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(MATH100)final_98.pdf
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FinalExamofMATH100(Fall1998)

Question1. LetSbethesurfaceofthesphereofradiusa.0withcenterattheorigin.Supposethe temperaturedistributiononSisgivenby
T(.....)..3(2+sin.)[2+sin(.;.)].(.....)2S.
where(.....)arethesphericalcoordinates,with0...2.and0.....Findthepoint onSwherethetemperatureismaximum.Whatisthemaximumtemperature.
Question2.
pp
LetRbetheregioninthexy-planeboundedbythecurvesy.4;x2.y.xandy.3x.
EvaluateRR R(x+y2)dxdy.

Question3.
EvaluateRRR GzdV,whereGisthesolidenclosedbythespherex2+y2+z2.4andthe
paraboloidx2+y2.3z.

Question4.
EvaluatethelineintegralR CF.dr,where

F(x.y).3x 2 yi+(x 3+ey)j.
andCistheuppersemi-circlex2+y2.9from(;3.0)to(3.0).
Question5. (a)Findthe.uxofthevector.eld
F(x.y.z).x 3i+2xz 2j+3y 2 zk
acrosstheportionoftheparaboloidz.x2+y2belowtheplanez.4,orientedbydownward unitnormals. (b)Findthetotal.uxacrosstheclosedsurfaceofthesolidboundedbytheparaboloidandthe planeinpart(a),orientedoutward.
Question6. Findthesurfaceareaofthepartofthespherex2+y2+z2.a2betweentheplanesz.0and z.a.2.
Question7. Let.betheportionoftheparaboloidz.4;x2;y2abovetheplanez.3andoriented upwards,andCbetheboundaryofthesurface.withthepositivedirection(i.e.whenwalking inthedirectionoftheboundarycurve,thesurfaceinontheleft).Supposethevector.eldis givenby
F(x.y.z).x 2i+z 2j;y 2k:
(a).DirectlyevaluateH CF.dr. (b).DirectlyevaluateRR (curlF).ndS.