MATH 253.504–506 An unsupported answerExam 3, version A is a wrong answer!12/1/99This test has 2 pages and 8 problems!Conversion from spherical to cartesian coordinates:x =ρ sin φ cos θy =ρ sin φ sin θz =ρ cos φdV =ρ2sin φdρdθdφ1. (15 pts.) Evaluate the line integralZCyds,whereCis the line segment starting at (0, 1, 1)and ending at (2, 2, 3).2. (10 pts.) Determine the equation of the plane tangent to the surface ~r(u, v)=hu2+v, u2− v, uviat the point (5, 3, −2).3. (15 pts.) For the following, determine if they are vector quantities, scalar quantities, orundefined. Here f is a scalar function and~F is a vector field.(a) ∇(div(~F ))(b) curl(∇f)(c) curl(~F )+div(~F)(d) div(∇f )(e) curl(div(~F ))4. (10 pts.) Write a parameterization for the part of the sphere x2+ y2+ z2= 4 which liesoutside of the cylinder x2+ y2= 1. Be sure to specifiy the parameter domain.5. (15 pts.) Findz of the center of mass of the region above the cone φ = φ0but inside thesphere ρ = ρ0, assuming that density is constant.6. (15 pts.) Let R be the region in the x, y plane bounded by xy =1,xy =2,xy2=1andxy2= 3. WriteZZRxdA as an iterated integral in the variables u and v,whereu=xy andv = xy2.7. (15 pts.) Use Green’s theorem to evaluateIC(x3−y3) dx+(x3+y3)dy,whereCis the picturedcurve, consisting of two circular arcs and two line segments.0.20.40.60.811.21.41.61.820.20.40.60.8 1 1.21.41.61.8 2Figure for problem 7.8. (5 pts.) If~F is the vector field shown, can~F be conservative? Why or why not?–1–0.8–0.6–0.4–0.20.20.40.60.81y–1 –0.6 –0.2 0.20.40.60.8 1xFigure for problem
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