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Deck 13: Vector Calculus
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Question
Let 
and let S be the boundary surface of the solid E bounded by 
, and 
. Evaluate the surface integral 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
and let S be the boundary surface of the solid E bounded by , and . Evaluate the surface integral .A)
e. b. f. c. g. d. h. 
Question
Find the flux of 
across the surface of the solid bounded by 
, and the planes 
.
across the surface of the solid bounded by , and the planes . Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Question
Let 
and let S be the surface of the rectangular box bounded by the planes 
, and 
. Evaluate the surface integral 
.
and let S be the surface of the rectangular box bounded by the planes , and . Evaluate the surface integral . Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
and let S be the boundary surface of the solid . Evaluate the surface integral . Question
Let 
and let S be the surface of the solid bounded by the spheres 
and 
. Evaluate the surface integral 
.
and let S be the surface of the solid bounded by the spheres and . Evaluate the surface integral . Question
Evaluate 
, where S is the cube bounded by the planes 
and 
, and n is the outward normal.
, where S is the cube bounded by the planes and , and n is the outward normal. Question
Let 
and let S be the surface with equation 
. Evaluate the surface integral 
.
and let S be the surface with equation . Evaluate the surface integral . Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Question
Use the Divergence Theorem to evaluate 
where 
and S is the surface of the cylinder 
, bounded by the planes 
and 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
where and S is the surface of the cylinder , bounded by the planes and .A)
e. b. f. c. g. d. h. Question
Find the flux of 
across the surface of the solid bounded by the paraboloid 
and the 
plane.
across the surface of the solid bounded by the paraboloid and the plane. Question
Let 
and let S be the surface of the tetrahedron with vertices 
, and 
. Evaluate the surface integral 
.
and let S be the surface of the tetrahedron with vertices , and . Evaluate the surface integral . Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Question
Use the Divergence Theorem to evaluate 
where 
and S is the sphere 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
where and S is the sphere .A)
e. b. f. c. g. d. h. Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
and let S be the boundary surface of the solid . Evaluate the surface integral . Question
Evaluate the flux integral 
over the boundary of the ball 
.
over the boundary of the ball . Question
Evaluate 
, where S is the boundary surface of the solid sphere 
and 
, where S is the boundary surface of the solid sphere and Question
Evaluate 
, where S is the boundary surface of the region outside the sphere 
and inside the ball 
and 
.
, where S is the boundary surface of the region outside the sphere and inside the ball and . Question
Let 
. Evaluate the line integral 
along the elliptical path 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
. Evaluate the line integral along the elliptical path .A)
e. b. f. c. g. d. h. Question
Let S be the outwardly-oriented surface of a solid region E where the volume of E is 
. If 
and 
, evaluate the surface integral 
.
. If and , evaluate the surface integral . Question
Use Stokes' Theorem to evaluate 
where C is the circle 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
where C is the circle .A)
e. b. f. c. g. d. h. Question
Evaluate 
, where 
and S is the sphere 
.
, where and S is the sphere . Question
Use Stokes' Theorem to evaluate 
where 
and S is the part of the paraboloid 
that lies inside the cylinder 
, oriented upward.
where and S is the part of the paraboloid that lies inside the cylinder , oriented upward. Question
Let 
. Evaluate the line integral 
along the elliptical path 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
. Evaluate the line integral along the elliptical path .A)
e. b. f. c. g. d. h. Question
Let 
. Evaluate the line integral 
along the rectangular path from 
to 
to 
to 
to 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
. Evaluate the line integral along the rectangular path from to to to to .A)
e. b. f. c. g. d. h. Question
Let 
. Evaluate 
over the surface S given by 
, with downward orientation.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
. Evaluate over the surface S given by , with downward orientation.A)
e. b. f. c. g. d. h. Question
Find the flux of 
across the surface of the solid bounded by the paraboloid 
and the 
plane.
across the surface of the solid bounded by the paraboloid and the plane. Question
Evaluate 
, where 
and S is the sphere 
.
, where and S is the sphere . Question
Find the flux of 
across the surface of the solid bounded by the paraboloid 
and the 
plane.
across the surface of the solid bounded by the paraboloid and the plane. Question
Let 
. Let C be the rectangular path from 
to 
to 
to 
to 
. Use Stokes' Theorem to evaluate the line integral 
, where T is the unit tangent vector to C.
. Let C be the rectangular path from to to to to . Use Stokes' Theorem to evaluate the line integral , where T is the unit tangent vector to C. Question
Find the flux of 
across the surface of the solid 
.
across the surface of the solid . Question
Evaluate 
, where the path C is the curve of intersection of the paraboloid 
with the plane 
.
, where the path C is the curve of intersection of the paraboloid with the plane . Question
Let 
. Evaluate the line integral 
, where C is the curve of intersection of the paraboloid 
and the cylinder 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
. Evaluate the line integral , where C is the curve of intersection of the paraboloid and the cylinder .A)
e. b. f. c. g. d. h. Question
Use Stokes' Theorem to evaluate 
where C is the circle 
.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
where C is the circle .A)
e. b. f. c. g. d. h. Question
Use Stokes' Theorem to evaluate 
where C is the triangle with vertices 
, and 
, oriented counter clockwise as viewed from above.
A)
e. 
b. 
f. 
c. 
g. 
d. 
h. 
where C is the triangle with vertices , and , oriented counter clockwise as viewed from above.A)
e. b. f. c. g. d. h. Question
Let 
and let S be the boundary surface of the solid 
. Evaluate the surface integral 
.
and let S be the boundary surface of the solid . Evaluate the surface integral . Question
A surface has the shape of the cone 
between 
and 
with the density function 
. Find the mass of the surface.
A)
B)
C)
D)
E)
F)
G)
H)
between and with the density function . Find the mass of the surface.A)
B)
C)
D)
E)
F)
G)
H)
Question
Consider the surfaces 
: 
, and 
: 
, and let F be a vector field with continuous partial derivatives everywhere. Why do we know that 
?
: , and : , and let F be a vector field with continuous partial derivatives everywhere. Why do we know that ? Question
Use Stokes' Theorem to evaluate 
, where C is the triangle with vertices 
, and 
, oriented counterclockwise as viewed from above.
, where C is the triangle with vertices , and , oriented counterclockwise as viewed from above. Question
Use Stokes' Theorem to evaluate 
where 
and S is the part of the hemisphere 
that lies inside the cylinder 
, oriented in the direction of the positive x-axis.
where and S is the part of the hemisphere that lies inside the cylinder , oriented in the direction of the positive x-axis. Question
Evaluate the surface integral 
, where 
and S is the part of the plane 
in the first octant with downward orientation.
A)
B)
C)
D)
E)
F)
G)
H)
, where and S is the part of the plane in the first octant with downward orientation.A)
B)
C)
D)
E)
F)
G)
H)
Question
Evaluate 
, where 
and S is the cube bounded by 
.
A)
B)
C)
D)
E)
F)
G)
H)
, where and S is the cube bounded by .A)
B)
C)
D)
E)
F)
G)
H)
Question
Evaluate the surface integral 
, where S is that part of the plane 
that lies above the square with vertices 
, and 
.
A)
B)
C)
D)
E)
F)
G)
H)
, where S is that part of the plane that lies above the square with vertices , and .A)
B)
C)
D)
E)
F)
G)
H)
Question
Verify that Stokes' Theorem is true for the vector field 
and the cone 
, oriented upward.
and the cone , oriented upward. Question
Let 
. Evaluate the surface integral 
, where S is that part of the plane 
that lies above the square with vertices 
, and 
and has upward orientation.
A)
B)
C)
D)
E)
F)
G)
H)
. Evaluate the surface integral , where S is that part of the plane that lies above the square with vertices , and and has upward orientation.A)
B)
C)
D)
E)
F)
G)
H)
Question
Evaluate 
where 
and S is the part of the surface 
that lies above the 
plane and has upward orientation.
A)
B)
C)
D)
E)
F)
G)
H)
where and S is the part of the surface that lies above the plane and has upward orientation.A)
B)
C)
D)
E)
F)
G)
H)
Question
Let S be the parametric surface 
. Use Stokes' Theorem to evaluate 
, where 
.
. Use Stokes' Theorem to evaluate , where . Question
Use Stokes' Theorem to evaluate 
, where C is the curve of intersection of the paraboloid 
and the cylinder 
, oriented counterclockwise as viewed from above.
, where C is the curve of intersection of the paraboloid and the cylinder , oriented counterclockwise as viewed from above. Question
An upper hemisphere is given by 
with the density function 
. Find the mass of the sphere.
A)
B)
C)
D)
E)
F)
G)
H)
with the density function . Find the mass of the sphere.A)
B)
C)
D)
E)
F)
G)
H)
Question
Evaluate the surface integral 
, where S is the triangle with vertices 
, and 
, where S is the triangle with vertices , and Question
Let 
. Evaluate the surface integral 
, where S is that part of the plane 
that lies above the square with vertices 
, and 
and has upward orientation.
A)
B)
C)
D)
E)
F)
G)
H)
. Evaluate the surface integral , where S is that part of the plane that lies above the square with vertices , and and has upward orientation.A)
B)
C)
D)
E)
F)
G)
H)
Question
Evaluate the surface integral 
, where S is that part of the cylinder 
that lies above the square with vertices 
, and 
.
A)
B)
C)
D)
E)
F)
G)
H)
, where S is that part of the cylinder that lies above the square with vertices , and .A)
B)
C)
D)
E)
F)
G)
H)
Question
Use Stokes' Theorem to evaluate 
where 
and C is the curve of intersection of the plane 
and the cylinder 
.
where and C is the curve of intersection of the plane and the cylinder . Question
Evaluate 
, where 
and S is the cube bounded by 
.
A)
B)
C)
D)
E)
F)
G)
H)
, where and S is the cube bounded by .A)
B)
C)
D)
E)
F)
G)
H)
Question
Evaluate the surface integral 
, where S is that part of the plane 
that lies above the square with vertices 
, and 
.
A)
B)
C)
D)2
E)
F)
G)
H)
, where S is that part of the plane that lies above the square with vertices , and .A)
B)
C)
D)2
E)
F)
G)
H)
Question
Use Stokes' Theorem to evaluate 
where 
and C is the triangle with vertices 
, and 
where and C is the triangle with vertices , and Question
A fluid has density 1500 and velocity field 
. Find the rate of flow outward through the sphere 
.
. Find the rate of flow outward through the sphere . Question
Evaluate 
, where 
and S is the upper half of the sphere 
, with upward orientation.
, where and S is the upper half of the sphere , with upward orientation. Question
Evaluate the surface integral 
for the vector field 
where S is part of the cone 
between the planes z = 1 and z = 2 with upward orientation.
for the vector field where S is part of the cone between the planes z = 1 and z = 2 with upward orientation. Question
Evaluate the flux of the vector field 
through the plane region with the given orientation as shown below. 
through the plane region with the given orientation as shown below. Question
Evaluate 
where 
and S is the part of the surface 
that lies above the rectangle 
and has upward orientation.
where and S is the part of the surface that lies above the rectangle and has upward orientation. Question
Evaluate the surface integral 
, where S is the part of the sphere 
that lies above the cone 
.
, where S is the part of the sphere that lies above the cone . Question
Evaluate the surface integral 
, where S is the part of the paraboloid 
that lies in front of the plane 
.
, where S is the part of the paraboloid that lies in front of the plane . Question
Compute the surface integral 
if 
and S is the piece of the sphere 
in the second octant 
.
if and S is the piece of the sphere in the second octant . Question
Consider the top half of the ellipsoid 
parametrized by 
. Find a normal vector N at the point determined by 
, and determine if it is upward and/or outward.
parametrized by . Find a normal vector N at the point determined by , and determine if it is upward and/or outward. Question
Evaluate the flux of the vector field 
through the plane region with the given orientation as shown below. 
through the plane region with the given orientation as shown below. Question
Evaluate the surface integral 
for the vector field 
, where S is the hemisphere 
with upward orientation.
for the vector field , where S is the hemisphere with upward orientation. Question
Find the flux of the vector field 
across the paraboloid given by 
with 
and upward orientation:
across the paraboloid given by with and upward orientation: Question
Find the z-coordinate of the centroid of the upper hemisphere with uniform density whose equation is given by 
.
. Question
Find the mass of the sphere 
whose density at each point is proportional to its distance to the 
plane.
whose density at each point is proportional to its distance to the plane. Question
Evaluate 
, where S is the part of the surface 
that lies between the cylinders 
and 
.
, where S is the part of the surface that lies between the cylinders and . Question
Evaluate 
, where 
and S is the part of the surface 
below the plane 
, with upward orientation.
, where and S is the part of the surface below the plane , with upward orientation. Question
Evaluate the surface integral 
for the vector field 
where S is the part of the elliptic paraboloid 
that lies below the square 
and has downward orientation.
for the vector field where S is the part of the elliptic paraboloid that lies below the square and has downward orientation. Question
Let 
. Find the curl of F.
A)
B)
C)
D)
E)
F)
G)
H)
. Find the curl of F.A)
B)
C)
D)
E)
F)
G)
H)
Question
Find the mass of a thin funnel in the shape of a cone 
, is its density
function is
.
, is its densityfunction is
. Question
Let 
. Find the curl of F.
A)
B)
C)
D)
E)
F)
G)
H)
. Find the curl of F.A)
B)
C)
D)
E)
F)
G)
H)
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Deck 13: Vector Calculus
1
Let and let S be the boundary surface of the solid E bounded by , and . Evaluate the surface integral .
A) e. b. f. c. g. d. h.
and let S be the boundary surface of the solid E bounded by , and . Evaluate the surface integral .A)
e. b. f. c. g. d. h. G
2
Find the flux of across the surface of the solid bounded by , and the planes .
across the surface of the solid bounded by , and the planes .
3
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
A) e. b. f. c. g. d. h.
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. A
4
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
A) e. b. f. c. g. d. h.
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Unlock Deck
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5
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
A) e. b. f. c. g. d. h.
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Unlock Deck
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6
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
A) e. b. f. c. g. d. h.
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
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7
Let and let S be the surface of the rectangular box bounded by the planes , and . Evaluate the surface integral .
and let S be the surface of the rectangular box bounded by the planes , and . Evaluate the surface integral . Unlock Deck
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8
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
and let S be the boundary surface of the solid . Evaluate the surface integral . Unlock Deck
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9
Let and let S be the surface of the solid bounded by the spheres and . Evaluate the surface integral .
and let S be the surface of the solid bounded by the spheres and . Evaluate the surface integral . Unlock Deck
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10
Evaluate , where S is the cube bounded by the planes and , and n is the outward normal.
, where S is the cube bounded by the planes and , and n is the outward normal. Unlock Deck
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11
Let and let S be the surface with equation . Evaluate the surface integral .
and let S be the surface with equation . Evaluate the surface integral . Unlock Deck
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12
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
A) e. b. f. c. g. d. h.
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Unlock Deck
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13
Use the Divergence Theorem to evaluate where and S is the surface of the cylinder , bounded by the planes and .
A) e. b. f. c. g. d. h.
where and S is the surface of the cylinder , bounded by the planes and .A)
e. b. f. c. g. d. h. Unlock Deck
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14
Find the flux of across the surface of the solid bounded by the paraboloid and the plane.
across the surface of the solid bounded by the paraboloid and the plane. Unlock Deck
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15
Let and let S be the surface of the tetrahedron with vertices , and . Evaluate the surface integral .
and let S be the surface of the tetrahedron with vertices , and . Evaluate the surface integral . Unlock Deck
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16
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
A) e. b. f. c. g. d. h.
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Unlock Deck
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17
Use the Divergence Theorem to evaluate where and S is the sphere .
A) e. b. f. c. g. d. h.
where and S is the sphere .A)
e. b. f. c. g. d. h. Unlock Deck
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18
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
A) e. b. f. c. g. d. h.
and let S be the boundary surface of the solid . Evaluate the surface integral .A)
e. b. f. c. g. d. h. Unlock Deck
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19
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
and let S be the boundary surface of the solid . Evaluate the surface integral . Unlock Deck
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20
Evaluate the flux integral over the boundary of the ball .
over the boundary of the ball . Unlock Deck
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21
Evaluate , where S is the boundary surface of the solid sphere and
, where S is the boundary surface of the solid sphere and Unlock Deck
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22
Evaluate , where S is the boundary surface of the region outside the sphere and inside the ball and .
, where S is the boundary surface of the region outside the sphere and inside the ball and . Unlock Deck
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23
Let . Evaluate the line integral along the elliptical path .
A) e. b. f. c. g. d. h.
. Evaluate the line integral along the elliptical path .A)
e. b. f. c. g. d. h. Unlock Deck
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24
Let S be the outwardly-oriented surface of a solid region E where the volume of E is . If and , evaluate the surface integral .
. If and , evaluate the surface integral . Unlock Deck
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25
Use Stokes' Theorem to evaluate where C is the circle .
A) e. b. f. c. g. d. h.
where C is the circle .A)
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26
Evaluate , where and S is the sphere .
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27
Use Stokes' Theorem to evaluate where and S is the part of the paraboloid that lies inside the cylinder , oriented upward.
where and S is the part of the paraboloid that lies inside the cylinder , oriented upward. Unlock Deck
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28
Let . Evaluate the line integral along the elliptical path .
A) e. b. f. c. g. d. h.
. Evaluate the line integral along the elliptical path .A)
e. b. f. c. g. d. h. Unlock Deck
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29
Let . Evaluate the line integral along the rectangular path from to to to to .
A) e. b. f. c. g. d. h.
. Evaluate the line integral along the rectangular path from to to to to .A)
e. b. f. c. g. d. h. Unlock Deck
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30
Let . Evaluate over the surface S given by , with downward orientation.
A) e. b. f. c. g. d. h.
. Evaluate over the surface S given by , with downward orientation.A)
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31
Find the flux of across the surface of the solid bounded by the paraboloid and the plane.
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32
Evaluate , where and S is the sphere .
, where and S is the sphere . Unlock Deck
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33
Find the flux of across the surface of the solid bounded by the paraboloid and the plane.
across the surface of the solid bounded by the paraboloid and the plane. Unlock Deck
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34
Let . Let C be the rectangular path from to to to to . Use Stokes' Theorem to evaluate the line integral , where T is the unit tangent vector to C.
. Let C be the rectangular path from to to to to . Use Stokes' Theorem to evaluate the line integral , where T is the unit tangent vector to C. Unlock Deck
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35
Find the flux of across the surface of the solid .
across the surface of the solid . Unlock Deck
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36
Evaluate , where the path C is the curve of intersection of the paraboloid with the plane .
, where the path C is the curve of intersection of the paraboloid with the plane . Unlock Deck
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37
Let . Evaluate the line integral , where C is the curve of intersection of the paraboloid and the cylinder .
A) e. b. f. c. g. d. h.
. Evaluate the line integral , where C is the curve of intersection of the paraboloid and the cylinder .A)
e. b. f. c. g. d. h. Unlock Deck
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38
Use Stokes' Theorem to evaluate where C is the circle .
A) e. b. f. c. g. d. h.
where C is the circle .A)
e. b. f. c. g. d. h. Unlock Deck
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39
Use Stokes' Theorem to evaluate where C is the triangle with vertices , and , oriented counter clockwise as viewed from above.
A) e. b. f. c. g. d. h.
where C is the triangle with vertices , and , oriented counter clockwise as viewed from above.A)
e. b. f. c. g. d. h. Unlock Deck
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40
Let and let S be the boundary surface of the solid . Evaluate the surface integral .
and let S be the boundary surface of the solid . Evaluate the surface integral . Unlock Deck
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41
A surface has the shape of the cone between and with the density function . Find the mass of the surface.
A)
B)
C)
D)
E)
F)
G)
H)
between and with the density function . Find the mass of the surface.A)
B)
C)
D)
E)
F)
G)
H)
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42
Consider the surfaces : , and : , and let F be a vector field with continuous partial derivatives everywhere. Why do we know that ?
: , and : , and let F be a vector field with continuous partial derivatives everywhere. Why do we know that ? Unlock Deck
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43
Use Stokes' Theorem to evaluate , where C is the triangle with vertices , and , oriented counterclockwise as viewed from above.
, where C is the triangle with vertices , and , oriented counterclockwise as viewed from above. Unlock Deck
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44
Use Stokes' Theorem to evaluate where and S is the part of the hemisphere that lies inside the cylinder , oriented in the direction of the positive x-axis.
where and S is the part of the hemisphere that lies inside the cylinder , oriented in the direction of the positive x-axis. Unlock Deck
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45
Evaluate the surface integral , where and S is the part of the plane in the first octant with downward orientation.
A)
B)
C)
D)
E)
F)
G)
H)
, where and S is the part of the plane in the first octant with downward orientation.A)
B)
C)
D)
E)
F)
G)
H)
Unlock Deck
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46
Evaluate , where and S is the cube bounded by .
A)
B)
C)
D)
E)
F)
G)
H)
, where and S is the cube bounded by .A)
B)
C)
D)
E)
F)
G)
H)
Unlock Deck
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47
Evaluate the surface integral , where S is that part of the plane that lies above the square with vertices , and .
A)
B)
C)
D)
E)
F)
G)
H)
, where S is that part of the plane that lies above the square with vertices , and .A)
B)
C)
D)
E)
F)
G)
H)
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48
Verify that Stokes' Theorem is true for the vector field and the cone , oriented upward.
and the cone , oriented upward. Unlock Deck
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49
Let . Evaluate the surface integral , where S is that part of the plane that lies above the square with vertices , and and has upward orientation.
A)
B)
C)
D)
E)
F)
G)
H)
. Evaluate the surface integral , where S is that part of the plane that lies above the square with vertices , and and has upward orientation.A)
B)
C)
D)
E)
F)
G)
H)
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50
Evaluate where and S is the part of the surface that lies above the plane and has upward orientation.
A)
B)
C)
D)
E)
F)
G)
H)
where and S is the part of the surface that lies above the plane and has upward orientation.A)
B)
C)
D)
E)
F)
G)
H)
Unlock Deck
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51
Let S be the parametric surface . Use Stokes' Theorem to evaluate , where .
. Use Stokes' Theorem to evaluate , where . Unlock Deck
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52
Use Stokes' Theorem to evaluate , where C is the curve of intersection of the paraboloid and the cylinder , oriented counterclockwise as viewed from above.
, where C is the curve of intersection of the paraboloid and the cylinder , oriented counterclockwise as viewed from above. Unlock Deck
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53
An upper hemisphere is given by with the density function . Find the mass of the sphere.
A)
B)
C)
D)
E)
F)
G)
H)
with the density function . Find the mass of the sphere.A)
B)
C)
D)
E)
F)
G)
H)
Unlock Deck
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54
Evaluate the surface integral , where S is the triangle with vertices , and
, where S is the triangle with vertices , and Unlock Deck
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55
Let . Evaluate the surface integral , where S is that part of the plane that lies above the square with vertices , and and has upward orientation.
A)
B)
C)
D)
E)
F)
G)
H)
. Evaluate the surface integral , where S is that part of the plane that lies above the square with vertices , and and has upward orientation.A)
B)
C)
D)
E)
F)
G)
H)
Unlock Deck
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56
Evaluate the surface integral , where S is that part of the cylinder that lies above the square with vertices , and .
A)
B)
C)
D)
E)
F)
G)
H)
, where S is that part of the cylinder that lies above the square with vertices , and .A)
B)
C)
D)
E)
F)
G)
H)
Unlock Deck
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57
Use Stokes' Theorem to evaluate where and C is the curve of intersection of the plane and the cylinder .
where and C is the curve of intersection of the plane and the cylinder . Unlock Deck
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58
Evaluate , where and S is the cube bounded by .
A)
B)
C)
D)
E)
F)
G)
H)
, where and S is the cube bounded by .A)
B)
C)
D)
E)
F)
G)
H)
Unlock Deck
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k this deck
59
Evaluate the surface integral , where S is that part of the plane that lies above the square with vertices , and .
A)
B)
C)
D)2
E)
F)
G)
H)
, where S is that part of the plane that lies above the square with vertices , and .A)
B)
C)
D)2
E)
F)
G)
H)
Unlock Deck
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60
Use Stokes' Theorem to evaluate where and C is the triangle with vertices , and
where and C is the triangle with vertices , and Unlock Deck
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61
A fluid has density 1500 and velocity field . Find the rate of flow outward through the sphere .
. Find the rate of flow outward through the sphere . Unlock Deck
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62
Evaluate , where and S is the upper half of the sphere , with upward orientation.
, where and S is the upper half of the sphere , with upward orientation. Unlock Deck
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63
Evaluate the surface integral for the vector field where S is part of the cone between the planes z = 1 and z = 2 with upward orientation.
for the vector field where S is part of the cone between the planes z = 1 and z = 2 with upward orientation. Unlock Deck
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64
Evaluate the flux of the vector field through the plane region with the given orientation as shown below.
through the plane region with the given orientation as shown below. Unlock Deck
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65
Evaluate where and S is the part of the surface that lies above the rectangle and has upward orientation.
where and S is the part of the surface that lies above the rectangle and has upward orientation. Unlock Deck
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66
Evaluate the surface integral , where S is the part of the sphere that lies above the cone .
, where S is the part of the sphere that lies above the cone . Unlock Deck
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67
Evaluate the surface integral , where S is the part of the paraboloid that lies in front of the plane .
, where S is the part of the paraboloid that lies in front of the plane . Unlock Deck
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68
Compute the surface integral if and S is the piece of the sphere in the second octant .
if and S is the piece of the sphere in the second octant . Unlock Deck
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69
Consider the top half of the ellipsoid parametrized by . Find a normal vector N at the point determined by , and determine if it is upward and/or outward.
parametrized by . Find a normal vector N at the point determined by , and determine if it is upward and/or outward. Unlock Deck
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70
Evaluate the flux of the vector field through the plane region with the given orientation as shown below.
through the plane region with the given orientation as shown below. Unlock Deck
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71
Evaluate the surface integral for the vector field , where S is the hemisphere with upward orientation.
for the vector field , where S is the hemisphere with upward orientation. Unlock Deck
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72
Find the flux of the vector field across the paraboloid given by with and upward orientation:
across the paraboloid given by with and upward orientation: Unlock Deck
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73
Find the z-coordinate of the centroid of the upper hemisphere with uniform density whose equation is given by .
. Unlock Deck
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74
Find the mass of the sphere whose density at each point is proportional to its distance to the plane.
whose density at each point is proportional to its distance to the plane. Unlock Deck
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75
Evaluate , where S is the part of the surface that lies between the cylinders and .
, where S is the part of the surface that lies between the cylinders and . Unlock Deck
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76
Evaluate , where and S is the part of the surface below the plane , with upward orientation.
, where and S is the part of the surface below the plane , with upward orientation. Unlock Deck
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77
Evaluate the surface integral for the vector field where S is the part of the elliptic paraboloid that lies below the square and has downward orientation.
for the vector field where S is the part of the elliptic paraboloid that lies below the square and has downward orientation. Unlock Deck
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78
Let . Find the curl of F.
A)
B)
C)
D)
E)
F)
G)
H)
. Find the curl of F.A)
B)
C)
D)
E)
F)
G)
H)
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79
Find the mass of a thin funnel in the shape of a cone , is its density
function is .
, is its densityfunction is
. Unlock Deck
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80
Let . Find the curl of F.
A)
B)
C)
D)
E)
F)
G)
H)
. Find the curl of F.A)
B)
C)
D)
E)
F)
G)
H)
Unlock Deck
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Unlock Deck
Unlock for access to all 240 flashcards in this deck.
