!"##"$%&'(!"#$%&'()*+(,-+((()*+,-).",)"/0123425" (

Hence we get the x- and y-components of the surface paramtzn fctn as:

05.01-1HWP 05.01Introduction: Constructing the Surface Parameterization Function

05.01-2

!

!""#""#"

2

05.01-3

2

(a) Infinitesimal increment vectors

2

2

05.01-4(b) Infinitesimal surface area vector

See drawing above, showing -vector in the F -plane, attached to a point on the paraboloid surface. The infinitesimal increment vectors are also shown at the same surface point.

Corr190401:

Corr190401:

05.01-5+x2 ]

+ ]

)+ ]

, , , , ,

, , , ,

,

, ,

, ,

(1-

(c) 05.01-5

05.01-6

[ + ]

[ + ]

0t 0e+

ci"rrt (a.u q5 q."tir

f ,- vec*ors

,ayq'sirr fi-dir".tinl l-rygyses fkc siJ t r i t4|.e J . S O

05.01-7

Route I:

(d)

(*)

(*) Note: After shrinking HQ to zero, the -vectors on the"flattened" paraboloid, P(HQ), are pointing in the (+z)-direction, i.e., into the volumeV, as shown in the drawing above. But the surface Q is required to have its -vectors pointing out of the volumeV. That's why the -vectors on surface Q must be reversed relative to the -vectorson the flattened P(HQ)-surface.

V

V

RouteII:Wewillusetheresultsforparameterizationofacirculardiskshowninthethein-classexamplefile,postedontheHWProblemspageofthecoursewebsite:

hw05R_P3900_SurfaceIntgr+Stokes_Pt1_CircDisk_V01.pdf.Pleasereviewthisexampleand,inparticular,payattentiontothefiguresshowninthisexample.Inthefollowing,wewillrefertothisfile,andtheexampleshowntherein,as“CD”,forshort.

ThesurfaceQisacirculardiskofradiusRinthex-y-planecenteredatthecoordinateorigin,i.e.,withacenter𝑐 = 𝑐#𝑐%𝑐& = [000].TheparameterizationfunctionforthisdisksurfacegiveninCDis:

𝑟 𝑠, 𝜙 = 𝑥 𝑠, 𝜙 , 𝑦 𝑠, 𝜙 ,𝑧 𝑠, 𝜙 = 𝑠 cos 𝜙 ,𝑠 sin 𝜙 ,0 ,withintegrationintervals

0 ≤ 𝑠 ≤ 𝑅and0 ≤ 𝜙 < 2𝜋.

Theinfinitesimalincrementvectorsof𝑟 𝑠, 𝜙 aregiveninCDas𝑑<𝑟 = 𝜕<𝑟 𝑠, 𝜙 𝑑𝑠 = cos 𝜙 ,sin 𝜙 ,0 𝑑𝑠

and𝑑>𝑟 = 𝜕>𝑟 𝑠, 𝜙 𝑑𝜙 = [−𝑠 sin 𝜙 ,𝑠 cos 𝜙 ,0]𝑑𝜙.

Wewanttheareaelementvectors𝑑𝑎topointinthe(−𝑧)-direction:thisisthedirectionpointingoutwardfromthevolumeV,enclosedbetweentheparabolicdomesurfacePandthecirculardisksurfaceQ.Therefore,asshownindetailinCD,wemustchoosethecrossproductoftheinfinitesimalincrementvectors(with𝑑𝑠 > 0and𝑑𝜙 > 0)as

𝑑𝑎 = 𝑑>𝑟 × 𝑑<𝑟 AsshowninCD,thiswillgiveus

𝑑𝑎 = 0, 0, −𝑠 𝑑𝑠𝑑𝜙whichdoesindeedpointinthe(−𝑧)-direction,asrequired.This𝑑𝑎correspondstoacrossproductofthepartialderivatives

𝜕>𝑟 𝑠, 𝜙 × 𝜕<𝑟 𝑠, 𝜙 = 0, 0, −𝑠whichwewillusebelowtocarryoutthesurfaceintegrationviatheparametrizationequation.

Compoundingthegivenintegrandvectorfield,𝐹(𝑟),withthetheparameterizationfunction𝑟 𝑠, 𝜙 ,weget

05.01-8

𝐹 𝑟 𝑠, 𝜙 = 𝑥 𝑠, 𝜙 𝑦 𝑠, 𝜙 ,𝑦 𝑠, 𝜙 𝑧 𝑠, 𝜙 ,𝑧 𝑠, 𝜙 𝑥 𝑠, 𝜙 + 𝑥 𝑠, 𝜙 G

= 𝑠Gsin 𝜙 cos 𝜙 ,0,𝑠GcosG 𝜙 ,makinguseofthefactthat𝑧 𝑠, 𝜙 = 0everywhere.

The 𝑠, 𝜙 -integrandintheparameterizationequationforthesurfaceintegral(seebelow)isthengivenby

𝑓 𝑠, 𝜙 ≔𝐹 𝑟 𝑠, 𝜙 ∙ 𝜕>𝑟 𝑠, 𝜙 × 𝜕<𝑟 𝑠, 𝜙

= 𝐹& 𝑟 𝑠, 𝜙 𝜕>𝑟 𝑠, 𝜙 × 𝜕<𝑟 𝑠, 𝜙&

= 𝑠GcosG 𝜙 −𝑠= −𝑠KcosG 𝜙 .

Notethatthevectorfieldcomponents𝐹# 𝑟 𝑠, 𝜙 and𝐹% 𝑟 𝑠, 𝜙 donotcontributetotheintegrand𝑓 𝑠, 𝜙 ,nortoitsintegral,sincethe𝑥-and𝑦-componentsofthepartialderivativecrossproductarezero.

Bytheparameterizationequationforthesurfaceintegral,wehave𝑑𝑎 ∙ 𝐹(𝑟)

L = 𝑑𝑠MN 𝑑𝜙GO

N 𝐹 𝑟 𝑠, 𝜙 ∙ 𝜕>𝑟 𝑠, 𝜙 × 𝜕<𝑟 𝑠, 𝜙

= 𝑑𝑠MN 𝑑𝜙GO

N 𝑓 𝑠, 𝜙 =− 𝑑𝑠M

N 𝑠K 𝑑𝜙GON cosG 𝜙

Usinganintegraltable,wefind𝑑𝜙GO

N cosG 𝜙 = 𝜋.Hence,

𝑑𝑎 ∙ 𝐹(𝑟)L = −𝜋 𝑑𝑠M

N 𝑠K

=−𝜋 PQ𝑠Q

N

M

= −OQ𝑅Q

05.01-9

(e) Integral over the entire closed surface S 05.01-10

By given Concatenation Theorem for surface integrals:

Insert results for P- and Q-integral from parts (c) and (d):

q.e.d.

!"##"$%&'(4"#$%&'()*+(,-+((()*+,-).",)"/0123425"

05.0205.02-1

the

This is illustrated in the two figures on the next page: Please review and make sure you understand them!

is again perpendicular to the F -plane, just as for the

are just concentric circles around the z-axis; and so are lines of constant s and constant z in the volume V.

. This is due to the fact that lines of constant s on the P-surface

05.02-2

05.02-3

05.02-2

+x2 ]

05.02-4

, ,

Substitute: u := s/R, -> s = Ru, ds = Rdu

-> zb(s)=H(1-u2),

->(zb(s))2 = H2 (1-u2)2 = H2 (1-2u2+u4)

see HWP5.01 (e)

, ,

0tI

05 or-f

4 g = 7

-4ii.,.

_? , :

0

.-tz l

-21r lr j

a

1"

6

**9: fr,T3 o,r A"'-+ { h.E 8, , ,:. : ,.-- -a'+d; KTe ou Qlgn) * (Bs tB,\ = (ctfr)

T4

- 5

_ _ . t4 1

t-

t 9 ( e t

- 7 i I. Ii

- t

,3- ,_i

0

4r--- l

C.- 9 :. Dz{q-3if I = 4

c7):t!t)

I

(

?.2)L

l-r t)

ae= t

0s,03-"-

Et

J0

0

Ft

o ? 2 i ) |U Q+zi) (- . )o (-l+r4 ?li ).o [v - r'1 Qt*ri/

r l(z) I(r,)(+ rs;) I

etr)(€+nil

frfl3 orr Er- U-ts)fr= F. tr=iiri

(-2il€-nti)

I4 +l3i4r :

t'|

- $.; l;J: .t) H, n - /\'Krl ' l or{ b+ + Grt-

. - :

"la = ut t u = l

t ft-ls;)= (-ror-a)*ri) (-tt tli; = (- al-r{ i)'[]{f t -(-{of-zri)= LL-ttl{

09,0< - rJ

3y hg\cr ev X utiot IQ,.' ben:

&d(U) = Ua U". : 4r U+r

t E , ( r , t 6 . 4 U

t l1 t r / |W @ t

a

L,_(4/9;) , :;) Qz- Ltti)

(a;) Qz- ut;). . t t

ddSI= --8t - qv i

,_ _,=:----*:. -

A f n z l rvJ t v5 -y

t

t 0 -il-:- 5 0

r- ?Tt l ( lt+i\r" +(g*i)z -

? el = /to,nk) fr,,K,,(*) frun (")@' ,,;R ,oru,V

Rr,' (a) ': An,n * {*oz5 o t

frr,r.(n) = /,r,rro]*[lr,r(n1 ?

$ot fl:l t(1 ,,, M

F&):\ $ n \ '=(tZr, . , t l

i^ii'k:kW + fury:T.'er *, -*'il! ltrxt;ln+d#; ;ji {: o' c"' rri bq{us tor $,:JiV, F[n*rfc{;'q, 1ra o;i+*ftil,';ll5|i5 qivst uy:' ' ! l ' , '^-( '1(o(rr; =o {uo t ' l rzr ' - 'M o

oS, o3-S

N

w

o W*f i

cou 51fis(, qulu- i*n* ,u, tC)turqtt iV'

'For q. pw,^afu.( io,t-{fec"r,lv, b,r ij ry .}+

? (+} i r

fuo 3o"{*) {-" ecr#i &*i e l-A,$i, etct, f".{tt Kr,,r(,,) 1ouf

tlri, Wq"us

{d' qLl

?M,

Lwu- f!", w*viv 4r*ntuet{tniq# Tn()r*

e , '

qu,z'lCo'r ff{ta

-W

T lrc lel2r,vq',a a41 o 3v3 +.rrfni,t, R ql)x ete = 1 . 2

Jf=

i:{i r

tte

$otl ar

( ;ou,^lq.ct lr r.. ?c(.q o lhu

I D: &z 3*3 R=,+ l ) . R . = R . . ? - -

It-= O"j J)

f i J = ( 3 , 2 , 1

tr- B€hd;

litee_ 3*

dlor2

0 t r ; - z) (- l l -3 i-32

- t+- ( i

+{41 ovd_u

,1 '11tnn"btvs

"Qi.'l

we'{e p uakix eltq

{ornrroi

= 4 0 l d r ? + z " + q ? Z

Q o : - 2 0 , = 2 . Q o = 3 . o t - = -

osos-:l4ee

t'/ ;tl

or vicL-

: p .€. g.

!

!!

!

!"#

!$#

!%#

!&#

!'#

(#

'#

&#

%#

$#

"#

!&)(# !')"# !')(# !()"# ()(# ()"# ')(# ')"# &)(#

!"#$%&'()

&)

!"#$%&'()*+,)&)))%-../0')

*+,-./01#23)#/#

!"#$"%

!"#&'%

!"#&"%

!"#"'%

"#""%

"#"'%

"#&"%

"#&'%

"#$"%

!&#("% !&#)*% !&#)+% !&#)(% !&#)$% !&#)"%

!"#$%&'()

&)

!"#$%&'()*+,)&)))%-../0')

,-./0123%45#%1%

!"#"$"%

!"#"&'%

!"#"&"%

!"#""'%

"#"""%

"#""'%

"#"&"%

"#"&'%

"#"$"%

!&#()"% !&#((*% !&#((+% !&#(()% !&#(($% !&#(("%

!"#$%&'()

&)

!"#$%&'()*+,)&)))%-../0')

,-./0123%45#%1%

ft q <qutt

ia o\ vq [4e

t < N 4 l - I a s -- - l / \ v . _ V -

ogos-F

Tc.1^4 ttt l t '

o,vt €iq vvtV<cfor <

on e.i grvgrir of t1 ryrllJlq sqtatts s ' c q q 5 e

0 * q A

0tf o[

V e c

gt lsns!.'!l shcq;. lsrr-s tb g'rd !1

llow-28

we*or is r , \

-- - - l

echv

-Zeto yLctar Ctwt dq2t4

9owtz V€.tw Coarvertoft 9nI 1y,

tgto '.= o

lsk ariv- al rcqrl Iqe+l

ov i5et = f g t zq Vec

09os^g

S-r

we can

: q "

awsvtr, t

' , 1 , . r A , -o t r we

l z l t t . , - ,tfz z l 4 z t " '

trfe 1

g Coltrr g\UeLtwakradw s ,

I .q,,e{.fVe *o.

c!*t L Le

Q

05 t q t

\,c,? Lgp thu o(4,

otf S"[u i;t (1,;{1" j t"

oqlFo

irr"v V {,

09, ot-6

0w44r "* +/.{go(tfr,'ouri lrj. oLeys

ff )o i

h e i V W I 0v.. u,t le4 ,

=

- ( { ' = 0,o ut{ndu vurtu,

tiyerw*t"t+

qVeal0€J Vol Rr i

ov- ei

owl bol4<,II b<- -ZerO = 2 . , w i ' .

t Qz=l aLtd,Youtrg f,, r ) , ' - lQw?L'e &I<iqwiie.t'1, tS Mt r 0 o S{ep

09,0r-7

.t+ *+h<ge+rafuLl4-

t[i-i,;ioYs ; daet not erfrL -lfsol

ot- u.*tl&o'

'";tt- ve'iiwziye2 1+ ': 'tyeqy:-'gd0-TQ,a> 3{{ q3; L ar,f cl y"y've Aore:

4c !=:

a

I

d f.'tbr ' - -

: A:Su[oul. i r t ,

: i {M-

05.05;-8

il,nri tti, ,ro eeAw<

0,., l"{ ! h

d", c

. g{0r^1Vectof "'.'.' :vl ei

2 r 3 i

i q t e N >3 o t o 9te

3,' !-cs

v. GsnUe 5e

b+e ;

t LHS=

= -'tf

I =O 2-' ', lf

1t + (-il-3i

Qpw'zf.r

-? i ltl-6 i

(",, rlr,) ikfr

, LHt =ft8).0ri.J$04'-> Ll4s=7r15 -> 6 ,r('

lff ? o-tae' l=- f

0F, or-lg

,fioLr.r'l

I \ 7 ) t =

? ) = - Z ' + l r z - 7

= -Zt+

, ' Q

) krd Tbul.box o.-6qrl"plril0lq;d f;.fu'iq{,x.,1 05..q5dloVf u

2 z ; 4

s+|1.(zz

- 2 2 -

bo, lb l1bg Q(z ) , R Tooltox ^'Co,npfct

Rorb 4(e) r 0=*(aop z"ith+t09,6r-lz.

=*2, €h:=-1.,

otor- lz

\'at l &

U-3i),rar. = {g-li) qa

:l- si

Vlv^"|9.0*.n

4 ; I i . .+ o

Eq. $ l l ) , 0 t 6in?3i

.Vle Qs

osos^lt. s Ft) to sdvs ft- fu.., il

is cbo ob

as,ohl9

o, r (3 )

r t 7.ta?9

3

=q, ?os), t*'lli nti,f Lv3 (u, + l-pq ilS t

\3-3i){ ," { ^ A; i \7, | 0.(c? = 3( l+i) L\, . (q.szo,C*{-.I)q,-Jfioil;\t ", +Ft,f+i),.r", = 3 (t+i\ uj, (E.sQ+t;\q* + (+*;;iu,l r-(-rr+itu- = > 1ri) ui Gq,9l-:..

)ft'u\l ) : tl sclve toh'q$o*irr 'lrrz = iq Ei. Sui) q iver covluq d; ciiuu :

a"-3 ; i h+sr

n|r;rh lJ ir .r ,r ' ;ble. reqaXl\ett of tol,rkd, +r-. TVtt

{,r( ' tfiv€ Cqr"(6), Sr',rcc ilI l, -'*)'

+u/gle

>tee G) : S"t 1te=01, 1r4,.=ll, -solv< Iov u=a.92(\-> MxJ +6+-r, h+s;'--) ei+-6i)f l.. r-i-zi

/^-- -l,t-6r - zl"rk u,7rt7,n (E(,$.o),(eo.g?_z\ are ob<ye,l t l l , = J .

fu€zo), Ltts' = o, f,rri=o =,ll{S I A',y.(c ,20 | LHs =(tilp t kB;),t+(4-::i) (+)= Z + Z;Ru-s =4( l+ i ) ( * )= ? + 7 i = l u< o o o*

9" all : €. Tvect-ov

osoE-16. f u) ,fr( , tzz

.=_L qe.Gse

 

og. a:>=wt co,,rr

ctov* o,\ fnoh Dv-o

r P l \ s

otr lhe.

q d i

I e \qc*vec to rsiu ( l /

k ix

c'|4,er ' l ) *4

.i/yl coln"

Covr(

s 'vtlttl,.i eleS o w i e €l

aqd \oi+t? (1r(')..va lue

o u ' 0

o, {u' ex qw "Io . i( q cowr r&{ e 3 * S wa+";

u hou- d?.nnu p,w *e \, So M=Lbe cl,llu

diagonalizable  

0Vevlv.toSovt*t ed

0F. os-f8

vql,a

".'q*(j) {o

vntov C?>

{' "{ i l(l}. ?ne wq

I=

? ) be l t te c"ra; +i

r.uil

iut ?va1na oY2

rr,r .l*-;lt \( f uw

Qr, + 1122 Q2, I ( { 2 ' , 3

tf<qt 6\

05.0t-tq

geckvg 4r0\ 'b"l a u -ter

u!A)_ ' it dueettvelw h.

iu€ YTka 4Utc \)(-

9 \coqV i v t

Lrfben

9 *

vc(*t)

eavv t vq ?-

qnv 14-'., Pnr av.v ',l.-+hDl/V o

VAlt/rt

-lbtprj'0 Ve+WsI t' n oss', bk fo co,r>tq"{,,Q.h ll4- luo+(."a

(a), . -. ,W<all i> *\o Schv,i

g 0 ) At

.5"i+ olarr{

Iittn*\i'

tle bwvq+;ort

".[ +k", \ *v 5.05_a

n ltor.foi qqttaor q . U

offowi>

te\b rqlll

I

(

i,'t

(

tI

tI

il

t"t w*Q" qoqd.Q- | a, \ew, PHr( [ ,i eig.ua t'ybvT ,?r,t4jt S *q'*y;fqe h cot i* c*.){,;,'tgf

?r.; Qra1.-.' ucit l' so'f +a-+ (wcrs,.... ',,,t'{,)rt 0h{Qoqor.,'f 1ig., guc_\ wrt Vrtgfwc^) -0kr i+vn

" r,yi lt' olf t^r ('i] 4 A fu irr, =l ,_, /,1 .

e(ean olon s ("'j'. u u'))

svl\o Jgqg( ll-fuptlavolw","h !

0s.o5n

Ltsod. ' lo l,qs*re '+q"duoged u vg f nor, ?d Q

<c ol-tw poteL4-' ;lt

ovfuoq o

9+rul beLvt

, R.$os). a,"1

,rh+q -544ti++ sqrt\ i" {it4 qg

i9, i-t"

= qft rt r -: - off-\''-1

1l

WtY€ s€4

wi>e S=$