Question

(a3, y3,4z3). Let Si be the disk in the 12. Consider the vector field in space given by F(x, y, z) xy-plan described by x2 + y2 < 4, z = 0; and let S2 be the upper half of the paraboloid given by z 4 y2, z 2 0. Both Si and S2 are oriented upwards. Let E be the solid region enclosed by S1 and S2 (a) Evaluate the flux integral FdS (b) Calculate div F div F dV (c) Calculate the iterated integral F dS (d) Evaluate the flux integral S2 Hint: Using parts (a) to (c), this point can be solved without performing any more integration. Use the Divergence Theorem and the fact that S2 S2 (1-1i)-1-i (sin a2)i + (e + x2)j+ (242.x2)k 13. If F use Stokes' Theorem to evaluate the line integral Fdr where C is the triangle that goes from (3, 0, 0) to (0, 2,0) to (0,0,1) and back to (3,0, 0) 14. Let R be a region in the plane with boundary curve C', where C is a piecewise smooth curve oriented in the positive direction (a) Use Green's Theorem to show that the area of R is given by the line integral 1 -y dx dy. 2 (b) Use part (a) to find the area of the ellipse y2 + 4 = 1.

Answer 1

12. F(x. y.)(x.y'.4 S, :x +y 4,0:S,4--y.20 JJF.ds=y'4')(k) dvex = [[ 4='cdv«hx = J 40 dycx 0 a D D (b) divF -3x2 +3y +122 2 2 C3( (e) divF)dV = 3(x2 + y2 + 4:2 )dV = + 42)rdzcrde E E 4(4-2 rdrde 3 -477 45-180216drde - 180 15 108 x2T 2 26180x24 2 28 15 +108 x22 4 -128T (d) Using divergence theorem, di + _ S S S divF dV = 128T [[F ds =0+ [[[(divF}dV S E [fF.ds 128T S