A. Consider the vector field F = (x, y,0). (a) Compute VXF and V.F. (b) Let...

  

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A. Consider the vector field F = (x, y,0). (a) Compute VXF and V.F. (b) Let C be the closed curve in the plane z = y + 3 lying over the curve r = 2 cos 0 in the zy-plane. Compute F.Tds. (c) Compute the flux of F out of the cylinder x + y = 4 for -2m sz 2. (d) Compute the flux of F out of the wavy cylinder r = 2+ sinz for -2 ss 2. 10.B. Let F(x, y, z)=G(x, y, z) = be two vector fields, where r = ri + vj + zk. (a) Let S be the sphere of radius 3 centered at the origin. Compute: (i) The flux of F out of S. (ii) The flux of G out of S. (b) Describe a surface T such that the flux of F through T equals the flux of G through T. Give an equation for T. 10.C. A wire along the z-axis carrying an electric current I generates a magnetic field B(x, y, z) = |r| 2/1/( 21 (-yi+zj x + y where c is the speed of light. (a) Describe the domain of definition of B (in R). (b) Show that x B is zero away from the z-axis. (c) Compute the circulation of B around the unit circle C, in the ay-plane centered at the origin, oriented counterclockwise when viewed from above. (d) Describe a closed loop in R around which the circulation of B is zero. (e) Compute the circulation of B around the ellipse C in the plane z = 2 given by equation 2 +9y = 9, oriented counterclockwise when viewed from above. 10.D. You are holding a crystal ball of radius 1, four feet above the ground in an energy vector field - y, 752) Find the and y coordinates of the ball's center at which the flux of F out of the ball will be the most positive. Express your answer in the form (x, y, 4). Find another location where the flux will be negative. 10.E. Let S be the "black hole" surface in R parameterized by r(r, 0) (r cos 0,r sin 0, lnr) (0 A. Consider the vector field F = (x, y,0). (a) Compute VXF and V.F. (b) Let C be the closed curve in the plane z = y + 3 lying over the curve r = 2 cos 0 in the zy-plane. Compute F.Tds. (c) Compute the flux of F out of the cylinder x + y = 4 for -2m sz 2. (d) Compute the flux of F out of the wavy cylinder r = 2+ sinz for -2 ss 2. 10.B. Let F(x, y, z)=G(x, y, z) = be two vector fields, where r = ri + vj + zk. (a) Let S be the sphere of radius 3 centered at the origin. Compute: (i) The flux of F out of S. (ii) The flux of G out of S. (b) Describe a surface T such that the flux of F through T equals the flux of G through T. Give an equation for T. 10.C. A wire along the z-axis carrying an electric current I generates a magnetic field B(x, y, z) = |r| 2/1/( 21 (-yi+zj x + y where c is the speed of light. (a) Describe the domain of definition of B (in R). (b) Show that x B is zero away from the z-axis. (c) Compute the circulation of B around the unit circle C, in the ay-plane centered at the origin, oriented counterclockwise when viewed from above. (d) Describe a closed loop in R around which the circulation of B is zero. (e) Compute the circulation of B around the ellipse C in the plane z = 2 given by equation 2 +9y = 9, oriented counterclockwise when viewed from above. 10.D. You are holding a crystal ball of radius 1, four feet above the ground in an energy vector field - y, 752) Find the and y coordinates of the ball's center at which the flux of F out of the ball will be the most positive. Express your answer in the form (x, y, 4). Find another location where the flux will be negative. 10.E. Let S be the "black hole" surface in R parameterized by r(r, 0) (r cos 0,r sin 0, lnr) (0