KUIS MATEMATII<A TEKN)K 1-~1 .rtr Se~ CQNr Apnono~ ~I MTpmc "Dr. If'. EkO Tjipto Raha tJV ,M· a Cnrf"Nh Karimah

'VI H Mada EhaSWQYi - ~I\-fAdhTt-yo S',qtriq PratamQ - e!)Qr_

1. Dil<etahul:f)itanya ..

JQV\lab :

.• > (~ x W) =

/\~ ~ . ~ ~·)(WXV)::: ~ J k

~'2. t:Z xl.

~ 1: 41;-X

"\ 2 '\ 1'\ t'\ 1\ 2. ":\( W x V) = l2 (4 to -X) i + x ~J + ';j2'l I< _ y"to 2 k - ~2( 42: -x) J _ X t: I

cW l\ ~) - (4t,.3 - Xl"" -If z ) i' -l- (x'2-~- 4~2e- xiX, + (~'2--l_ Y1:'2..) ~ _J" ..,? '0 -7 -7 ~ ~ ...., .., ~

a. 01 V ( V x w) := _ (v X w), + ~ [V x W )2 + -.§ Cv x V\/)3.~x ~~ 8~

• ., ~ 2. "-dl V Cv X. 1I'i) =..£. ( x 2.", -4:l'-+ ~~) + ~ <. 4:l't: _x y'Z:.... x~) -t 1 UFc -!J 'l).'Ox (J~ e-t-

divcJx~) _~ -7div t.v X W) ~...., ~

b fCUd .c v x W) =

I -!> ~0) cur C. vV X v ) :=.

.1\

tse-x..

4i:3_X'l.'Z._ 'f,.2:t:;

'7'" /'Cttrl (vV X v) = t Jl. ~

O-t: '(7X'h:.--~2 4 7!_X':l2-h

+ k \ ~ (7~X ~~4't:1-X't?_ r=l ~ ~2..:c'-X~'Z-'-X~

WrI cZZ XV) ::= (2~:C _~2 +1~'l-)~ _ (\2~?'- 2X=t-.->f)J + (~'2.-{- 2.X~) ¥.)

rl \ .., -). 'Z. '2.. ~ 1\ ~o (VIr cv x \IV)::> (r - 2~rt; --41d )1 -cx.~-12.:c:'2-+2X -e-)J ..•.. l- 'j2- - 2-X ~) ~

cur l (\!J x "J)::. (2U t:- '2.. 2-)/.\ r\ <\- c J -1:. + 4:i (.- (t'2=l'Z. -2X=l:->f")J + c. ~2-+2X'3) J<. +

-7~ -7,-? - " 1\ ~curl c v xv'J) + curl cw xv) = 0 , - OJ + 0 "" z: 0 //

2. EvaluQsjlah I cien90n :f dQn c diben'kan, detl9qn rv1etode ~ang dianS3 qP~aliY\9oxok _\Y\9at bahwa f adalah JO:JQ I rnqk.G I(\teg ral tenebLl+- rYLen~-

ha~ilkqn KeIJU -~qn9 dilakl/kan clen90fl perpindahoh sepan]ong C.

a. D i1<eta h UI :. ~ = [)( -:J 0 eX] .I I '

C : ~::. 3)(2. :t. :=. 2X I uY'\Nk X darl 0 sampa,'!)..r

• MisQI x::: t .y~3X:2-:;:. 3t'2 dan Z:~2X::>..2..t

orLt) :=.(t,3 t 2.r 2.tJ

r r ( 1;) =- [ , I Gt f 2]

• Fe r (t-)).. r' t-t) ::= [t -3i 2. I 0let J . [ 1 ( G of; I 2 ]t

:::-t -~-t'2- + 0 + 2et . -= t -3t 2 +2e .

· f f c1), rJ.? . ~. f 1> F c7 Lt)). ~ 'C-t) .J .LCa' ' q'l.r-

J -7 -" ~ J2 tf ,(r) .d r = C - 3t 2+ 2e - dt =c . 0

.'b. f= [srn n.y/C05~XrS(I)n..x]/c.adQlah batas: OLX6~/O~.Y~2,

l := 2)<.11

. Berdqsarkan +eoremCf " Path iY\de pendenc.e r bohWQ :

~<Jah.l integral EarlS den!]on -f1 1 F2 r F3 ,,,on1\nu pada suaN dOfYlaiA Ddolam dim ens, ada/ah "~ath rndep81denT~clalqlY) D JiKC1 dan han~q

Jil<a F =- [f1r n,f3] adq[ah 8radfen dar; heberapa f4ngs (tdi DI?F ,..srad f / demfkfqn Pt =-!f / t=:l-::: tf , r3 => af'()~ o~ ~ .. ~

• Jadt; F ==' [S;( n rt.-~ I cos I\.-X I 9n fL.X]

-:? Ft =- ~ ~ sin fly -7 f = X· s;ln rt~ + 3 c 3 11=) .9-"A

-? f1.:== ~ = ~ C9, ~n= CoS- rt, X -7 f ==' X. srn f\.,~ +- ~. eo s rL-->\ + h C'i:-)'<7y i7~

~ f3 :=. Of .:= .2 ch c:c))~ ~in n-x -7 f :=. x. sin rL~ + ~ COs- n.x -l-:C .Sin rt-x .'01; 0l1::

KemudiOh :

~ (F, elx + h ~ + f3d'b) = r C ~ d &fJ C 8-X X + --. cly 1- £...f. d=t::)

fb ( C7~ Ol~- ~ - OX 8-f.- Cl ().)( d't- + - ·1::1 + ~f - d~V

b ~(j dt ;)9::- cl t:::.r df d at

Jo -' t-~ f [ XfL.) j-b-bdi: . u, ,.J (-t-) ,-l:: (t)] I-(;-=~ .

== t= C h))1 :}Cb), r;Cb») -fc>(cClf CB)

),y caJ 1-r Cq))=- - f (4) .

·JC{dl,

fee f1 dx: + f2 dy t 8 d=b)

o s. d. '/zo s.d.2o .s . d. 1 .

Ft B

-leT d

~ jCB) -fW~ f'= (2:,2.,1) -- fCo/%)

=- (J sfrY."'1.n. + 2 CO~fL + . )~ . / z.. f SIT) ~ n, - t» I 0/0 )

- l. sin 1n. :=.2. fY'

zc , ¥ -= [.:z2.fX~ ~2.J ,e; x2.+~2.~4 r X. +~ + ~ = o .

• M1Sa I : X == 2 eo 5 -c x + y +:c. :::-0~ ~ 2- S.i n 1:; :t: -=- - x - ~~ ::::. - 2 cOS t - 2 S, Yl -c . 1: ::. - 2. co S t -2 S tr-«-7 - - _

..., ,..tt-) = [ 2J;.o'St I 2.S>in t, -2 cost -2 .siY\t)• F ( t (t)) .1-' li:) = [( 4 -t '6sint. CoH), 4 C05 2 t,4S:ir>2..j,J •

[ - 2. s\nt I z.co s+ I 2. sint - 2 C0St 1-7

fcttt)), 1'<.tJ.::. -gsint -\~sln2.i-cost + 8cos 3 t+ 8~in3t _ 8sin 'ZtCOl-t

== - 8 Si nt - 2..4 Sin't ccs-e + ~ cos>t- -\-<6Si Y) >t .

. f f(r'? (t)), r'[t) dt = f - '6Sint - 24 sln'2.i~5t +'3 CO,>t + 8 ~in~tCc.

: ~ Se (CoS' ~t -\-sin ~t - Sint -3sin 2t COs -C) cl-/:;.

r ~. ~.. Y'-': . f1,-/2-= 3'2..J ~S3t dt + 31 '>-Sin?>-/: clt - ?2f ~iYl+dt- .3(:>S~ln'l:Ct.!stclt

o 0 0 0

~ 3?..l ~ Cl+ CoS~)Sint J;+ ,-~ C~ t - j Sin\ C(~t Lt.4- 31co,t L~[

,3 ] .Tv-- '!f, . S I~ e -;--.

".. ~ [ ~ 1 + 32 [~1 + 32 [ 1 1 - Bb I~J?.

_ ~3 ~ 4~ ~ [.3

- ~ + 32 -3.b3 -3

3. £valuasi 'int\gral inl secara lan.9sun.9 QtqU j!kO f'{)tln9Kif) den.ggn +eorem adjvergen~i .

et. f ::[ Y I -x I 0 J , s: 3x 1" 2~ +;z. .•...b , X? 0 " y ~ o/ =l Z 0 .

M -6101\.1; Metode L~n9svn.9

- Misal : x =- l./ / 9:: v~ 3X + 2~ t :e ::. 6 .

3u + 2 V + ~ = G --7 ~::=. 6 - 3 LI "- 2V~ .

~ rcu,V) = [U, v , 6-3<.J-zvJ

• Dj hidang a'Q~ r dqpat dibenf'-lk pef5amoan .!Ioris :

~- !fa z: X-Xo-v. -'::10 x, -Xo

Y-D X-2 ~ X -2..- -7 :=:3-0 3 -~

0-2 -2

•,

-2Y e 3X -E;"

Y:: -1x + 3 -7 V = -~ li +32 2

.., ~e r x ar =d-4 ev

i • [3/2/]

·f CS) ~ .•Fc r cu,v)) => [V, - u , 0 ]

• PeS) • N '" [V, - V r /;] • [ 3,2, 1] _

J f f. if dA ~

3 V - 2.U

'7(\ cfA

Jf Fe? Lv/v)) • ~ tu/v) duo dv.r:2 -~U+3

z: f J~V-2Ll dv.dv = J2..iV2._'ZLlV -iUT3JV.

o 0 0 Z2.::.f 3 (-3 ) z. 3

o 2' 2 U +3 - 2 U (-"2 u + 3) cl LJ2._ J (27 U2 _ 27 27 '2.

c) s 2 Lf. + 2 + 3U - GLl) d tf

J2.

::: 0 S1 Lt 2._ ~ LJ + 27 d lJ~ 2 Z

o

5

•

3 ., 102- 17 LI - 39 u'2.. + 92 '-4g 4" ;2.

- 17 - ·39 -t- 27 ~ 5 1/

Melok,; TeorefY)Q Divefgensi GaU5S

Sf f. ri clA -:::

s

Jf f. if .dA --S

5JF.'hdA -~

J3 2b~- 2:)2. dj - J 6X- 3x 2 dx

o 0

(3J 2- ~l J~)- (3X2~ x3/:)

27 - t3 - 12. +~ = ~ ~

~b F [:z. 2 '2:z.J 5 pe mL/}<oon x2....,..u2. f:: 4- / 0 ~ z:c:::.S-- .. := ~rX r'- , r J

Berda.sarkan Teoremo Gauss:

I J f. n .dA~

JJf.hdA5

ke koordlnot- polor '.5" 2- J4:-~2.'

~ J J J J.1:: dx a-'l d 10 .o 0 0

Jf!- Z S-

- t iJ J nrd'lo.dr.d-&.o 0 0

n. 2. s !!::- 2.

f £2. J :;.2r/ Jrdf> '" 4 f2. J .2sro 0 0 0 0

cir cI-e. .

J r F . ({ cl A :::. ~. ~, 0 ~ 2-5 n.. fJ >x ~.>

c. f '" [Sin 2X , -.'J sin 2X ,5 ~ L s perrYll//'.Qon Ix I L.. a I 111) ~ h I 1'1:) ~ C .

.z. .I ' r.:c..- • Berdasarkan feoremo lIive\genSJ V'4USS',

f f F . "it d A = f5 S div F dVS T

'J • Batas -haws !

~ \xlL.a -7 -Cf~ X~Q

-7- I.Y/~ h -7 -b ~ !J~b

~ I cl c. c. ~ - c ~ C ~ C .

h