Hence we get the x- and y-components of the surface paramtzn fctn as:
05.01-1HWP 05.01Introduction: Constructing the Surface Parameterization Function
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:
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,asโCDโ,forshort.
ThesurfaceQisacirculardiskofradiusRinthex-y-planecenteredatthecoordinateorigin,i.e.,withacenter๐ = ๐#๐%๐& = [000].TheparameterizationfunctionforthisdisksurfacegiveninCDis:
๐ ๐ , ๐ = ๐ฅ ๐ , ๐ , ๐ฆ ๐ , ๐ ,๐ง ๐ , ๐ = ๐ cos ๐ ,๐ sin ๐ ,0 ,withintegrationintervals
0 โค ๐ โค ๐ and0 โค ๐ < 2๐.
Theinfinitesimalincrementvectorsof๐ ๐ , ๐ aregiveninCDas๐<๐ = ๐<๐ ๐ , ๐ ๐๐ = cos ๐ ,sin ๐ ,0 ๐๐
and๐>๐ = ๐>๐ ๐ , ๐ ๐๐ = [โ๐ sin ๐ ,๐ cos ๐ ,0]๐๐.
Wewanttheareaelementvectors๐๐topointinthe(โ๐ง)-direction:thisisthedirectionpointingoutwardfromthevolumeV,enclosedbetweentheparabolicdomesurfacePandthecirculardisksurfaceQ.Therefore,asshownindetailinCD,wemustchoosethecrossproductoftheinfinitesimalincrementvectors(with๐๐ > 0and๐๐ > 0)as
๐๐ = ๐>๐ ร ๐<๐ AsshowninCD,thiswillgiveus
๐๐ = 0, 0, โ๐ ๐๐ ๐๐whichdoesindeedpointinthe(โ๐ง)-direction,asrequired.This๐๐correspondstoacrossproductofthepartialderivatives
๐>๐ ๐ , ๐ ร ๐<๐ ๐ , ๐ = 0, 0, โ๐ whichwewillusebelowtocarryoutthesurfaceintegrationviatheparametrizationequation.
Compoundingthegivenintegrandvectorfield,๐น(๐),withthetheparameterizationfunction๐ ๐ , ๐ ,weget
05.01-8
๐น ๐ ๐ , ๐ = ๐ฅ ๐ , ๐ ๐ฆ ๐ , ๐ ,๐ฆ ๐ , ๐ ๐ง ๐ , ๐ ,๐ง ๐ , ๐ ๐ฅ ๐ , ๐ + ๐ฅ ๐ , ๐ G
= ๐ Gsin ๐ cos ๐ ,0,๐ GcosG ๐ ,makinguseofthefactthat๐ง ๐ , ๐ = 0everywhere.
The ๐ , ๐ -integrandintheparameterizationequationforthesurfaceintegral(seebelow)isthengivenby
๐ ๐ , ๐ โ๐น ๐ ๐ , ๐ โ ๐>๐ ๐ , ๐ ร ๐<๐ ๐ , ๐
= ๐น& ๐ ๐ , ๐ ๐>๐ ๐ , ๐ ร ๐<๐ ๐ , ๐&
= ๐ GcosG ๐ โ๐ = โ๐ KcosG ๐ .
Notethatthevectorfieldcomponents๐น# ๐ ๐ , ๐ and๐น% ๐ ๐ , ๐ donotcontributetotheintegrand๐ ๐ , ๐ ,nortoitsintegral,sincethe๐ฅ-and๐ฆ-componentsofthepartialderivativecrossproductarezero.
Bytheparameterizationequationforthesurfaceintegral,wehave๐๐ โ ๐น(๐)
L = ๐๐ MN ๐๐GO
N ๐น ๐ ๐ , ๐ โ ๐>๐ ๐ , ๐ ร ๐<๐ ๐ , ๐
= ๐๐ MN ๐๐GO
N ๐ ๐ , ๐ =โ ๐๐ M
N ๐ K ๐๐GON cosG ๐
Usinganintegraltable,wefind๐๐GO
N cosG ๐ = ๐.Hence,
๐๐ โ ๐น(๐)L = โ๐ ๐๐ M
N ๐ K
=โ๐ PQ๐ Q
N
M
= โOQ๐ Q
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.
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
05.02-4
, ,
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)
, ,
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