Section 13.2

Jim Lambers
MAT 280
Summer Session 2012-13
Homework Set 3
This homework set is due on Friday, July 29. Each problem is worth 10 points.
Section 13.2
1. Evaluate the line integral
Z
x ds
C
where C is the portion of the parabola y = x2 from (0, 0) to (1, 1).
2. Evaluate the line integral
Z
yz cos x ds
C
where C is the curve defined by the parametric equations
x = t,
y = 3 cos t,
z = 3 sin t,
0 ≤ t ≤ π.
3. Evaluate the line integral
Z
ye−xz ds
C
where C is the curve defined by the parametric equations
x = t,
y = 3t,
z = −6t,
0 ≤ t ≤ ln 8.
4. Evaluate the line integral
Z
y 3 dx + x2 dy,
C
where C is the arc of the parabola x = 1 − y 2 from (0, −1) to (0, 1).
5. Evaluate the line integral
Z
y dx + (x + y 2 ) dy,
C
where C is the ellipse 4x2 + 9y 2 = 36, with counterclockwise orientation.
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6. Evaluate the line integral
Z
F · dr,
C
where F = hy, z, −xi and C is the curve defined by the parametric equations
x = cos t,
y = sin t,
z = 4,
0 ≤ t ≤ 2π.
7. Find the work done by the force field
F(x, y, z) =
(x2
hx, y, zi
+ y 2 + z 2 )3/2
in moving a particle along the path
r(t) = ht2 , 3t2 , −t2 i,
1 ≤ t ≤ 2.
Section 13.3
1. Find the work done by the force field
F(x, y, z) = hyz, xz, xyi
in moving a particle from the point (0, 0, 0) to (1, 1, 1), using the following paths:
(a) A straight line
(b) The helix x = sin t, y = 2t/π, z = sin t, 0 ≤ t ≤ π/2
2. Show that the vector field
F(x, y, z) = hsin y, x cos y, − sin zi
is a conservative vector field. Then, find a function f (x, y, z) such that F = ∇f .
3. Show that the vector field
F(x, y, z) = h(1 + xy)exy , ey + x2 exy i
is a conservative vector field. Then, find a function f (x, y) such that F = ∇f .
4. Evaluate
Z
2xe2y dx + (2x2 e2y + 2y cos z) dy − y 2 sin z dz
C
where C is the curve with parametric equations
x = cos t,
y = sin t,
z = sin t,
0 ≤ t ≤ 2π.
5. Show that the vector field
F(x, y, z) = ez hy, x, xyi
is a conservative vector field. Then, find a function f (x, y, z) such that F = ∇f .
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Section 13.4
1. Use Green’s Theorem to evaluate
Z p
1 + x3 dx + 2xy dy,
C
where C is the triangle with vertices (0, 0), (1, 0) and (2, 3).
2. Use Green’s Theorem to evaluate
Z
x2 y dx − xy 2 dy,
C
where C is the circle x2 + y 2 = 4 with counterclockwise orientation.
3. Let
F(x, y) =
x2
1
h2x3 + 2xy 2 − 2y, 2y 3 + 2x2 y + 2xi,
+ y2
and let C be a simple, closed, positively oriented, smooth curve that encircles the origin.
Compute
Z
F · dr.
C
4. Use Green’s Theorem to evaluate
Z
(−3y + x3/2 ) dx + (x − y 2/3 ) dy,
C
where C is the boundary of the half disk {(x, y) : x2 + y 2 ≤ 2, y ≥ 0} with counterclockwise
orientation.
Section 13.5
1. Let F and G be vector fields defined on R3 that have continuous first partial derivatives.
Show that
curl(F × G) = F div G − G div F + (G · ∇)F − (F · ∇)G,
D
E
∂
∂
∂
where ∇ = ∂x
, ∂y
, ∂z
.
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Section 13.7
1. Evaluate the surface integral
Z Z
F · dS,
S
where F(x, y, z) = hx2 , xy, zi and S is the part of the paraboloid z = x2 + y 2 below the plane
z = 1, with upward orientation.
2. Evaluate the surface integral
Z Z
F · dS,
S
where F(x, y, z) = hxz, −2y, 3xi and S is the sphere x2 +y 2 +z 2 = 4 with outward orientation.
3. Evaluate the surface integral
Z Z
F · dS,
S
where
hx, y, zi
F(x, y, z) = p
x2 + y 2 + z 2
and S is the sphere x2 + y 2 + z 2 = a2 with outward orientation.
Section 13.8
1. Use Stokes’ Theorem to evaluate
Z Z
curl F · dS,
S
where F(x, y, z) = hx2 yz, yz 2 , z 3 exy i, and S is the part of the sphere x2 + y 2 + z 2 = 5 that
lies above the plane z = 1, with upward orientation.
2. Use Stokes’ Theorem to evaluate
Z
F · dr,
C
where F(x, y, z) = hxy, yz, zxi, and C is the triangle with vertices (1, 0, 0), (0, 1, 0) and
(0, 0, 1), oriented counterclockwise as viewed from above.
3. Use Stokes’ Theorem to evaluate
Z
F · dr,
C
where F(x, y, z) = hx2 − y 2 , x, 2yzi, and C is the boundary of the plane z = 6 − 2x − y in the
first octant.
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Section 13.9
1. Compute the outward flux of
F(x, y, z) =
(x2
+
1
hx, y, zi
+ z 2 )3/2
y2
through the ellipsoid 4x2 + 9y 2 + 6z 2 = 36. Hint: You can use the Divergence Theorem, but
not on the solid enclosed by the ellipsoid. Explain why.
2. Use the Divergence Theorem to evaluate the surface integral
Z Z
F · dS,
S
where F(x, y, z) = hx3 , y 3 , z 3 i, and S is the sphere x2 + y 2 + z 2 = 9.
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