Deck 17: Vector Calculus

Question 1

Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle x² + y² = 4.

-F = i - j

Answer 1

Answers will vary.

Question 2

Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle x² + y² = 4.

-F = i + j

Answer 2

Answers will vary.

Question 3

Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle x² + y² = 4.

-F = - i - j

Answer 3

Answers will vary.

Question 4

Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle x² + y² = 4.

-F = -xi - yj

Question 5

Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle x² + y² = 4.

-F = -xi + yj

Question 6

Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle x² + y² = 4.

-F = xi - yj

Question 7

Find the gradient field F of the function f.

-f(x, y, z) = +

A)

B)

C)

D)

Question 8

Find the gradient field F of the function f.

-f(x, y, z) = +

A)
B)
C)
D)

Question 9

Find the gradient field F of the function f.

-f(x, y, z) =

A)
B)
C)
D)

Question 10

Find the gradient field F of the function f.

-f(x, y, z) = ln ( + + )

A)
B)
C)
D)

Question 11

Find the gradient field F of the function f.

-f(x, y, z) =

A)
B)
C)
D)

Question 12

Find the gradient field F of the function f.

-f(x, y, z) = z sin (x + y + z)

A)
B)
C)
D)

Question 13

Find the gradient field F of the function f.

-f(x, y, z) =

A)
B)
C)
D)

Question 14

Find the gradient field F of the function f.

-f(x, y, z) = ( - )

A)
B)
C)
D)

Question 15

Find the gradient field F of the function f.

-f(x, y, z) = ln +

A)
B)
C)
D)

Question 16

Find the gradient field F of the function f.

-f(x, y, z) =

A)
B)
C)
D)

Question 17

Evaluate the line integral of f(x,y) along the curve C.

-f(x, y) = , C: y = , 0 $\le$ x $\le$ 1

A) 1/4
B) 1/5
C) 0
D) 1

Question 18

Evaluate the line integral of f(x,y) along the curve C.

-f(x, y) = + , C: y = -2x - 4, 0 $\le$ x $\le$ 3

A) 55
B) 165
C) 255
D) 165

Question 19

Evaluate the line integral of f(x,y) along the curve C.

-f(x, y) = y + x, C: + = 4 in the first quadrant from ( 2, 0) to (0, 2)

A) 16
B) 8
C) 0
D) 4

Question 20

Evaluate the line integral of f(x,y) along the curve C.

-f(x, y) = + , C: the perimeter of the circle + = 16

A) 8 $\pi$
B) 32 $\pi$
C) 128 $\pi$
D) 64 $\pi$

Question 21

Evaluate the line integral of f(x,y) along the curve C.

-f(x, y) = x, C: y = , 0 $\le$ x $\le$

A) 215/12
B) 215
C) 215/8
D) 215/3

Question 22

Evaluate the line integral of f(x,y) along the curve C.

-f(x, y) = cos x + sin y, C: y = x, 0 $\le$ x $\le$

A)
B) 2
C) 0
D) 2

Question 23

Evaluate the line integral of f(x,y) along the curve C.

-f(x,y) = 3 , C: y = , 0 $\le$ x $\le$ 3

A)
B)
C)
D)

Question 24

Evaluate the line integral along the curve C.

- ds , C is the straight-line segment x = 0, y = 3 - t, z = t from (0, 3, 0) to

A) 9
B) 9/2
C) 0
D) 9

Question 25

Evaluate the line integral along the curve C.

-

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

Question 26

Evaluate the line integral along the curve C.

-

A) - 19/3
B) -19
C) 29
D) 29/3

Question 27

Evaluate the line integral along the curve C.

-

A) 1/0
B) 5/0
C) 1/0
D) 2/0

Question 28

Evaluate the line integral along the curve C.

-

A) 13/25
B) 1/25
C) 169/25
D) - 91/400

Question 29

Evaluate the line integral along the curve C.

-

A) 3/2
B) 3
C) $\pi$
D) 0

Question 30

Evaluate the line integral along the curve C.

-

A) 48
B) 24
C)48
D) 24

Question 31

Find the work done by F over the curve in the direction of increasing t.

-F = 10zi + 2xj + 9yk; C: r(t) = ti + tj + tk, 0 $\le$ t $\le$ 1

A) W = 42
B) W = 21/2
C) W = 7
D) W = 21

Question 32

Find the work done by F over the curve in the direction of increasing t.

-F = -8yi + 8xj + 3 k; C: r(t) = cos ti + sin tj, 0 $\le$ t $\le$ 6

A) W = 96
B) W = 0
C) W = 48
D) W = 144

Question 33

Find the work done by F over the curve in the direction of increasing t.

-

A)
B)
C)
D)

Question 34

Find the work done by F over the curve in the direction of increasing t.

-F = xyi + 8j + 3xk; C: r(t) = cos 8ti + sin 8tj + tk, 0 $\le$ t $\le$

A) W = 0
B) W = 25/3
C) W = 209/24
D) W = 193/24

Question 35

Find the work done by F over the curve in the direction of increasing t.

-F = 5yi + j + ( 5x + 2z)k; C: r(t) = ti + j + tk, 0 $\le$ t $\le$ 2

A) W = 47 + 20
B) W = +
C) W = + 20
D) W = 0

Question 36

Calculate the circulation of the field F around the closed curve C.

-

A) 2/3
B) 4/3
C) 8/3
D) 0

Question 37

Calculate the circulation of the field F around the closed curve C.

-F = xyi + 5j , curve C is r(t) = 3 cos ti + 3 sin tj, 0 $\le$ t $\le$ 2 $\pi$

A) 16
B) 4
C) 10
D) 0

Question 38

Calculate the circulation of the field F around the closed curve C.

-curve C is the counterclockwise path around the rectangle with vertices at

A) 0
B) 72
C) 108
D) 36

Question 39

Calculate the circulation of the field F around the closed curve C.

- curve C is the counterclockwise path around

A) 32
B) 16
C) 0
D) 4

Question 40

Calculate the circulation of the field F around the closed curve C.

- curve C is the counterclockwise path around the triangle with vertices at (0,0) , (2,.) and (0,4)

A) 8/3
B) 56/3
C) - 8/3
D) 0

Question 41

Calculate the circulation of the field F around the closed curve C.

-F = (-x - y)i + (x + y)j , curve C is the counterclockwise path around the circle with radius 2 centered at (2,1)

A) 16 $\pi$
B) 8(1 + $\pi$ ) + 24
C) 8(1 + $\pi$ )
D) 8 $\pi$

Question 42

Calculate the flux of the field F across the closed plane curve C.

-F = xi + yj; the curve C is the counterclockwise path around the circle + = 16

A) 64 $\pi$
B) 8 $\pi$
C) 0
D) 32 $\pi$

Question 43

Calculate the flux of the field F across the closed plane curve C.

- the curve C is the closed counterclockwise path formed from the semicircle 0 ≤ t ≤ π, and the straight line segment from (-4, 0) to ( 4, 0)

A) 64/3
B) - 32/3
C) 0
D) 32/3

Question 44

Calculate the flux of the field F across the closed plane curve C.

- the curve C is the closed counterclockwise path around the triangle with vertices at

A) 70
B) 70/3
C) -10
D) 0

Question 45

Calculate the flux of the field F across the closed plane curve C.

-F = xi + yj; the curve C is the closed counterclockwise path around the rectangle with vertices at

A) 101
B) 20
C) 0
D) 99

Question 46

Calculate the flux of the field F across the closed plane curve C.

-F = (x+y)i + xyj; the curve C is the closed counterclockwise path around the rectangle with vertices at

A) 52
B) 60
C) 96
D) 124

Question 47

Calculate the flux of the field F across the closed plane curve C.

-F = yi - xj; the curve C is the circle + = 36

A) -144
B) -72
C) 0
D) 72

Question 48

Calculate the flux of the field F across the closed plane curve C.

-F = xi + yj; the curve C is the circle + = 121

A) 242 $\pi$ + 11
B) 242 $\pi$
C) 0
D) 2 $\pi$

Question 49

Find the mass of the wire that lies along the curve r and has density ?.

-r(t) = i + 7tj, 0 $\le$ t $\le$ 1; = 3t

A) 169/15 units
B) 57 units
C) 169/10 units
D) 3/2 units

Question 50

Find the mass of the wire that lies along the curve r and has density ?.

-r(t) = 7i + ( 9 - 4t)j + 3tk, 0 $\le$ t $\le$ 2 $\pi$ ; = 5(1 + sin 7t)

A) 10 $\pi$ units
B) 50/7 + 50 $\pi$ units
C) 50 $\pi$ units
D) 100/7 + 50 $\pi$ units

Question 51

Find the mass of the wire that lies along the curve r and has density ?.

-r(t) = ( 7 cos t)i + ( 7 sin t)j + 7tk, 0 $\le$ t $\le$ 2 $\pi$ ; = 8

A) 112 $\pi$ units
B) 16 $\pi$ units
C) 784 $\pi$ units
D) 14 $\pi$ units

Question 52

Find the mass of the wire that lies along the curve r and has density ?.

-

A)
B)
C)
D)

Question 53

Find the potential function f for the field F.

-

A)

B)

C)

D)

Question 54

Find the potential function f for the field F.

-

A)
B)
C)
D)

Question 55

Find the potential function f for the field F.

-

A)
B)
C)
D)

Question 56

Find the potential function f for the field F.

-

A)
B)
C)
D)

Question 57

Find the potential function f for the field F.

-

A)
B)
C)
D)

Question 58

Find the potential function f for the field F.

-

A)
B)
C)
D)

Question 59

Find the potential function f for the field F.

-

A)
B)
C)
D)

Question 60

Find the potential function f for the field F.

-F = (y - z)i + (x + 2y - z)j - (x + y)k

A)
B)
C)
D)

Question 61

Evaluate the work done between point 1 and point 2 for the conservative field F.

-F = (y + z)i + xj + xk; (0, 0, 0), ( 3, 10, 7)

A) W = 30
B) W = 51
C) W = 9
D) W = 0

Question 62

Evaluate the work done between point 1 and point 2 for the conservative field F.

-F = 6xi + 6yj + 6zk; ( 4, 4, 5) , ( 6, 9, 6)

A) W = 630
B) W = -288
C) W = 0
D) W = 288

Question 63

Evaluate the work done between point 1 and point 2 for the conservative field F.

-F = 4 sin 4x cos 7y cos 4zi + 7 cos 4x sin 7y cos 4zj + 4 cos 4x cos 7y sin 4zk ;

A) W = 1
B) W = -2
C) W = 0
D) W = 2

Question 64

Evaluate the work done between point 1 and point 2 for the conservative field F.

-

A) W = - 1
B) W = + + - 1
C) W =
D) W = 0

Question 65

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.

-F = ( + )i + (x - y)j; C is the rectangle with vertices at (0, 0), ( 3, 0), ( 3, 9), and (0, 9)

A) 0
B) 216
C) -216
D) 270

Question 66

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.

-F = sin 3yi + cos 9xj; C is the rectangle with vertices at (0, 0), , , and

A) - 4/3 $\pi$
B) 0
C) - 2/3 $\pi$
D) 2/3 $\pi$

Question 67

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.

-F = (x - y)i + (x + y)j; C is the triangle with vertices at (0, 0), ( 3, 0), and (0, 10)

A) 30
B) 60
C) 300
D) 0

Question 68

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.

-F = xyi + xj; C is the triangle with vertices at (0, 0), ( 7, 0), and (0, 7)

A) 0
B) 245/3
C) 343/6
D) - 98/3

Question 69

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.

-F = ln ( + )i + j; C is the region defined by the polar coordinate inequalities 1 $\le$ r $\le$ 7 and

A) 96
B) -12
C) 50
D) 0

Question 70

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.

-F = - i + j; C is the region defined by the polar coordinate inequalities 8 $\le$ r $\le$ 9 and

A) 0
B) 9
C) 34
D) 145

Question 71

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.

-F = - i; C is the region defined by the polar coordinate inequalities 1 $\le$ r $\le$ 4 and

A) 65/96
B) 21/32
C) 0
D) - 21/32

Question 72

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.

-F = ( 6x + 6y)i + ( 9x - 2y)j; C is the region bounded above by y = -5 + 112 and below by in the first quadrant

A) -1332
B) - 2180
C) 896
D) -

Question 73

Using Green's Theorem, calculate the area of the indicated region.

-The area bounded above by y = 5x and below by y = 7

A) 125/147
B) 25/294
C) 625/588
D) 125/294

Question 74

Using Green's Theorem, calculate the area of the indicated region.

-The area bounded above by y = 10 and below by y =

A) 160/3
B) 0
C) 640/3
D) 320/3

Question 75

Using Green's Theorem, calculate the area of the indicated region.

-The area bounded above by y = 3 and below by y = 5

A) 27/250
B) 27/500
C) 27/1000
D) 27/125

Question 76

Using Green's Theorem, calculate the area of the indicated region.

-The circle r(t) = ( 10 cos t)i + ( 10 sin t)j, 0 $\le$ t $\le$ 2 $\pi$

A) 10 $\pi$
B) 2 $\pi$
C) 20 $\pi$
D) 100 $\pi$

Question 77

Using Green's Theorem, find the outward flux of F across the closed curve C.

-F =( + )i + (x - y)j ; C is the rectangle with vertices at (0, 0), ( 3, 0), ( 3, 10), and

A) 120
B) 60
C) 330
D) -270

Question 78

Using Green's Theorem, find the outward flux of F across the closed curve C.

-F = sin 6yi + cos 6xj ; C is the rectangle with vertices at (0, 0), , , and

A) 0
B) - 1/3 $\pi$
C) 1/3 $\pi$
D) - 2/3 $\pi$

Question 79

Using Green's Theorem, find the outward flux of F across the closed curve C.

-F = ln ( + ) i + j; C is the region defined by the polar coordinate inequalities and

A) 0
B) 58
C) 80
D) 40

Question 80

Using Green's Theorem, find the outward flux of F across the closed curve C.

-F = - i + j ; C is the region defined by the polar coordinate inequalities and

A) 0
B) 68
C) 60
D) 120