Poligrafo Física Manuscrito

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·F410 - Métodos da Física Teórica I
Thrma A - 1. sem. 1996
Márcio José Menon - DRCC, sala 209
-Datas das Provas e Exames:
PI : 12 de abril
P2 : 24 de maio
P3 : 28 de junho
E : õde julho
,l
I. Programa
1. Análise Vetorial
.3 ~i Série e Integral de Fourier
.2 .IP. Solução de Equações Diferenciais por meio de Série
4. Equações Diferenciais Parciais
.l
IV. Horário de Atendimento de Dúvidas
~~e Quintas das 18:00 às 19:00 hs .
111/ ncl.J~ cfC~j li' ()ú "'"<I l/g :Ot!)
1.,(..A.-vv U
11. Bibliografia
- E. Kreyszig, "Matemática Superior" (Livros Técnicos e Científicos, 1976), Volumes 1, 2 e 3.
I
~. G. Arfken, "Matematical Methods for Physicists" (Academic, 1970).
- E. Butkov, "Física Matemática" (Guanabara Dois, 1978).
- F.W. Byron e R.W. Fuller, "Mathematics of Classical and Quantum Physics" (Dover, 1979).
- M.J. Menon, "Tópicos de Análise Vetorial - Notas de Aula", Biblioteca IFGW (1996).
1 de março de 1996
IH. Critério de Aprovação
- Três provas (Pi, i = 1, 2, 3) e nota média de lista de exercícios (L):
Márcia José Menon
-Se A ;::'7;O '* dispensa de exame e Nota Final:
-Se- A <"7, O '*. exame obrigatório e Nota Final:
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qu2tndo t V..{y'Z(
de r CD.-, f .
e' D 'reto..-
. t-JQ~h O,So
e' o te"", PD--r = Ylt)
'P0 ~ic.l.I.".:, cI e. u T>'\~
D f'4Y~Y>"Ietyo e5-
Impoft't'lte (mo~tYll'"): se Âtt;). e.~ u-'" ve.'I'SOY que. ele-
r~"'.!e do te-..qJo J Âtt) e' re~peY1di(vl,,-VCl À/r)"r==l~ t~~Tti e O~ VQ-rsoY(~.s
.!.~ tlxos (o .•.•5bl'1te~ vic te""".,):
-,
I (' lI4.YI~ I rodeTYto~
! ---;> r+Ai
r~YrÚ' ) 'oj<tnà(~" z .11v 4. ~ I ~~-.J .: ..1r= frttAt) - ?tt-)
-,
Yl tU t) - I(t-)
~)
(n_ '<Y)"- 'M os Fo Y\ c.'; e s
e) Je.peY)d~YI--) d"
\f D S E S CJ.. t ~ v C' ..5
'1fleOt_ tUI{~ Ta)}
e -c6 L \ P o te Y\ c l"'- I
'. --"?
rY)rD~ Ve.t-DY-I~\) V-
o Cl cl'"l..cl(' l..f?.:- i? (i)
=-> -,-,
E z: e ci )
-l"1
7
d c, P OY\ t,:) ""\"2..':'
fDSic(~.::. no
~ Y It Y) d e ;su c~d oj
e .s p">( 'ío. Po Je "'I
f= tp(~)
Jey\s id "l.. de ?lY7 ')
U Ci() I
-)
V (y )
fD Y'f'r F z: F (r:J ) C2z 'fY) r ()
I c~'(Y)ro ~Y2tUd~(\·QY\ •. 1 Ô'::- ~ (Y.)
I
I
! I
1/
d~ 'pY',:::,S\e'M."
e.--..., c:ll'l;-cveV\tej 0S C~'{vllaoj
~ LoS t<?·YVt1,5 J c:
:"7 S 1ST E lYl rt S J) E coo n DE N f1 DAS e
. -S1) IfVTI10DU,c1'l0
Nz... ClneY>1.th(l< esh{''n105 i'1fc.yessad05 em descyev('y e lYJO-
Vl~ento, IndepeodenTeTYtente. d~s CólUSZl.5. (o.,.""" VI~OS, no
CltSo de U1Yl2{ 'f4Ytlcul~ í ss o sf~n-lf("lf espe.qf-!eZ(y a.. pc-
stçio e. l( vdoétdz..de e'lY) +unà~o de teTYtI'O (e!.b.do meci{-
nico d2( partt-(IJI~). De 11-= dr/dt 1 a tYà.ndql{ fun dl{'rr1<?I1_
tltl 112( des.(yú/t·u e- a rOSlç~o . AH1'-n, D l'rohleTYllt se re.su-
'Yl1e YI<\, espe.clfl·O{(/.tu d2t f'0SI~1t-u . CO'(YJO r e' UTY\ Uttov~ VZ(-
'rY)os tyzdi-Io COYnc -ele...-.,eYlto de llY'l t=Sf'n-~a V8T
D
lêlI<1L.
rJUTY) e~rltso v e toviltl d~ c! \Yne Yl.5.fo N) Pzty« Se esre-
Clfl'ou' l)Y>1 ve lo r e:..- Yle(e~~~X"lu UTYl2r. b~5(' (LO'1r
uYdc
de
rv lic.toYe~. linellnYleYlte Indcpe"ldcyde5 e 9.ve ~eY'ó. ..•.•• c es ,
Fltço) €.2t5 cooydeYll<cÍ-u dtl ve t c- l1eSSl{ b<ise_ DCl1ofztYldo
a b1t!ic 1'0"- L e. J ez., .-_., 7f" }) as coovdl"Y\l,Jl<5 de \.l'tV)....., -") -} V ~ V ~
vetoY' V e=« VJJVz-,···./VtJ; V-=V,el+ z.e.zt"---I-- v C,.,
f s t 4. V'CYh o 5 1YJ t c y c: S.5 4.C o S nu Yn C<\ .5 o rA y t 1'( u I ~ y- no 9 u<t I
OS ve.toY('s d:lo: b~se t2rvr 'fYl~dulo und.i.v-io (Vc.rsovo) -e Szt"à
Y>1ú tuà. 'Y'tlel1 te pe..- pe n d I (vI. yc.j :
e"'z. I
p 12I.Y10: _. _ ~ __
i e,
ê,.e"z=D
e, - e, s: e~. €. = I
"" ""'\ ""\ '""\ -v .,
e,o e2.: e., t's = eZ.e3 = o
e.. e~ = é"•. c:'~ = e~_e~ = I
As b2t5e5 de VCVSOVCJ h1vtuaYYle'1te pefpe'ld'lCúl~Ye5 S.tc de
l)oYYlI·nà.d~j cytoI10YlVI21:lS
o c.o>idvY\to toYrrn.do pelà P:t5e e 2.5 coordenz..JI!:S e-dCnon-ú-
I'\<tc!o S 1$ te Yl1q de Coor d e Y\.à J 2, s •
UM Ynes-",o espàço vetorl2tl rode cont(v
scs Y1;ts q\)~l) U.tT) v~to...- e. esc";tt
o
, at
Y
4lJé)
d \vc f s 4.5b <t -
ch.t. S .su<l.s Co-
'\
~tr'19 do' frobleml{ V"lY1 si:;teo,q pode Ser TYl.:tI:; LDn\Je-
YJtc. do que ouho (ISto e ) to -c nã. .• as e-l:y:lY1.'55oes <tl"à"e'-
(llS 'rl121IS s-lY>'Jples). A$Ir.,elf(4 es t z \Iiadz,. ~ r
OYTl1
4
Dm~t\'I('t das COOV-rle'(là.d1<.5 do veiay- Y)2.t bltse _ por- LSSo
tuclh~C'lYl05 dt-(-C\'C'1TC5 SiSt-CYY'\à5 de (ool'denõ.dz....s e..
~'f'II(A~';es l'los rnost~4Yà-" StJàS cOY\l.JcY\le'ndas_
DIS-cult'I'("TYl05 os StStc'f'14S-'
ta.\') "à'ub.v } p laYlo rerà.~tuloty }
I - espa,(o'o Zl y- e s re\- ico
ci \iYld r ico
.orcla%eTYl COYYJo ií eshn-Y\o~ ta~d\1<Y:I~21do5 COY'\ o s ís ;
re.YYl2t Yeià.Yl'aul~v- J f'a·d\Y('rY\o.s dele e D6te-
r~yYJoscO'r)ce'tto5 'rY\ltl5 'R'ev-<tls '5obye Coo\"~e-
'n à. d Zts e b 4. s e . f't s e ~ v I.... p a Y' h é () I<\:Y I ~a re _
'IY'lOS esses CO\'l(edo5 pltYl(. l;IS+CYY\qj CO'r>f
t - J' P I' t 1 e-s tu d'l ve-DU \'a.s s 1m e I \'"I<t5 ay3. CÃ<l~ SIS Cm.. d
"""05 ar. ·~e.t, •.•\(,; •.) ?{. fO~ \ (li:.., (. o ele""ento e a re a
(PIllY\o) ov vÇ>lu,..,~ - ( e~Pll'c.o) ,
SISTEcYlflS f'lD Pl-iIltJO
Se.i4 r UTYt y>Ol1 to no f'lãno.
!SisrelYl~ de COOVdell,idZts Re.ta\19!ul2t'fes ((/l.v-te.sloino)
10
5.2..\.3) Ele""""ento de. A'ro.
O e.le.l'YI ~nto de.
Ynl hd~ re.l~s cUV'Uà.S
re.~ lllo de.li-
Constll.nt~
<!H COOTae 1'12<.J~ ç YlUM pon r" e n V1'V\ Duh-o ponto J dlS t,~,-de
do 2lYl ferlo\" Je.. UT'fl cIi rue "'I eu I de co 'lYl P Y I :;,.,e V1 to . ASSlv" I
e.Y\1 cooV'c:len~d7,.S reJ2< Y) !lu I<l r e ~ o ele.,.-"ento de 2t-y-e "l co r r es pOY) -
1
d~ 2< z.."'0 d1.. re &'lllo
de lh .•.j I t À. d l I'el Zl S eu r lias ;
)( = cre
JLrdx.~ cte
~ rd'3' ---I-----f.,..,I1~=+-J,-!5--
~ dx
1-:cte
"àt-J"õ =cte
f\~~y2t te TY[-'se) dA-
-o~--------4-~------~x
= d.)( d~
.ire ?( ~ 1y.. ih '.
J dlt:: JJ J)(d~A=
.) Detín'I(O-eS
~OYlúJefZt1'1t1", doú e.1"o~ que se ('(ulzt...., nu'lYl POYl to o (ol'i-
.i • J, !i"·,~o'U ''Ío,.): ~.~ H u. :,1 p
.e; 1 ~ , i T
'-'e-.tà ~lIS : ')( I 'tr i ;
-t L to." -v >)(: 1"7 'Po r 2<111t~ no It '(" ~ .•.•..•
e s s z, b2ts(" te...--., CllI'lé.\')bqZ;u rll:'t Ylo rlZtvlo, ,d-o €~
o fonto r Se 'TYlouiP'\C!\1tq 2tS LoovdeYlà.Jz.s "YYluc!<'("'1
) 2<.> d\V'·e.~';e~ de s \tCv5oY'e~ S:«.::' tLI\.q5, 'PoY' e u t v e
as fl'rltthS do~ c!oi$ IJeV'SOyn Fode"" ou n~;:; (DI",-d-
(0'<>1 2.t o Y'l'te..... dos e.r.(q$ {'Y"1e~'rnZ( b4.!>e).
~) Posiciu
r c. o vc t-ov pos ipt:.... do -pOY\ to P, co ..•.....o :>( e.' a. coo« cIe _
i~ n a, d.l"",·~.cq;) do ve.·II"SOV -.>l.
~ Zt coovde'f\z..d2( 'nl{ .dLV-~(~-u
lI~Y.sOV .q- ;
Ir =- '"J()1 t- J.f i [
d~ F~urq 'lh'li.(o , de lado) L
~
L
AV'I~hsi>....,do as Je.H\')i(,õe~ da b~se. r~t~Y)~ul4tY' ven,c~
'lvt) eHoll,iJ~ a I9rt~~...., e, E>S .tolo!: e t-Ce.s J z..s coorde-
Y\~ ~z..s 9. ve. J eJ- 1 'n e.YY1 o veto Y' r tu" y J~ ..., UYn z.. s I ..;, e t rI ~
Y'e..rúY\plar CO",,", os e i r e.s esco\~,'doJ (rêYY'\ 2{ fOYYY'\l< de
t- I) O f- P (p1t •.·'.•..•l .>e~ui"te) e..' detey-l) 'fY\ re u. Y) 'a lJ o, f0"<1 I o u
'-"''''2tdo:> pe.lo cYv~z..'rVleYltü das retas que. co."te~ os
t f F 0p cs,~r "-etl.t..s pod.e-o-, sec eS-se t Y>'> e "'I Q S I :li e I ...., '- J«' ,
CoYY)o ~ b'2c.se te.;, DY'lé..,'\zqk' ...• 1l''('2t no e~\pl.tC;o /
:5t I ~ ,l)~~~eY1dP""",,", do -t~\'Vl~~: I ~~ JJl . r
Pe.~l f! úd"l<s afrz.vc) hts
(00 r J e Y\ ~ cl 'l- s :
r.dz, '1 ve COl') ~e:'" r P'l --õ) -=1::-= c.te .':~{reh. q"l.' Co'1~'..., ('fi ~ 1r :cte
As
II
- - - - -- :i
/
/P
;f !
~=cte x
CUVO\).lS (que Y!e!.te. C2tso f'4yhevl.ly" Si<" Y"ehs) deter""-
por V4tlores Consr.l.ntes J2tS Coorden2..d'ts; .s~;;- d
e
'1o
eu Il. v rt oS Coo R. D E IV n- D ~.s •
'POY outvo b:do o CO"YYJf'l(.\""4Ylc!c e s s a s
ve:ysove'~ do?<. b~se v e....-., os q.LJQ:
{ "t ys o y51 e - tA.YJ~ e -rt c 'a (" 'f V4. z.oo v de Y12t d <\ % z: (te
~YSo" i e- t-Jtnte'1.i~ ~ (Ur"V4. COOydt:',",k.,J~ "";)l.-=- (re
beY\e-(l<lt~Àndo eS~lt~ obseYlJlI./õe~, pode'rY\oJ) CD't>1 b;ue
\'\\)1rrl2t ~l.td7{ Sl'rrleh·ü \'to pl2t'r1a } de.fi"'·IY U,<"">--\ ~.lstC''''"'lt
de. (OOYc:!l''rtl{(:~~SJ es(oll.-te'VIdo:
l). on~e.."" e""Ct-eye'>1(\q
~ du~~ Looydpn~d4s
o·~ "llloye~ cOVlsh'rttes J7t5 COoYde'l)l{dl{5 (curvkS C"DOY-ele."lla ~·4S) deve...., a p re se Yl ht V" a,. S I r.--. e h ...\ ~
cleH(i1<.dq.
~e. lll:(tC e u:z.=ctc S~~ a.s CUl"'-.
Vq~-cooV"de.'na:das I Os VCY'Sorcs da b~5e
52tÕ olotido.5 tOYYJ<tl1do-se ..
.!lI:h n~eYltc 'ct C.UV'\li\ coofde'O<td
4
.llz.= d~
..u"z,:hn'1tl1~c ~ CUYU4 (()orc:!eVlltdz.. Ál
l
= de
for t°Y"t"'1i\ teolVlétt·, (~ dz,. cu'("-
114. (ot>vd~T\acla pode SeV"' Y'e. tz.. I c..1Y'cunf-eve-01(il{ I pa.."t"2t_
bolll, e..It)6se ,eJc .... é- <t $\~eiYi~ de. \"I"\te. .•.e.t.!>e' qve.
d..c.h::t\'VI,· •..•~ q totntll t(.DYl'l~lyj{1( d z, CUYlJ~ (ooYde"à.d~.
:Ele"Y>1el'lTO de a:-
y
e4l.: no C<U;o 'à"e'o"'<ll 1 o e./e"nle'lto de 2l
r
ez. e.' deie •..-
'mlllüo pe.1Z\ 4tY'e.z,. Je.h "':'lhd~ 1'!./:itS Curvz, s :
C o r v a j 05
..u 1= cte
...lItt"d.u/ =ch~
J.ll.:cte
JJ.t t-c:! 4z. = c I-e
.:besse modo, dA:: cl.Q/Jl2. I onde d.l, e d~~ 54.;; os cor>Jpri-
"MentDS dos ele'Men [os de. <lYco (clepende......, do ~Iste»t<t de (/)o(dç>,,~c1~s)_
Ve.re .•.•os ~ '>C'ilU1Y I>T.•.•~ rirttcu/'lr·164C~-;' JeS!.<lS ·lc:Je,~..s f~-
t"'A o CHo Je Sn •..•ct-riz, CI':Cu l'tl""' .
S:.2.3.1) De'hnl(p-eS
·0 ri ~ e"YYl e r e. te "("e~nCt li ~
tJUlYJ~ Y'eh. (el·,{o ~) tOTrla.'mosr
ceY\ho d~ Cl'fCun reYe'"c·l~
'Coorde na Jz<s:
r: dis.rot""\,Clq de rOI1~o p 'a orl~e'O"J
-e-: àYl'tu1o e."<lhe Df' e o elto ::ll.
'CLl~V4S Coordcnàda5
r=á~: c.'Y(unrCfenÚ4 CoM Ce"l"\ ho e"\"Vl O e '('4lo I
e-"c.t-e'. tefl{ que c.OVlte~ D r
'Vc'('SOycs dlt bà5e
r: tltYl,!Cnt~ -<\ C\.lYuZ( (oolde .•..•4c14 fj" =-de
& :. r~'1~e n t e -ã cU V'Ul<. looYde"'~c:I iI. -c = crc
As d\)à~ cU'O<\:S CoO"(dCn4J><s poJc..r., Se...
·IYlh~YFef'hc."" ~......., o(\Jl.tl'1.vev f'0Yl 1"0 d.o e.!>p<'<f-" e 4.S:'I\--, Obtc-
"rYlQ s <'<5 Coor.te nl<. dz. s .
Pe.l~ debYJlc.lt'.., ..a. baserol2tY"· ··h:~ U......... c.~""",pc>yt'-l..,.,..,e'1t-.:,
dt\ ..évc.vd(' dll... bl<.5e -c e.ta.Y\'tu1z..v no C~50 e:.Y"'r) 'tu~ o 1'01"1_
to P se "YY10lJl-yYJ('nt4 . V\h-toJ 0Lue. » e s s e (4..s0 a ndQYl-
,-ul:lt'C 'fYIA..Ylte;,.., . 'huc. on€Ylt<tp: .•..u f-lX« no e...sp<tçu .. Fore-.,..,
Yl:t b"lt)e f'Dlav I Co"rYJo ° vc.vso'( Y, 4poYlh-t d"2t o"([.cr
e
,,",
p2tVl(. o pOnto P, Se. 2t COOydeYll<.J", -e- van·?{ I il O'<leYl-
r1«(it""., d2.. b~se fll..">Y\be"''"'1 V2{Vl2{. ft b~.)e Sc- 'o1<tI1n:...,
"2t o~leYlllU.,lt-:" +["4 se. o vYt.ouíY'o-le'Oto fOY rch \í
l1
eo,is_
to e.~, se &- ~<.o"'st4.vttc
a oqie.n, (o) COll'J.o
( po (0)
p
o ;)(
-t
" &
",,
.e.:.. :.----
o
(0"""'0 ~u"'-V1do &- '-':\.Y(4 ,4. or,êY\t,<~Z;..>
v:lt'riq v c .•.•..•os qu<:. + e.e sa-o
tJ O C l{, <; O '6 Q r 4. \ e- de' pe .., d '" do
dos vCYSOYIi'5,
f-\Jyt~"'e~ C4 (~(ct:
tc"",po e poy--
t-Zl.'Yv\be~
'Y):lth {7
t/(,Y\to 0 e. e :
I B- = &( t)
'5"'2,3,2) ?os·,ç.\o
Se Y e' o ve.tOY ros,''i~o do pon~o FI
Ve.Y'n05 que. "t C.OY>1po'f'lel1.tc de. r n a di-
r~q .•.-~ do uel"SOY -e- e. Y\ul'l( pof~ e- ~".
~o pc •• pe.l1d,·(ulz.y 2... y. I c:. e.~ ta,..., be;:"
a. Y'. Por o oÍ vo 12•.d"" à. CoMroY\e'1-
l e d.(' r Yl-l c:hvuzZ'-' d..", U(YSov )lr'
e. o s e v rVOfy'lo '(Y)odulo IYI ='(.
C}{.:> e.,.,. c.oovdc Y1à J4 ~ l'0 \ à. Y~.5
17~ r?1
5,2.,3.3) Ele'h)e"to de. A~el(
E" Jde •......,ú,<\c.l.:> pe.l~ <lh~" d-l., ""-nlio \i~d'l.ch
tu('u~s "
'Y'= de.
'rtdl"= cte:.
-fr ;. de.
e h!'õ :: c:tc..
D:t f,'~U'(4 ã"C\"'-"1., (",..,0 ed4»1oS ir.d~V1do C0h-] chfe:-
'l'e. ., ci~is
e.
A: J dA- = 11 r Jl'J-rõ
,'nde peY1ce,>"hne Co<>,\o
Ca.\c"le"'1o) C\yculo
~.3,1) SI~tCYl1lj, de (oofhnJths fi'dz(Y\tu!hes
-:; 3.1·1) De ti T)"cóes
()Y\s·,dua.Y\do t.•.ê s e,~o.s 'CY\uru1t~e •..•te fe-<~eY\d,('ut'l{-
'r e.!> '9 •.•e: s, e c '( u ~ 4.."'1n \J ...., I:>°'" to D ( o r I i e..,..., ) l d e l- \ V\ i", os
b 4. se; ~::l[,~ I S r à
coo<dcy\l<h,.5:.:>\ I ~ , o
r~.......,\'e."; e,"'" tv(~ chr.,eY)-
'i"e '> 2t b 2{ ~ e 'l'e.t ll. VI t u t lt ,,-
te,,,,,, o r t e Y'\ t«. $ ~ o t zX-"I no
e s rz...~'"
5.3, /.'2.) P()slOZ0
Po-r eX. re'V1s4,-'" dDIr:)()( \-1i~ ~~ 1
>:3) SI'SfEMNS tio ESPAço
Seil( f' \ln, pOllto do e s fi< ,LU
. ' No C:\.So do e s pAÇo
luY'V1 e ~'u"""''I. e -\~~'1' '\.'.., do
YV1e"lto ~Q a~c~ "0 "IÃY\o.
. c" l" V ~ ,} ~ e."" o S S u p e.y t 1-( i e)
e -z.. 'i'q \,,-'~ele. b~-<.....,\",~h
:>l ~ de
:nrcx: C tc
1",d •.
"f r-d(f = d-e
li :cl-\,
'5 H.s : c te
1(1.. . 2.lT
2..
','- - - - ... ,.
e da" = o
clt-
. a. d c. t c..•.."., 1''1 1c ç ~o d D e.l e ""·e n ~o J I: lJ o
p re c e s s e d.e. Je h: 'c'v1l nll(4... d" ele. _
A J;~c~c:..,,(~ e" <t~~ ao l"lu(!') ./.e
- A .s5 ,~ o e. 1e ''''H'''' b d Q. V c \ u"" e
,
s'-'pe-<r\Ciel
I II I
: I
.:>(.
( p IQ.., o~ ) :
dv::-,bLdlQdo e p1tQ u....., uo\(.,,.,.,~
v = J J v = J f J d)( J '~ d ;r
IS
r~e"""p1o: plY'1< o 1''íI:,r'l~1 d"\ ~Ituq,
de. lud",s LI)L2. e L~',
V'O fl) hl (lt J ~ J (J J ~
o ,)D o LI)C....---_
f...., tvé:s d\'rrl~Y\~õe.5 I ?C=crc.,
S2t.Ô re,tas -rtllt~ pl~Y\o~, t:=s:, 7.{ s,
nomi'r'12tc2<.J' svpe(tl'éle~ COl)YdeT1l<cl4s.
A pl.te~secc.é\õ de. du~ç surutlci~l co-
D r d ~ 'I 2t d <'<.s Q.' u r» à.. CU Y"\i g Co o '< de. _
't'l2tdZt. Po'( €..{e'rrlrlo 2t úüen:c<~~o
do pl~Y\D :::(=0 COM o fl~Ylc .14''=0 e.
O e..d..0 '25
~ = (te. e. '6 = c te ....•2<...--::'
Su fe"{-'ICI'€) Sltc de
1)
I,
J'
- ~ L-
I
I,"
" -;'''0
\te.Y'Y\ os 9.0 e, Ds, v ~y S o r e 5 d z.. b leI: e 54. õ pe r \",e Yl c! [ _
CVI4..ye~ "a.S sUpe-eft-Cl'es c.ooydeYl~d4:S COr're.SFo(1d~'ltes:
IJI:"cSor :t : norro2t\ It SUpevrl(ie CoOrdeVl2l.cl<'{ ::>(.: cte
V(YSOY i .-Y101T>"lltl ~ supc.-t-,cie C.OO(deY\4c1q J.ó':c h:
HYÇOf 3"; norrrqj 2< Supewt{-cie COoydeY\lt~l( '6" .=-ctC
Pã,Ylc, UM dltdo pOVlto f': c.. I
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q"dllJ" ve tOl'li.. \ e.' ;r=-dY/dt. CO'MO ldv\-= ds ~"J-
tà.5 l-Ü=/= 11lc.1 • po'n:,.." I e.,.." te""""o~ de "'lllov~s
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f "t-dr. Se If/: Irtdr! o~ IIl1for(Jr ~ :-.
e r-t-Jr iltô rupe"d"vlll'r(i
a Jr(f"ls ldr/=ds=- dih:ye'lci_
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e na" ~a' sc..,ti<!;:, ~t P1lt.'f1te ••. UJY1G1 vJ.Yi~ve\ (o'rJst<l,,te e
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f'rp',)E<IYO, E .(e"""'plo: ~:':;t,..Jí{/dl05 it f-orrot '.
F = ~~ •."t. J( I- '6 J( õ' i t S ~ 3 I Q t 6 =i c.
e. U~lT'lOl c~\,,,(~'~(6~) intq·ul d(i\U6fl»010) a (J/oll3'°1 a-)
J F.dr = r""("~'\a. "53 x •..bJ( Õ i \-01'(1J . [dJd t'J'õi thà]=
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= r>·'I~·~1líd::x.t b)(~ J'a' r (X 'õ d Y
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foi"" rc1o\vev' d l·nt~t ••• 1 tfec·15lt.\'Y')o5 e.sf~cd,íny
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a YC'f'1'orntl<.cl(·" F.tl"{-
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A) I"os ci..\,,,los ,,\"t fllv~""'OI d~ l"te1";tI~ de. [1'.,1,4 , df-
f'1:.",de"do, do tIfo de 1'•.obl~.""4, ut,lli;\Y(""os à. tO're.s,,~o
9ve. define o rroduto es(~b, •.. (f'f.I!= ,~11Il\(D<,,) ou' "- e..ti>r~~~~õ do f'r'od..i"':
to 5('1121 y e....., a>olh"l\"",S «t"Ln'tul~le~i s,« = 11•. B .• t Pt~ B.r) :
\d'f \2 ::: d? d~ I tCfYlOS
ds:: ldrl= tdf'-dY'
tA..Y"'I 'b(...., elerio roi "11 ti o de 'fY1 e_'1,---t_o__ J...::e_Ji,!\'.~_. g,u_ ..~i.k.-
de LoT'l 'f'YI •..•.•~"to ele <tYC'o.~
Se te '0'\0>
eu YV" I I~to
co..•.•..•l' y ,',..., li! " to
= J l.a.d'Y
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Jlb Jt Il I ~t~~I):; Vs!f.-ll Jt '" { (ir. ir) '\~r ~ [~,lj~ Jt J:t JJ- J.-
~
B) U.•.•.... do; ro,,~os fUY1J~"""~'1ti\i1 1'\0 (itlcul" de oJ'P1<\ ('t1-
te.1iu.\ de. \I''1~~ e.,- z,. eJ.rec·,HCHi ..• do (1<.Y>1i'flho J~ \'1relylt-
~ ,~~;e. (~'O'Ji'1\'o e: e.~re(lhl1{~" p0>' UY"11{ ~lJY\(~'~ Je =:t I
~_cõ' Po>, e-tC""I'f'lo ,SI! '0 (2\."""\ '1\) o ter' lJ"'M~ (\''':Cú'''!-~-
Y'c."lÚI( J~ Yklu f\ no rl't'10 ~ '1' e (0-'" (eYlt •.o 1'1:1\ ortt-cl"i·
)( t t- "O" Z = ~L
c.
faYA r ese lvcv ~ ','1 te".1 deve· ••c
ou F::-Ft •• ), $:=St,,) ~ \'1te"r,-t .•..
• Coofdf .•..•~hs Re. ~.•.<\tu\~yn;
O=:df;i lI::"::\ tFt 1)' ('!;d tJ'«1) =
c. c
s: J F.•dx t F; J 'f
c
Do ""~""'o
ZI. lYlh>'6u.1
c't·... de.
ModO, 1'''' y~ roolur: .•..
~~ve·s{ es(y~vt'" foI
oo tud", e:......, f""'l,l\'"
r:SSÀ~ re\a(;>·e.s C~fe(d'ü.'<>1 o c~....,\;,~~ (vl·.•..•cul.) e re'n"''-de"''1
<tul Se. eS(V{vl{ do a~ 1/.\ "'l2i:ve',~ e'TVI f'-'" (I{'" d e o..., ~ ("'o~ r
q"l ')(0,1'_ ~i'i" s.tõ ,h~.:>} 1\0 fYoble,",~ ).Po"e.\~"'rlo,~m +U"Ilf:-l"io J.e.:t:
;t ;'-~~o',)l/ e -~7~ = -'i;~J( /
e d~;'~I;'''-:-odo' d ~ = !:. dJ{i c d t = ~ dl (
71~ ~u"
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'ou t ido o o vn \ ;" '-1 o
v.l.-<i2i.ve.l (x.). [0....,0
:{:>/.of F. dY: f. f\ (~o Jd('5o )() h
c -t,,:> \::(0 )l '-*'
F
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re c , '~I( "-..., o o, T<'/ I'" I, o C ?
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"I"{ o C"""'..,\'o c c' re..-coniJo 05 ÃY'J-
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ta.,., ~ = ,!o = 't tA"~.:- ::{o = l
x- ;I( 1)0 o
e n tA:~ e s r e \.\ l'x';' ~.,t-,.c:><,
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"VI te. '\1".\ I.
l'edú!\, a.
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<t ciJY1>t ty.t~ e"TYl-
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e e:= e f..) I
c..
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C, x
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elE! U'ro1>t SO- v~ ill..ve I
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2'C~ ••.•itlho: re la s de to,o,o) it (~o,O,o) } Je (:(0,0,0) Zt (~., '6"0,0) e de
(~o,~o,o) Zt (J(~,K~,õo)
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los he~ CÀn,·,.,~o.s C. t-C~t-C~· ,,\dl(lcd~
0.) D(hr.icz~" do f'roJvto es.ca l~r.
'IV;..( F.d7 := Sc P (o), e ,dr I s: fc F (OS e eis
I F.d y" J F. J y t f F. d r t- t F. cI y
~ CI CL
f.dr= ().HJ)(.!)J(ctJ'l tc=t'ódt"
ele .e. (., •.~J~clo) a lbt~·"C·I~ J" f'<tit;cvla ôt> F'o,,to
C/J.~" ';"s tot"1 rc.: (Obs: ~,.;; ,hJ~, ~p~n>s)
jF/=F=Ke " «:.0....
'N .: /I; fc ~ (D~ e d S
VA.MOl esCYCV(y I'bdoJ o), te.",O)
lJmlt 50' ".\il~Ve I e, Jc "Moclo a
<! li!o , E'~(cl\'~"".s o àY1tvlo
o< ,ndÚllJ" .
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D~ fl~u'rq
J() :
J, Z=C\. Z t-o...1. _ 2o...l(o:' 't' =
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do i'ltq".,,<1:.
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e •....• ~"" (.,<-ú
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d<!.h"i~It·~ .tt.. ~"'tvlo ~.,." ~-\dllthO'
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~ r~i.{ :(::-::(0 ,J:>( = ~ } ~;3.:>C3 -3':l!" , .1'1:-';> J(~ F. dY:: f c:xo~o d~ z ( l:'o '6" ,!,o1i:o~~.;,. 8:;)
, , IJf.J( z: C~o'roOO
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Sf.""(lrcu"~er~,,ú.t Jt: '('/tlO o, Y\o pl.tno ::t't.
Calcule'o tr~b.tllto v-eall&-tJo fel1t ~or~~ F
que 21.tqj 2t f.td,'cvl'l r~Y~ o /,o"to (0..
1
0)
e. curo n-tódu 10 e' I< \/e rc s a. d IÜ il1Cl'l d4
pllV'tléul'l 2t ~S.le f'""to. rte.~olv ~ o rro-
blen-tl{ 2th~\lt; d~ ~nt'e,s.aõ ,\vt deti'1e. o
rrodvto e.!>c~llty ~'lt~".lveS cl.z.. ~q're.ssã: • ./.0 pl'odut-" ~~c.~\a" ~"1 .
c.oo r d<!. Ylz..c!O\s retl<Y1,!v l",y~s. (c .•.rte~li"l1U),
A5 c:.t'J'rOJo(:) (I) ,(L) c: (J) h~~<:.Y>1 ~.,...bu t..~'l.s
~!t~o. SLJ6stit"i"1~~ ..,,~ ·\"tc.plt/:
'N=- Kl-YiQ..lll-(Ol~)' (OS ~. o, d ~
c ••:0
W=-'l.l\Q1..] SeYl~(o.5:f d~ =
2.. 2-o(:/T
e de. fI- (os~) ~ 2 Se.., 1.0(
Z.
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ú
o 5(';' a< J a(
1\1.1= "2.1<0..'1
o
b) c00 y,lc ...,li cI 11.} r e.t 2; ..., ~ LJ\ ã •e.!l
W'.: fc -r. d1-:; Jc F~d )l t F,.. J Iff
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e eXrt:vCY tudo e.n-o tCf,,",o~ d~ UYY1~ va.'riztvel
d2( 8'C.O.,..,c=tf,·q
(a,-n't"/o ,,<).
47
F te. Y) \-n
c:kd \
1
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-v, z:
.t=1
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1'..;- ~ se '1 "" -? d~\o..cos-.' Jo( (51
~l= rco~ 4'
F'l s: - F5:c.., r-
) 10 -~ """
cp .z: f F. d~ ~ i
s s
p~,(l, UI'rl'i S.Urevfl~I·( fectn.ch \l1d\(~lY105 ~ _- ~ F-d~
CQ."O \J(Y\ç ~J :;ob 'ré'. S e.Yl-\-\'<! '" de ri',
S\Jpc"f;cie' fecn\.dl, -"J s;-e.-·üich P'<.Y"l< fO(~
Sllpe"fl'Ll(' VYl<.~fech),0~ e f\Y\\t~ -::> rt'<'J,. fU50 de ro~(l. cl~"cd~
(CO'n-t-<>fl'10) .
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""~Y)h\ "'- e!>re.C\·hc ••.c.~.., ~s:> o.."",,';'\,<> Je. \Ylh--a-V4f-\....> I''-v.''-
as I~-\{'\"~t~ J<c, SU'f'e,(\(\e. o Fo<lr.::> _c~"t,,'<.' -e..- 4. c~l'e(l-
t\(X(~c d.,,- 'tye-.. '1)-'" "'l ...•••.\ 2.... \·nt(~v-../'<.o -e" 'rc'<l,~"c!'t
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to J.Q.~ye", ("do .•.) da= ""d~ e. pel05 hTYtI\-e) ck.''1-
r e 't y l,. ( ,::" 'n 1.. s \J ~ V' \ 't v e I.) ~ ~ c.0\'" ',~ '< ) (j \V'1 '<:. t ; I ." J ,::;,f' v::> _
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7 2. 2. E t\.c.'<"1 r to
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pO'()tvx \ ~ 2..tr~v~::. d~ u'<.... ':>ufe"t"((C
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F'((,,) ::--Z-K"'- !.Q"-1 5e.'1{f[ -1) .z: -k~~('..,-{ (7)
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w= J~-Q[-j(o,.l'{(-(Os«')se"'-<_kQ..{. se'7-.'CQJ",j c!-<.-=-
,<,;-V IT r J
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de. "'''IVl c~ -,...,f o IJ n \ to r 'r">c=:
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