## Transcribed Text

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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