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Unformatted text preview: 14-75SECTION 14.7..The Divergence Theorem118956.Syz dS, whereSis the portion ofx2+y2=1 withxandzbetweenz=1 andz=4y57.S(y2+z2)dS, whereSis the portion of the paraboloidx=9y2z2in front of theyz-plane58.S(y2+z2)dS, whereSis the hemispherex=4y2z259.Sx2dS, whereSis the portion of the paraboloidy=x2+z2to the left of the planey=160.S(x2+z2)dS, whereSis the hemispherey=4x2z261.S4dS, whereSis the portion ofy=1x2withy0 andbetweenz=0 andz=262.S(x2+z2)dS, whereSis the portion ofy=4x2be-tweenz=1 andz=463.Explain the following result geometrically. The flux integralofF(x,y,z)=x,y,zacross the conez=x2+y2is 0.64.Ingeometricterms,determinewhetherthefluxin-tegralofF(x,y,z)=x,y,zacrossthehemispherez=1x2y2is 0.65.For the conez=cx2+y2(wherec&gt;0), show that in spher-ical coordinates tan=1c. Then show that parametric equa-tions arex=ucosvc2+1,y=usinvc2+1andz=cuc2+1.66.Find the surface area of the portion ofz=cx2+y2belowz=1,usingtheparametricequationsinexercise 65.67.Find the flux ofx,y,zacross the portion ofz=cx2+y2belowz=1. Explain in physical terms why this answer makessense.68.Findthefluxofx,y,zacrosstheentireconez2=c2(x2+y2).69.Find the flux ofx,y,across the portion ofz=cx2+y2belowz=1. Explain in physical terms why this answer makessense.70.Find the limit ascapproaches 0 of the flux in exercise 69.Explain in physical terms why this answer makes sense.EXPLORATORY EXERCISES1.Ifx=3 sinucosv,y=3 cosuandz=3 sinusinv, showthatx2+y2+z2=9. Explain why this equation doesntguarantee that the parametric surface defined is the entiresphere, but it does guarantee that all points on the surfaceare also on the sphere. In this case, the parametric surfaceis the entire sphere. To verify this in graphical terms, sketcha picture showing geometric interpretations of the spher-ical coordinatesuandv. To see what problems can oc-cur, sketch the surface defined byx=3 sinu2u2+1cosv,y=3 cosu2u2+1andz=3 sinu2u2+1sinv. Explain why youdo not get the entire sphere. To see a more subtle exampleof the same problem, sketch the surfacex=cosucoshv,y=sinhv,z=sinucoshv. Use identities to show thatx2y2+z2=1 and identify the surface. Then sketch thesurfacex=cosucoshv,y=cosusinhv,z=sinuand useidentities to show thatx2y2+z2=1. Explain why the sec-ond surface is not the entire hyperboloid. Explain in words andpictures exactly what the second surface is.14.7THE DIVERGENCE THEOREMRecall that at the end of section 14.5, we had rewritten Greens Theorem in terms of thedivergence of a two-dimensional vector field. We had found there (see equation 5.3) thatCFnds=R F(x,y)dA.Here,Ris a region in thexy-plane enclosed by a piecewise-smooth, positively oriented, sim-ple closed curveC. Further,F(x,y)=M(x,y),N(x,y),0 , whereM(x,y) andN(x,y)are continuous and have continuous first partial derivatives in some open regionDin thexy-plane, withRD.1190CHAPTER 14.....
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This note was uploaded on 10/04/2010 for the course CHE2C 929102 taught by Professor Carter during the Spring '10 term at UC Davis.
- Spring '10