1. Compute the integral R C F .d ~ ~r where C is the spiral x(t)...

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1. Compute the integral R C F .d ~ ~r where C is the spiral x(t) = et cost, y(t) = et sin t, 0  t  1, and F~ = hxy2, x2yi. 2. Is the vector field F~ = hyz2+ zexz , xz 2 , 2xyz + xe xz i conservative? If yes, find its potential function. 3. Let F~ = h1, y, z2i, S1 and S2 be the following surfaces. Compute both of the flux RR S1 F .d ~ S~ and RR S2 F .d ~ S~. (a) S1 : is the closed surface consisting of the cone z = px2 + y2 for 0  z  2, together with its “lid” given by z = 2, x2 + y2  4, all oriented outward. (b) S2 : is the surface of the cone z = px2 + y2 for 0  z  2 where the lid z = 2 is NOT included and it has the same orientation as part (a). 4. Compute the flux Let F~ =RR hy, x, zi and S is the part of the paraboloid z = 4 x 2 y 2 where z 0. S Curl(F~ ).dS~. 5. Find the integral R C(ex33x+4y)dx+(ln(y2+6)+x)dy where C is the circle x2+y2 = 5 that is oriented counterclockwise. 6. Find the absolute maximum and minimum of f(x, y) = xy(1 x y) on the closed triangular region with vertices (0, 0), (2, 0), and (0, 2). 7. Let f(x, y, z) = e xyz 2 . Find the maximum rate of change of f at the point P(1, 2, 3) and the direction in which it occurs. 8. Find the equation of the plane that passes through P(1, 2, 0) and contains the line x = 2t, y = 3 t, z =1+3t

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