Unformatted text preview: 1. (10 points) Find the ﬂux of xi + y j + z k
3 (x2 + y 2 + z 2 ) 2
radius 1 with outward orientation. Explain. through the sphere centered at (2, 0, 0) of 2. (20 points) Suppose a vector ﬁeld is deﬁned by F(x, y, z ) = 2xy i + (x2 + 2yz )j + y 2 k.
a) Determine whether there is a scalar function φ(x, y, z ) deﬁned everywhere in space such
that φ = F. If there is such a φ, ﬁnd one; if there is not explain why not.
b) Determine whether there is a vector function A(x, y, z ) deﬁned on all of space such that
× A = F. If there is such an A ﬁnd one, if there are none, explain why not.
c) Compute the integral C F·T ds, where C is the curve parametrized by r(t) = (2 cos t, sin t, 0)
for t ∈ [0, π ].
3. (20 points) Verify stokes theorem for the vector ﬁeld F(x, y, z ) = xi + y j + xyz k and the
surface S , the part of the plane 2x + y + z = 2 that lies in the ﬁrst octant, oriented upward.
4. (10 points) Find an equation for the tangent plane to the catenoid x(u, v ) = cosh v cos ui +
cosh v sin uj + v k at (u, v ) = (0, π/2) (recall cosh t = (et + e−t )/2 and sinh t = (et − e−t )/2 so
that (cosh t) = sinh t and (sinh t) = cosh t).
5. (20 points) a) Use the divergence theorem to compute the ﬂux of F(x, y, z ) = (1 + y 2 )j
out of the curved part of the halfcylinder bounded by x2 + y 2 = a2 (y ≥ 0), z = 0, z = b,
and y = 0. Justify your answer.
b) Suppose that S is a closed surface that lies entirely in y < 0. Is the outward ﬂux of
F(x, y, z ) = (1 + y 2 )j through S positive, negative, or zero? Justify your answer
6. (8 points) Write the complex number z = (3 + 3i)6 in polar form (ie. as reiθ or
r(cos θ + i sin θ).
7. (10 points) Describe all complex solutions of z 3 = −3 in rectangular form. Sketch the
solutions.
8. (10 points) Find a smooth closed curve C which maximizes the line integral
2x3 dy . Explain. 1 C (y 3 − y ) dx − ...
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This note was uploaded on 10/18/2011 for the course MATH S1202Q taught by Professor Peters during the Spring '11 term at Columbia.
 Spring '11
 Peters
 Calculus

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