Deck 16: Vector Fields

Question 1

Find the gradient vector field of f(x, y) = ln ( + 3y).

A) i + j
B) i + j
C) i + j
D) i + j
E) i + j

Answer 1

i + j

Question 2

Find the gradient vector field f(x,y) of f(x, y) = .

A) - i - j - k
B) i + j + k
C) i + j + k
D) i + j + k
E) - i - j - k

Answer 2

i + j + k

Question 3

Describe the streamlines of the given velocity field v(x, y, z) = - yi + xj.

A) concentric circles centred on the x-axis in planes perpendicular to the x-axis
B) concentric circles centred on the y-axis in planes perpendicular to the y-axis
C) concentric circles centred on the z-axis in planes perpendicular to the z-axis
D) hyperbolas in planes perpendicular to the z-axis
E) parabolas in planes perpendicular to the z-axis

Answer 3

concentric circles centred on the z-axis in planes perpendicular to the z-axis

Question 4

Show that r(t) = i + 2t j + k for t ≠ 0 is a streamline for the velocity vector field .

Question 5

Find the family of field lines of the plane polar field F(r, $\theta$ ) = 2 + $\theta$ .

A) r = C $\theta$
B) r = + C
C) r = C
D) r = C
E) r = ln $\theta$

Question 6

Find parametric equations of the streamline of the velocity field v(x, y, z) = y i - y j + y k that passes through the point (2, -3, -4).

A) x = 2 + t, y = -3 - t, z = -4 + t
B) x = 2 + t, y = -3 + t, z = -4 + t
C) x = 2 + t, y = -3 - t, z = -4 - t
D) x = 2 + t, y = 3 - t, z = 4 + t
E) x = 2t, y = -3t, z = -4t

Question 7

Find a vector parametric equation of the field line of the vector field F(x, y, z) = -y i + x j + k that passes through the point (2, 0, 0).

A) r = 2 cos t i + 2 sin t j - t k
B) r = 2 cos t i + 2 sin t j + t k
C) r = 2 cos t i - 2 sin t j + t k
D) r = cos t i + sin t j + t k
E) r = 2 cos t i + 2 sin t j

Question 8

Define carefully what is meant by: The function V(x,y) is a Liapunov function in a domain D (containing the fixed point at the origin) for the autonomous system associated with the vector field F = P(x,y) i + Q(j, x,y).Assume that P and Q have continuous partial derivatives in D.

Question 9

The function V(x,y) = 3 +3xy + 5 is a Liapunov function for the autonomous system associated with the vector field F = (y -7x) i + (3x - 5y) j in any domain D containing the fixed point at the origin.

Question 10

Use a suitable Liapunov function to show that the fixed point at the origin for the autonomous system is at least stable.

Question 11

Use the Liapunov function V(x,y) = ( + ) to determine the stability of the fixed point at the origin of an autonomous system associated with the vector field F = (- y - x ) i + (x - y ) j.

Question 12

Show that the fixed point at the origin for the autonomous system is unstable.

Question 13

The Liapunov function V(x,y) = 2 - xy + 3 is suitable to confirm that the fixed point at the origin for the autonomous system is at least stable.

Question 14

Let V(x , y) = A + B , where A and B are constant real numbers, be a Liapunov function in a domain D (containing the fixed point at the origin) for the autonomous non-linear system .Find A and B if the derivative of V along the trajectories of the system is equal to - .Is the origin stable, asymptotically stable, or unstable?

A) A = 2, B = -3; asymptotically stable
B) A = -2, B = -3; unstable
C) A =2, B = 3; unstable
D) A = -2, B = 3; stable
E) A = 2, B = 3; asymptotically stable

Question 15

Is F (x,y) = (3 y + 2x + 1) i + ( + 2 y + 1) j conservative? If so, find a potential for it.

A)
B)
C)
D)
E) No, it is not conservative.

Question 16

Is F (x,y,z) = 6xy sin(2z) i + 3 sin(2z) j - 6 xy cos(2z) k conservative? If so, find a potential for it.

A) yes, (x,y) = 3 y sin(2z) + C
B) yes, 11ee7ba1_26b3_e973_ae82_2f72847b9867_TB9661_11 (x,y) = 3 y cos(2z) + C
C) yes, 11ee7ba1_26b3_e973_ae82_2f72847b9867_TB9661_11 (x,y) = 6xy sin(2z) + C
D) yes, 11ee7ba1_26b3_e973_ae82_2f72847b9867_TB9661_11 (x,y) = 3x sin(2z) + C
E) No, it is not conservative.

Question 17

An equipotential surface of a conservative vector field F is given by + ln( ) = 12 for y, z > 0.Find F.

A)
B)
C)
D)
E)

Question 18

For what values of the constants A, B, and C is F conservative?

A) A = 2, B = - $\pi$ , C = 1
B) A = 3, B = - $\pi$ , C = -1
C) A = 3, B = -2, C = -1
D) A = 2, B = $\pi$ , C = 2
E) There are no values of A, B, and C that will make F conservative.

Question 19

The gradient of a scalar field expressed in terms of polar coordinates [r, $\theta$ ] in the plane is (r, $\theta$ ) = + . Use the result above to find the necessary condition for the vector field F(r, $\theta$ ) = P(r, $\theta$ ) + Q(r, $\theta$ ) to be conservative.

A) =
B) = r
C) = -
D) - r = Q
E) - = r

Question 20

Find the equipotential surfaces of the conservative field F(x,y,z) = (i + x j + 2x k).

A) x = C
B) x = C
C) = C
D) x = C
E) y = C

Question 21

(a)In terms of polar coordinates r and $\theta$ , describe the field lines of the conservative plane vector field F(x,y) = x i + y j.
(b) In terms of polar coordinates r and $\theta$ , describe the equipotential curves of the conservative plane vector field F(x,y) = x i + y j.

A) (a) radial lines $\theta$ = (b) circles r =
B) (a) circles r = (b) radial lines $\theta$ =
C) (a) circles r = sin( $\theta$ ) (b) circles r = cos( $\theta$ )
D) (a) lines r cos $\theta$ = (b) lines r sin $\theta$ =
E) (a) lines r cos $\theta$ = (b) radial lines $\theta$ =

Question 22

Which of the following three vector fields is conservative?
F = (4xz + 4 ) i + (xy + ) j + (2yz + 2 ) k,
G = 7xy j
H = F + G

A) only F
B) only G
C) only H
D) only F and G
E) All three are conservative.

Question 23

Let F = i + j + K k. For what value of the constant K is F conservative?
If K has that value, find the family of equipotential surfaces of F.

A) K = -1, z = C( + )
B) K = 1, = C( + )
C) K = -1, z ( + ) = C
D) K = -2, = C( + )
E) K = 2, z = C( + )

Question 24

If F and G are plane conservative vector fields with potentials and , respectively, then the vector field H = 3F - 2G is also conservative with potential 311ee7ba1_f0bb_6cc5_ae82_7109d7d2775a_TB9661_11 - 2https://d2lvgg3v3hfg70.cloudfront.net/TB9661/11ee7ba1_ff03_9556_ae82_45b7421486b4_TB9661_11.

Question 25

The gradient of a scalar field expressed in terms of polar coordinates [r, $\theta$ ] in the plane is (r, $\theta$ ) = + . Use the result above to find the constant real numbers a and b such that the vector field F = cos(2 $\theta$ ) + a sin(2 $\theta$ ) is conservative.

A) a = 1 , b = -1
B) a = -1 , b = 2
C) a = - , b = 2
D) a = - , b = 2
E) a = , b = -2

Question 26

The gradient of a scalar field expressed in terms of polar coordinates [r, $\theta$ ] in the plane is (r, $\theta$ ) = + . Use the above result to find a potential function for the conservative vector field (expressed in polar form) F = 3 cos( $\theta$ ) - sin( $\theta$ ) .

A) 4 cos( $\theta$ ) + C
B) - 8r sin( $\theta$ ) + C
C) cos( $\theta$ ) + cos( $\theta$ )
D) cos( $\theta$ ) + C
E) cos( $\theta$ ) + cos( $\theta$ ) + C

Question 27

A potential function of a vector field F is given by , where (r , θ) are the polar coordinates and C is an arbitrary constant. Find F.
Hint: The gradient of g(r , θ) is given by ∇g(r , θ) = .

A) r sin(2 $\theta$ ) + r cos(2 $\theta$ )
B) r sin(2 $\theta$ ) + r cos(2 $\theta$ )
C) r sin(2 $\theta$ ) - r cos(2 $\theta$ )
D) r sin(2 $\theta$ ) + r cos(2 $\theta$ )
E) r sin(2 $\theta$ ) - r cos(2 $\theta$ )

Question 28

Describe the family of equipotential curves and the family of field lines for the conservative vector field F = x i - yj . Sketch at least four members of each family.

Question 29

Evaluate the integral ds once around the square C in the xy-plane with vertices (± 1, 1) and (± 1, -1).

A)
B)
C) 8
D) 11
E)

Question 30

Use a line integral to find the mass of a wire running along the curve y = from (0, 0) to (1, 1) if the density (mass per unit length) of the wire at any point (x, y) is numerically equal to y.

A) -
B) -
C) -
D) -
E) +

Question 31

Evaluate the line integral along the straight line from (1, 2, -1) to (3, 2, 5).

A)
B)
C)
D)
E)

Question 32

Evaluate the line integral along the first octant part of y = , z + y = 1 from (0, 0, 1) to (1, 1, 0).

A)
B)
C)
D) 18
E)

Question 33

Evaluate the line integral where C is the curve y = x, z = 1 + , from (-1, -1, 2) to (1, 1, 2).

A) 2
B) 1
C) 0
D) -1
E) -2

Question 34

Evaluate the line integral ds, where C is that part of the line of intersection of the two planes 4x - y - z = -1 and 2x - 3y + 2z = 2 from (0, 0, 1) to (1, 2, 3).

A)
B)
C) 6
D)
E) 5

Question 35

Find the integral ds, where C is the first octant portion of the curve of intersection of the cylinder x² + (y - 1)² = 1 and the plane x + z = 1.

A) (1 - 2 )
B) (1 - 2 )
C) (2 - 1)
D) (2 - 1)
E) 0

Question 36

Let C be the curve of intersection of the paraboloid z = 6 - x² - y² and the cone z = .Find the mass of the wire having the shape of the curve C if the line density function is given by (x, y, z) = z .

A)
B) 81 $\pi$
C) 16 $\pi$
D)
E) 8 $\pi$

Question 37

Find ds along the curve r = i + j + t k, 0 $\le$ t $\le$ l.

A) 5e -3
B) e + 1
C) 2e - 1
D) e - 1
E) 2e

Question 38

Find ds along the entire line 3x + 4y = 10. (Hint: Use symmetry to replace the line with a horizontal line at the same distance from the origin.)

A) $\pi$
B)
C) 2
D)
E)

Question 39

Use the definition of the line integral to evaluate dx, where C is the graph of x + y = 5 with initial point (1, 4) and terminal point (0, 5).

A) -24
B) -25
C) -26
D) -27
E) -23

Question 40

Find the work done by the force field F(x, y, z) = x i + 3xy j - (x + z) k on a particle moving along the line segment from (1, 4, 2) to (0, 5, 1).

A) 2
B) 10
C) 12
D) 8
E) 16

Question 41

Find + z dy + x dz where C is part of the helix r(t) = sin t i + cos t j + t k, 0 $\le$ t $\le$ $\pi$ .

A) 2 +
B) 2 -
C) 2 -
D) 2 +
E) 1 - $\pi$

Question 42

If F = -y i + x j + z k, calculate where C is the straight line segment from(1, 0, 0) to (-1, 0, $\pi$ ).

A)
B)
C) 1
D)
E) $\pi$

Question 43

Let = , = and = be sets of points in 3-space. Determine whether the set is a connected or a simply connected set.

A) S₁ is simply connected.
B) S₂ is not connected.
C) S₁ is connected but not simply connected.
D) S₃ is simply connected.
E) S₃ is connected but not simply connected.

Question 44

Let F be a smooth conservative force field defined in 2-space with a potential function , and let C be the curve shown in the figure below. Find the work done by the force field F in moving a particle along the curve C from P to R given that φ(1,- 2) = -17, andφ(4, 1) = 3.

Question 45

How much work is required for the force field F = y i + 2x j to move an object along the upper part of the ellipse from (3, 0) to (-3, 0)?

A) 2 $\pi$
B) 9 $\pi$
C) -9 $\pi$
D) -2 $\pi$
E) 0

Question 46

Let F be a vector field such that F = ∇φ - y i for some smooth scalar function φ. Evaluate counterclockwise around the ellipse + = 1.

Question 47

Evaluate the line integral + x dy + z dz along the curve C from (1, 0, 1) to(-1, 2, 5) with parametrization with

A) 9
B) 10
C) 11
D) 12
E) 8

Question 48

Evaluate the line integral dx + 2y dy + (x + 2z) dz along the curve C with parametrization with

A) 14
B) 15
C) 16
D) 17
E) 18

Question 49

Find the work done by the conservative force F = (2y + z) i + (2x + z) j + (x + y) k in moving a particle along the elliptical helix from

A) 28 $\pi$
B) 39 $\pi$
C) 4 $\pi$
D) 7 $\pi$
E) 16 $\pi$

Question 50

Evaluate the line integral for F = y, 0, zy , where R is the helix

A) -20 $\pi$
B) -17 $\pi$
C) -18 $\pi$
D) -19 $\pi$
E) -21 $\pi$

Question 51

Evaluate the line integral for F = (2y + z) i + (2x + z) j + (x + y) k, where C is the curve

A)
B)
C)
D) 0
E)

Question 52

For what value of the constant A does the line integral I = , where have the same value for all paths from the origin to the point For that value of A, what is the value of the integral?

A) A = 3, I = 6
B) A = 2, I = -2
C) A = 2, I = 1
D) A = 3, I = 1
E) A = 2, I = 2

Question 53

Use the fact that the field F = 2x sin(z) i - sin(z) j + ( cos(z) + y) k is almost conservative (except for the last term) to help you evaluate around the circle

A) 2 $\pi$
B) $\pi$
C)
D)
E) 0

Question 54

Use the fact that the field is almost conservative to help you evaluate along the curve from

A) -
B) -
C)
D) -
E)

Question 55

Around what non-self-intersecting closed curves in the xy-plane will the line integral have a zero value?

A) circles that pass through the origin
B) circles that are centred at the origin
C) curves bounding domains that contain the origin in their interiors
D) curves bounding domains that do not contain the origin in their interiors
E) any non-self-intersecting closed curve in the xy-plane

Question 56

What conditions must a domain D in the xy-plane satisfy to ensure that the value of the line integral + dy should depend only on the initial and terminal points of the path C that lies in D?

A) D must be simply connected and not contain the point (1, -1).
B) D must be connected and not contain the point (1, -1).
C) D must not contain the point (1, -1).
D) D must be simply connected and must contain the point (1, -1).
E) D can be any domain in the xy-plane.

Question 57

Evaluate the surface integral , where S is the graph of z = for

A) 2
B) 3
C) 4
D) 5
E) 1

Question 58

Find the area of the surface cut from the paraboloid + - z = 0 by the plane z = 2.

A) $\pi$ square units
B) $\pi$ square units
C) $\pi$ square units
D) $\pi$ square units
E) $\pi$ square units

Question 59

Evaluate the surface integral where S is the portion of the plane z = 2x - 4y above the region bounded by y = 0 , x = 1 and x = 3y in the xy-plane.

A)
B)
C)
D)
E)

Question 60

Find the area of the region cut from the plane x + 2y + 2z = 5 by the cylinder whose walls are and

A) 2 square units
B) 3 square units
C) 4 square units
D) 6 square units
E) 1 square unit

Question 61

Evaluate the surface integral where S is the portion of the plane z = 2x - 4y above the region bounded by y = 0 , x = 1 and x = 3y in the xy-plane.

A) -
B) -
C)
D) -
E)

Question 62

Find the area of the ellipse cut from the plane z = cx by the cylinder (c is constant.)

A) $\pi$ square units
B) $\pi$ square units
C) 2 $\pi$ square units
D) square units
E) square units

Question 63

Evaluate the surface integral where S is the portion of the plane x + y + z = 1 inside the cylinder x² + y² = 1

A) $\pi$
B) $\pi$
C) $\pi$
D) $\pi$
E) $\pi$

Question 64

Evaluate the surface integral , where S is the portion of the plane above the region R bounded by and in the xy-plane.

A)
B) -
C)
D) -
E) -

Question 65

Find dS, where S is the part of the paraboloid that lies above the ring in the xy-plane.

A) (25 + 1)
B) (391 + 1)
C) (391 - 25 )
D) (391 - 25 )
E) (1235 - 5 )

Question 66

Evaluate the surface integral where S is the entire surface x + y + z = 1 lying in the first octant.

A)
B)
C)
D)
E)

Question 67

Find the area of the part of the surface - 2 ln x + y - z = 0 inside the prism bounded by the planes x = 1, x = 2, y = 0, and y = 1.

A) 2 + 2 ln 2 square units
B) 3 + 2 ln 2 square units
C) 4 + ln 2 square units
D) 5 + 2 ln 2 square units
E) + ln 2 square units

Question 68

Evaluate the surface integral where S is the portion of the cone above the quarter disk in the xy-plane.

A)
B)
C)
D)
E) 0

Question 69

Integrate f(x, y, z) = x + y + z over the surface of the cube cut from the first octant by the planes x = a, y = a, and z = a.

A) 8
B) 9
C) 10
D) 11
E) 6

Question 70

Integrate g(x, y, z) = x²y²z² over the surface of the rectangular solid cut from the first octant by the planes x = a, y = b, and z = c.

A) (ab + ac + bc)
B) (ab + ac + bc)
C) (ab + ac + bc)
D) (ab + ac + bc)
E) abc(ab + ac + bc)

Question 71

Find the value of the positive constant real number a such that the area of the part of the plane inside the elliptic paraboloid z = 3x² + ay² is equal to square units.

A)
B)
C)
D)
E)

Question 72

Find the area element at a point r (θ, φ) on the parametric surface , .

Question 73

Evaluate where S is the part of the paraboloid 2x = 8 - - lying between the planes

A) 2 $\pi$
B) 2 $\pi$
C) 2 $\pi$
D) $\pi$
E) $\pi$

Question 74

Find the surface area of the part of the cylinder x² + y² = 2y that lies in the first octant and under the paraboloid

A) $\pi$ square units
B) 2 $\pi$ square units
C) 3 $\pi$ square units
D) 4 $\pi$ square units
E) 6 $\pi$ square units

Question 75

Find the surface area of the part of the sphere x² + y² + z² = 36 between the planes z = 1 and z = 5.

A) 12 $\pi$ square units
B) 288 $\pi$ square units
C) 48 $\pi$ square units
D) 96 $\pi$ square units
E) 144 $\pi$ square units

Question 76

Find two unit vectors orthogonal to the parametric surface S given by r (u, v) = ( - 4 - ) i + 4v j + 2u k at the point on the surface corresponding to (u, v) = ( 3 , - ).

A) ±
B) ± (- 1 , 0 ,0)
C) ±
D) ±
E) ±

Question 77

Compute the flux of F = x i + y j + z k upward through the part of the plane x + y + z = 3 in the first octant of 3-space.

A)
B)
C)
D)
E)

Question 78

Compute the flux of F = x i + y j + z k outward through the whole surface of the tetrahedron bounded by the coordinate planes and the plane

A)
B)
C)
D)
E) 0

Question 79

Compute the flux of F = upward through the the part of the cone = + lying inside the first octant of cylinder

A) 1
B) 2
C) 4
D)
E)

Question 80

Compute the flux of F = 2x i - j + (z - 2x + 2y) k upward through the part of the plane in the first octant of 3-space.

A)
B)
C)
D)
E) 0