+uo.

KEOAAAIO VI

W=71 .

W=18n I .

W = mdaP/ 8.

W = (trr + p)mg L/2.

a) W = kngP/ 2 - k)'xa sinq/4.F\ u = tAx3cosE / Qm) + H,x4 sinE/ eO - rcx2lt2.

W = k/tLq = arctan (l/p), N=nep'ffi .

u = W\to t < t0= N/ @p282),

u = ̂ l4il;---EJ, yuc r2'0.1 m \ 2mPztzl'

6.1

6.26.36.4

6.5

6.6

6.7

6.8

AnANTHzEIt AtKHzEaN

6.tt r =Z*sfi +s?F, tunq=Fn/Ft=s/R.

6.12 q = 6E/E = 4p / G+tSz.ro 4 ylverar p6yroro xar ioo U€ lwEo \tdk=1.

6.ll o) P = uccos (g/a2 | 1.

i lw=Ay\ P.=!p6zf s i1z6.' " 2

6.14 a) L=utr7@-u( .n .

( l,tzwl

G

nlTY k (M+m)

294.

I r -M\ m

2

t It(."$ n=1-

6.15 u) Ax = mu

P) /x=

2hnG

cra O u=fffs(.sin(qD,

$ n- t -n&' t .

6.17 x = M.

6.18 s =2'tE7E-ElITnt .

6,19 c) u= 6nott 0 = arcta11f(t + fi)anv] n wvto

rrcu oppcr(el q rq6ir4tc pe to oprf,6wro enLze6o.p) V =mucosq.

fiq m<<M e+E xct; u -+"[2E8.

tw m>>M 0--+fi2 y.st, u -+r[Zfi.

6,20 u,u= t 2gh + mg2/ k.

6.21 cr) W = mMth / (m+M). p) No,L.

ANANTH2EI2 AzKH'EAN 295

2g (m l+mr l ! ) 2

6.22 a = a=sE.

6.2r h = 3 [. /5.

a.zn ,=VEE ttg .mcos (E ! 0)

s.zs u = r[4]141 + mz/ n) .

6.26 d) Yra x = 0, Ep = 0,06oq euorc$otiE looQQotri,oE.

0) i) fta x = 0 Ep = 0, gta\ ooroeolls tooQQorr[oE.

ii) Fra x = -f '[ +*h, P" = - *21 2],,0eaetgusma0orig rooeQorlag

6.27 0<xs3L, t =2L/ \Ba/n +L/1[a/n.

6.2s e) F=al-+a,.( ; , . : )n,-+n,), p, w=-a/3.

6.29 a) Ncr. Ep=-a x2 /2 + bf /2 -czz /2, [3)'OXL, \)'OX..

- , 2 t t t ^ rh = u \tv' - mt . ov To fiolo|ra droTs],eiral o][6 u],rx6 nou ei,vcr nuo

2ME

oxlnQ6 oro ro ulrx6 tou oga[e[6iou.

, u2 (M-ml, = ffi.

cv ro u},r,xo rou oeorQr6[,ou e(,vc[ fim oxtrqe6 oro

xo u1"r1t6 Tou fiQlolrcros.

h =R [1 +cosq+ 1/ (2cos@].

t = l / u o , s = 2 M 1 . / @ + M ) .

a) h = l1mgk,(l) H = 3ns/k.

6.31

6.32

6.33

6.34 ̂ t =( 1 * r[t * zW.e) re t r.

6 . 3 5 c l \ t = l + 1 t - A 1 1 2 hr - , [1 - I Y t

B \ s=2- t r h .' ' I

296 AIIANTH'EI' A2KH'EAN

6.36 Qt=2tO* ("- t t [9) .

6.17 8r = I

x= 1 .1 R .

q=n /2 .

n=GW2tap@+M)].

aK = Ek / QltpNR) = 0.1 ryrsecz.

q u=M1fd77GD, g u=M1fdI7GD

7.1

7.2

KEOAAAIO YII

Lo=zi - t t l -s&.Lx = m [ax (f + z2) - a, xy - ary xzlLy = m [- ax xy + a, (x2 + *) - azlz],Lz = rll [- ax xz - a, fz + ttt, (x2 + f )J.Lo'= 5(2 + 30 *

L^= -I mu^ n2casoi .- 2

t = 2b {a/b.

AL = -40 i .

o.\ t = -! meh sin2o i.' 2 -

B L = - L mPJrt sin2a G.' ' 2 -

I aLl = !

nsrt, - 245 kgrm2/s.

a) 'OIa ra oIF€[a rIS afelcg nor etvcr, rdOerrtpeto O.$) IAA = 2muL cosq.

u^ cos@. A^,= 2 -----!!---------

"t---r---------t v6 -zgn

6.38

6.39

6.40

6.41

6.42

7.t

7.4

7.5

7.6

7.7

7.9 oro rolxoud qto oq-

AIAJ'TIH'EI' A'KH'EaN 297

7.L0 o.) Lo = muR ft,F )%=marRf t .

7.ll o.) Lo = nR (hR)tt2 tz *,

7 .13 u2 = u tr t / 11t 5inE1.

7.14 L = m (1.lGMrRrl/2.

7.15 up1 = uE ,

7.16 d) LA = ma t2 sin2Eft,

% = 2mR (1JUI2 t t.

A\ Lo=mal,2 sinE @osE cosatf + sinE sinoti + cosqft).

7.r7 n=2b? /ut.7. lE Au/u=0.15

T$ u=E uo-0.41uo.

7.20 r.* = 3P2,

. _ - 27.2r r=;h l t r ' ,1 I - (2-Dnsin2q), 6rwu ,=t iX.

7.22 rr,n = G M

E.2

KEOAAAIO VIII

a r=( -2 , {1 i+s j )n /s .

E " =L uG, ." 2 ',-

u't=,[2tt x/mr, u2='[2py*lpr, oror p n cvrryp6q pdlc.

a) p', =pr= 11'tG;8,

$ n'* = t

p G + tb, 6roup q dwtvp6q pd!a.

E'K=tp @? + uil, 6nou p n auryp61/n pdEo.8.5

t .6 o.) u = (mpl +m2u2)/(mrrm2)=us'

9) E'K =, p @t - u) 2, 6rou p 1 cv4vpevn pd(c'

8.E L = L'+ Mr"x uc,6fio! Mn ol'r'rq UdEc'

8.9 tlL' / dt = t" '

t.lo r" = xcf + Yci' 6no"s

. l p . i " \ t " 2 \

* = ̂i *\t ; * ̂ ,t), v" = aJ.,\oi * ̂ 't')'p= (n+F2 ) t .

t.11 u=-u V (A - Ail / @V + AoVo)'

t . tz u"=v' [6 / @t + mt).

8 .13 R=2Ro=3 '108kn '

8.14 F =mB Qm1-m)/(m1 +m).

8. tS T = 2muz/ 30, a = f i / 3!"

8.16 0^u = arcsin (m/IuI)

u l = u

u,ux=ull +' tTA4)= 1.0kn'/s '

u, = H, r[@, n *', = \m' * = ̂"(t * ry. )

r o = ̂ffi^ !.a'. r,, = ̂f1ffi^ t'a',T - . = m 3 m t 0 @ 2 .' ' '

m 1 + m 2 + m 3 -

c r Fx = , , " " t : , a t = ; . i - ; : '

k lmt + m2l "= *#i;.m - I I . I I l r U

8 '22 h t= nBGin) ' n t=62t lmt+m2l

t,l1 U=u 1--;1--"=-'

8 .23 x=u{Ex.

ANANTH'EI2 A'KH'EAN 299

8.24 a, = Sg, az = at -- 0.

t.25 o) Kux],[xri pe ro i6r,o xtvrqo xor orr[voR" = (dt R2, + r*, n2, )'/, / {m r + m)(3) u" = atR" .

8'26, = lrrgrorql + { t -i(t - 1*, y;y ) 1 2r

6xwtanq<p<3tanEx=2mgsinE/k 6r0v p> 3 tanp.

9 . 2 7 y = ( 4 / j ) \ r g R / 3 .

KEOAAAIO IX

9.1 a) I , = m62124, F) Ix=mh2/2, \ ) Ix ,=mh2 / 18, 6) Ix , ,=mh2/6.

9.2 I *=pP74, Iy=mP/4.

9.3 I= 3mR2 / 10.

9.4 I = mL2/ 12, cve(dqqrq rqg q.

9.5 Av x q 0r6or0on cin6 rq pdlc m1 ;a1 x = m2 (. / (m 1 + m2 ), [1) x = (2m2 + M) [. / 2 ( M + m | + m2 ).

9 . 6 u ) u " = J / m , $ ) a = I a / m k 2 , \ ) b = k 2 / a .

9.7 Q = mu2/ 10.

9.8 a) To oriorlp.a 0c xlvq0ef, 6rol d)ors ro K.M. rou vc 61el yeotlprxn rd-Tbxlqa uc = u0/ 2 xo.r, yorvutrxrl lfgrrr dfi6 our6 at = 6uo / 5 [.. To x€vrqopd(cg C cn€1el [, / I anb ro r11116l. 0) Axlvqro 0o pei,vel ro olpreio noudfitxlL 2 {. / 3 wtd ro 11tr16[.

9.9 a = 12mu /1.(m +M)

9.10 s) u.= u /2 , b uB= (10- 9{2) /20, ^t) AEK= 7 mu2/4o.

9.ll a) ot = 3mua / (M0.2 + 3nE), F) O1r,, yrcri oro O r;ndglor-rv e!<oreql-x6g bwdpe r,E. 'l) ltct a = 2 (./ 3, ffiet,64 ro O 6€v r€lvsr va xlvq0e(, xol e-fiot-r6vog 6ev epcpcv(,(ovrcr, elorreqlx6E 8rndpe r,E.

9.12 t= (M+ jm) L0/3mu.

9.13 F=mg/4.

AIIAN|H'EI' A'KH'EQN

9.14 h=H t3m/(M+ 6m)l2.

9.tS u- = ,J1* 3as2q - I . , =2 '[2eh 3cc{'v- -

t-tzE+ l [ 3as2q+ I

9J6 a =flE , oxlu.cri(er. yowlc E pe ro eirine6o,6nox q = arctan (1/3).

9.17 u" = L-2gm2 (uo - ur)2 * ig

M , L 2 L

9.18 u = 2M \t3SL / ( im + M).

9.19 S=N[2L/(6n20.

e.20 q=a'ss65l1 -gL (4M + 3m) (M + m)

9.21 H =(M + 2m) (M + 3m)

f . . r . 1

9-22 o = arccoslS @'-o ' )g l .lz 12s + bJ ) o2)

9.23 H = ttNU4.

9.24 t = Loa/3pg.

9.25 u=2twRM/(2m+M).

9.26 d) a = (I pr + I2o) / (I ! + I).

B) w = I[2 @1 - a)2 / 2 (I 1 + I).

9.27 s) AM = 2Am / (3 + An/m), P) AM = An / (3 + Am/m)

9.2E x=e12/3, u=2gt / i , Q-gtz/3R, a=2$/3R.

9 . 2 9 t ) a = a e / 5 , i l T = n C , / 5 \ ) d t = a 2 = 2 t / 5 R .

9.t0 a) ac= (M +2m) g/ (M + 4m + I/r2), $) a=2a".

9.ll a = 3 (M + 3m) t / (M + 9m + I/R2).

9. !2 o=2m,[zsh /n M+, [Z @, a=2mg/(2n+M)-

9.33 o.) a = 2mub / (M + 2m) Rz ,^ 2 , , 2 ^ 2

Fl Q = I^ mu' - --:::--:--"- .' 2 ( M + 2 m ) R 2

3m2 u2 lh .

AIIANTH2EI' A'KH'EAN 301

s.34 t=u0/3pt, S=sua/ $ps, W=nufr/6.9.3s S = 12u6 / 49u8 .9.36 u7 = uo- 7Mu / 5m .

9.37 h=4R/5.

9.3E t = uo Q + 5h/R) / 7W. IIqoE rc 6e!rd.

9 . 3 9 u = 2 t [ a L / 3 .

e.40 u=3Fq;o# ) ,

9.41 @c xuvq0ei rrQoE rtr 6€Etd.

a=zLR- r . y=2F EJ7 . y= F R - r r z .3MR3MRJMR

g.4z a =1114i2rnt, 6 =F2 t2 (3m +2m,)m(m+m1) 2m(m+mr )

u =lr *rv - 3F (R"go^q - 'l)/ @t - FsinE,

2F (ml + m)9.41 o.) d =

(mt + m2 + m) mR + 2(mt + m) m2R

a , = - - E R o . . a 2 = a t + a R .' 2 ( m t + m )

^ 2F(m, +m)b l a = - 4'

( m t + m 2 + m ) m R + 2 ( m r + m ) m 2 R

a , = - & . 6 - p g , a 2 = a t + d R .z (mt + m)

9 . 4 4 u 2 = 2 v , / j . W = n u l t 6 .

^ ^ , , '2 P29 . 4 5 Q = " ' t " ' 2 - ^ t

4 ( m r + m 2 )

9.46 h-'" = 27 R.10

9 . 4 7 y = R a B ( t + p 2 ) . 7 = g p ( l + t t 2 ) R8 n u g , ( l + D 2 ( l + D p g

302 ArrAN'fHzEIz A2KH'EaN

e.4E t=yH,

9.49 F = 3Pmg2-3 t t

v,J = MRaa2SltttFL

. a= 2ltg2 - 3 l l

9.50 u =

9.59 t =

t.t "=;^,rl).f;1, h9.52

9.s3

9.54

x = gt2sinE / 3, u = 2etsinq / 3, Q = gPsinE / 3R, o = 2CtsinE / 3R.

s=gsinq/2, F =mgsinE /2, s =gt2sinE/4, u=gtsine/2.

at u =Zmn@. Blu= r lTeh.7 " U 7 -

9.55 p=tanq/2, u=1[gh.

9.56 a = (m' + m:) stltg - wnzcosq r.3mr/ 2 + m2

9.57 7=mgsing/7, d = 4gsinE / 7R, 6XL.

g.st at = a2= a =lm,si49^t,;I l#rsinw s , o, =fi. or=f,

, = 2,[-ssotin@.

u2 l q r l l= rcl i- r4)

r---;-^-'\l-!+,I SsnE

9.69 u= (M+m)gsnE3M/2+m(1 + cosq)

9,6L q = arccos 0.8,

s.62 u=(txyz),[tnn.

9 .63 a \ p= tanE /2 , P ) p= tanE /3 , \ ) p= tanE /7 ,61 u= , [gh

9.64 W=65.8J.

9.65 a= l0rt .

9.66 a=1[Dt/6h).

AIlAlrfHzErt A2KH'EaN 30i

2xtM12 eg

t0.2 T =

10.3 T = 2n

10.4 a) T=2n

10.5 T = 2n

10.6

KEOAAAIO X

g T=2n

1 ass

It =

" \

r t ^ , t f t70. t r = ln +2arc * ' lY l l ^ { ;L t o r - 1

I f n f t ) - . g \r 0 .8 T=n l ^ l | + ^ ,1 ' - " I .

\ t 5 r 5 |

-na"1 0 . 9 f r 0 B o > p , . T = 2 n \ l - - -

I t Qo -Q t ,

rlo po = qu, T = - , 6ev €Xoupe rc]"dwoo1, ro odlpo pploxercl orqv

xordordon cbLdgogtlg tooQQoni'trs.

^ f-a.I t d a . < Q D T = 2 n \ l - -

v r a,_ a,O ouvufio]"oyropdE tqE cvrioraolS tou uypot ouEdvst rnv tlsQiobo.

10.10 x ={2 Scos (@t + 3n/4), 6.'If,,l) a='[e/2s.

16.11 2a = gT2/ 4f l , t= ' I /4 .

10.12 1t. = xo09 I g.

10.13 O0 0uEl|eei.

3U ANA^',:E'EI' A'KH'EQN

L0.14 T = 2tt

10.15 u^ = xo

,o.ro , = ,o( ----!' -\' 'o

\gz + a2 - 2ag sinEl

10.19 a)

fJ)

y)

s.s r=zr ffi. q

rV2

!0.!8 E = arc cos (S / a2l),0I. oX. 4.6.

,=rr'\[W,

tn b=(./ , [D,

L=2A/3, Or=2nlE.

10.20 T = 2n

L0.ZL T = 2n

L022

t (M+3mt+3m2)

r=nlmu'

r=211 [8 .T J g

- rc-zr, .- R'v 4, ' '-o'

f r n, l ;, a=-2L$ino-- jft 'u

L0.23

o Jut,dpa4.6

t (M+3mr+ im2)

- _ o * l L M + i m t + 3 m 2Y Og ntz - IIt t

s=- 6 (mz-m) Es ine " - -

6 (mz-m) * .

r0.24

i6

< I,[-se

AIt ANr HzElz-:Pt'KHzEd2N

tn,zs T = n

L0.26

L0.27

10.2t

L0.29

r--n{@ '

r=n'1ff '

Ta/ Te- O'9'

o.) u=

"=-ffi'no--ffie'- \! t .

p r=n

,r.sn 7=zn{#.

to: t17=zn{T, 1=2R'

r;-tun+zrqL-'=^^rl1t6Tzwcnqtqt*z^*'qD10.32

10.33

10.34

T=To /9 .

T=2n

':u.fS T = 2n s=lnst d.

10.36 d) u='[@ffi, 91 '={M-tl(v-'ffi'

f i .y r=n( ' [ t , { l ) t ' rg '

10.38 F = /ngcos (q0 c}s,,t) * ^1fi Ef silPott'

(M + zm) R' + mr"

M+!+zmR'

2mg@sE

tE,

306 AIIANTH'EI' A2KH2ErJN

KEOAAAIO XI

l1..l l4tl = uue = 1.23' 10-ta s.

ll.2 At = 10-8 s.

t1..3 x = 0.74ct, y = 0.3ct.

lL.4 t=20min.

ll.5 Et=34.68",I

11.6 II = all + 2\

117

11.E

tt.9

11.10

11.11

IT.L2

11.13

tt,t4

11.15

rt.1611,17

11.1E O) t =L1[ / +uz .

I " . 2 \ l t ^ . 2 \( .=( .^ l t -v= | l l t +v=1.

" \ c z l t \ c z l

S '=S /2 , v ' =V /4 .

u =.[V3 c = 0.816c .

II = 2.64a.

q' = 118.07", S' = 3.3 rfl-

0' = 49", L' = 3.8 m, S' = 3.6 rfl.

u=2(.0/ {At t l + ( to/cAt)21}.

o) t = 7.5 6ny, P) t' = 10 txtt, \) At' = 1.25 tx\.

a) u = 0.98c, F) At = 16.2 grr.

S=5 .2m.

I = 6.27 ' 1F kn.

q t=c l [ t ' - r ' z ,

11.19 @0 rraold$er. >Tp,qovotv.

t1.20 u' = 0.66c, oxlpariEer y(l)vld E' = 172.4" Vt to"t 0.Eo,ta x.

11..22 u' = 0.345c.

1L.23 u = 0.268c.

1L.24 ct) ue = 0.833c, P) ue = - 0.76c, y) ue = 0.8U2c, E = 7.16".

AIANTH'EI' A'KH'EQN 307

t.ao cotq=ll- E , l"ot!\ m o c ' l z

o.1 v=u/11 +(m, /mr\ ' tT- t t /c l ,

g1 m = amo /'[1 = 2.i mo > 2mo .

e=' [hnlz=zzc"v.

11.25 u" = 0.23|c-

f1.26 p = 5.0 ' 10-22 kgr m/s, T = 0.66 MeV.

11..27 u=2.8'108m/s.

ll..zE &) u = 6.25' tO -sc, F) u = 0.4 1 3c, \\ u = 0.947c.

1L.29 p2/p1=2.82 .

13n 7=mocz ( ' {2 - t ) .

l ' .31 u={3/4 c.

11.92 p =t'[1 + zm6cTf /c.

11.33 t r ) r=e '^ f ! -n7! , g u=3c/s .' U l +u / c

1r] . . r4 t . '= E-P^vt-------::T] t V '

V c2

l r . - rsr ,=--4-' - o " l 1 + @ B R / m o c ) 2

LL.36 F = mo c2/ a.

u.38 a) r" = znoc'(,[1i 17 zne - t ) .

0't p"='[wr/z, Y) v"=c'[r/(TJZmoO '

L1.39 T, = Ez ( E = \cos2 E

\moc 'l

(ia.')'-?^,)'"""

11.42

Blpholqcqic

Avoe6Qoupe nro xdto cl,eaprplrd rc xugu6reqa pf]'tc rou xrrx]uoeo-Qo$! UfiaeeaoF6va oiry olllvuni ayoQd liai rc onolc regdlow cvclwl-mi raQouolaoq qE 0etrlelog qg MrguvlxrlE, o]r].d xrr fio]ul6E aol(ioerg, 16-oo ),r4,l6veE, 6oo xcl dl,ureE.

- ALONSO / FINN . <@epe),td$qE llavemmqpnxfi .puolxi>, T . 1 .

- "Ma4tiqara iDuotxrig rcu llavetaotrlpi,ou tou BERKELY>,T. 1.

- HALLIDAY - RESNIK. <<Puolxi>, M6aoE A' (H sdw1 Oe<oqic tlg>Xettl6rqrcE reerfiercr oro M6OoE B ' ).

- OHANIAN. <.fuotxtl>, T. A' (H e6rr(i @€oQia rnE txerfi6TrFog r€-erfx€rdr mov T. B').

- SERWAY. <Physics for Scientists & Engineers>,T.I.

- H. YOUNG. <Ilavemonlptaxfi tVwxi dmteulthn pt ZtiDeovtt <ptfrt-,ct t r ,T, A' .