Question

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rough the surface 4. o pm) What is the value of the flux of the vector field F(x,y)j+z ioriented with upward- pointing normal vector? (A) 0 (B) 2n/3 (C) π (D) 4T/3 (E) 2π Use Stokes, Theorem to evaluateⅡcurl F.dS, where F(x, y, z)-(x2 sin Theorem to evaluate Jceun F'.asS , where Fl.e)(', ») and 5. (5pts.) F,y, sin z, y', xy) and s is the part of the paraboloid : - 1-y that lies above the xy-plane, oriented upward. Use Green's Theorem to find the exact area of the region bounded by the curve r)-(), 05151. (5pts.) . 2. 5 pe) Evaluatds where S is the part of the plane or +4y +3:- 12 lying in the 1 octant. (A) 4 61 (B) 4V61 (C) 3/61 (D) 61 61 3. (4 ps) Find an equation of the tangent plane to the surface given by r(mv)-(u + v, Зиг, u-v) at the point (2, 3.0).

Answer 1

4. Ans>E).

T1Ven S: x2 +y2+2=1, z20 and F = (x,y,z) Flux of F through the surface Sis Fn ds Consi der Fn ds Let φ= x2 +y2 + Z2-1 If Ris theprjection of the surface on the zxy-plane, weget '부의ds = 511 (1) dxdy dx dy dxdy 12 (On changing into polar coord antes) (1-r2)-n r dr dθ 1-/2 n +1 -(27)