(12 points) 1. Give a parameterization for the plane containing the points (1, 2, 3), (2, 3, 5), and (0, 3, 2).
1.
(4 points) 2. Below is a sketch in the xy-plane of a three-dimensional vector field F! . Use the geometric
definition of the curl to decide whether the curl of F! at the origin points, up, down, or is the
zero vector.
2.
(12 points) 3. Compute the line integral
!
C
!3y 2 + 2x, 6xy" · d!r,
where C is the path consisting of the line segment from (0, 0) to (1, 2) followed by the parabola
y = x2 + 1 from (1, 2) to (2, 5).
3.
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(14 points) 4. Use Stokes’ Theorem to compute the line integral
!
(x!i − z!j + y!k) · d!r,
C
where C is path around the triangle in the yz-plane with vertices (0, −1, 0), (0, 2, 0), and
(0, 0, 3), traversed counterclockwise when viewed from the positive x-axis.
4.
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! = !z + 2xy, 3y + x2 , x − 5z#.
5. Consider the vector field G
(4 points)
!
(a) What is the domain of G?
(a)
(8 points)
!
(b) Compute curl G.
(b)
(4 points)
! a gradient field? Explain.
(c) Is G
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(14 points) 6. Use the Divergence Theorem to compute the flux of
! 3"
! 3"
y !
x !
!
F =
i+
j + (x + y)!k
3
3
out of the cylinder x2 + y 2 = 1, 0 ≤ z ≤ 5
6.
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(14 points) 7. Compute the surface area of the portion of the graph of z = 2x − 3y + 10 lying above the
square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 in the xy-plane.
7.
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(14 points) 8. Calculate the flux of F! = 2x!i + 2y!j + 3z!k through the portion of the cylinder x2 + y 2 = 1,
0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1, oriented away from the z-axis.
8.
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