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Unformatted text preview: 14-51SECTION 14.5..Curl and Divergence116516.C(ysec2x2)dx+(tanx4y2)dy, whereCis formed byx=1y2andx=17.Cx2dx+2x dy+(z2)dz, whereCis the triangle from(0,,2) to (2,,2) to (2,2,2) to (0,,2)18.C4y dx+y3dy+z4dz, whereCisx2+y2=4 in the planez=19.CFdr, whereF=x3y4,ex2+z2,x216y2z2andCisx2+z2=1 in the planey=20.CFdr, whereF=x3y2z,x2+z2,4xyz4andCisformed byz=1x2andz=0 in the planey=2In exercises 2126, use a line integral to compute the area of thegiven region.21.The ellipse 4x2+y2=1622.The ellipse 4x2+y2=423.The region bounded byx2/3+y2/3=1. (Hint: Letx=cos3tandy=sin3t)24.The region bounded byx2/5+y2/5=125.The region bounded byy=x2andy=426.The region bounded byy=x2andy=2x27.Use Greens Theorem to show that the center of mass of theregion bounded by the positive curveCwith constant den-sity is given by x=12ACx2dyand y= 12ACy2dx, whereAis the area of the region.28.Use the result of exercise 27 to find the center of mass of theregion in exercise 26, assuming constant density.29.Use the result of exercise 27 to find the center of mass of theregion bounded by the curve traced out byt3t,1t2, for1t1, assuming constant density.30.Use the result of exercise 27 to find the center of mass of theregion bounded by the curve traced out byt2t,t3t, fort1, assuming constant density.31.Use Greens Theorem to prove the change of variables formulaRdA=S(x,y)(u,v)du dv,wherex=x(u,v) andy=y(u,v) are functions with continu-ous partial derivatives.32.ForF=1x2+y2y,xandCany circle of radiusr>0 notcontaining the origin, show thatCFdr=0.In exercises 3336, use the technique of example 4.5 to evaluatethe line integral.33.CFdr, whereF=xx2+y2,yx2+y2andCis any posi-tively oriented simple closed curve containing the origin34.CFdr, whereF=y2x2(x2+y2)2,2xy(x2+y2)2andCis anypositively oriented simple closed curve containing the origin35.CFdr, whereF=x3x4+y4,y3x4+y4andCis any posi-tively oriented simple closed curve containing the origin36.CFdr, whereF=y2xx4+y4,x2yx4+y4andCis any posi-tively oriented simple closed curve containing the origin37.Where isF(x,y)=2xx2+y2,2yx2+y2defined? Show thatMy=Nxeverywhere the partial derivatives are defined. IfCis a simple closed curve enclosing the origin, does GreensTheorem guarantee thatCFdr=0? Explain.38.For the vector field of exercise 37, show thatCFdris thesame for all closed curves enclosing the origin.39.IfF(x,y)=2xx2+y2,2yx2+y2andCis a simple closedcurve in the fourth quadrant, does Greens Theorem guaranteethatCFdr=0? Explain.EXPLORATORY EXERCISES1.EvaluateCFdr, whereF=y(x2+y2)2,x(x2+y2)2andCis the circlex2+y2=a2. Use the result and GreensTheorem to show thatR2(x2+y2)2dAdiverges, whereRisthe diskx2+y21....
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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