Deck 15: Topics in Vector Calculus

Question 1

Use Stokes' Theorem to evaluate $\int$ _C 28z dx - 14x dy + 14x dz where C is the intersection of the cylinder x² + y² = 1 and the plane z = y + 1.

A)
B)
C) 0
D)
E)

Answer 1

Question 2

Use Stokes' Theorem to evaluate where F(x, y, z) = 5x j and $\sigma$ is that portion of the ellipsoid 4x² + 4y² + z² = 8 for which z $\ge$ 0.

A) 5
B) 0
C) 12
D)
E)

Answer 2

0

Question 3

Use Stokes' Theorem to evaluate $\int$ _C 2(x + y)dx + 2(2x - 3)dy + 2(y + z)dz over the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0), and (0, 0, 6) traversed in a counterclockwise manner.

A) 2
B) 12
C) 0
D) 24
E) 18

Answer 3

24

Question 4

Use Stokes' Theorem to evaluate $\int$ _C 11z dx + 11x dy + 11y dz over the triangle with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1) traversed in a counterclockwise manner.

A) 16.5
B) 11
C) 22
D) 44
E) 1

Question 5

Find the outward flux of the vector field across the sphere .

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

Question 6

Verify Stokes' Theorem if $\sigma$ is the portion of the sphere x² + y² + z² = 1 for which z $\ge$ 0 and F(x, y, z) = (2x - y)i - yz² j - y²z k.

Question 7

Use Stokes' Theorem to evaluate $\int$ _C -4yz dx + 4xz dy + 4xy dz where C is the circle x² + y² = 2, z = 1.

A) 4
B) 16
C)
D)
E)

Question 8

Use Stokes' Theorem to evaluate $\int$ _C -21 dx + 21x dy + 7z dz over the circle x² + y² = 1, z = 1 traversed counterclockwise.

Question 9

Use Stokes' Theorem to evaluate where F(x, y, z) = 11(z - y)i + 11(z² + x)j + 11(x² - y²)k and $\sigma$ is that portion of the sphere x² + y² + z² = 4 for which z $\ge$ 0.

A)
B)
C)
D)
E)

Question 10

Use Stokes' Theorem to evaluate $\int$ _C 3(4x - 2y)dx - 3yz²dy - 3y²z dz where C is the circular region enclosed by x² + y² = 4, z = 2.

A) 3
B) 24
C)
D)
E) 0

Question 11

Use Stokes' Theorem to evaluate $\int$ _C 30y²dx + 30x²dy - 30(x + z)dz where C is a triangle in the xy-plane with vertices (0, 0, 0), (1, 0, 0), and (1, 1, 0) with a counterclockwise orientation looking down the positive z axis.

Question 12

Use Stokes' Theorem to evaluate $\int$ _C (x + y)dx + (2x - 3)dy + (y + z)dz over the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0), and (0, 0, 6) traversed in a counterclockwise manner.

Question 13

Use Stokes' Theorem to evaluate where F(x, y, z) = 3y i and $\sigma$ is that portion of the ellipsoid 4x² + 4y² + z² = 4 for which z $\ge$ 0.

Question 14

Use Stokes' Theorem to evaluate over the circle x² + y² = 16, z = 4.

A) 0
B) 64
C) -64
D)
E)

Question 15

Use Stokes' Theorem to evaluate $\int$ _C 3sin z dx - 3cos x dy + 3sin y dz over the rectangle
0 $\le$ x $\le$ $\pi$ , 0 $\le$ y $\le$ 1, and z = 2 traversed in a counterclockwise manner.

Question 16

Use Stokes' Theorem to evaluate where F(x, y, z) = (z - y)i + (z² + x)j + (x² - y²)k and $\sigma$ is that portion of the sphere x² + y² + z² = 4 for which z $\ge$ 0.

A)
B)
C) 0
D)
E)

Question 17

Use Stokes' Theorem to evaluate $\int$ _C 8z dx - 4x dy + 4x dz where C is the intersection of the cylinder x² + y² = 1 and the plane z = y + 1.

Question 18

Use Stokes' Theorem to evaluate over the circle x² + y² = 1, z = 1.

Question 19

Use Stokes' Theorem to evaluate $\int$ _C 3(z - y)dx + 3(x - z)dy + 3(y - x)dz where C is the boundary, in the xy-plane, of the surface $\sigma$ given by z = 4 - (x² + y²), z $\ge$ 0.

A)
B)
C) 0
D)
E)

Question 20

Use Stokes' Theorem to evaluate where F(x, y, z) = 8y k and $\sigma$ is that portion of the ellipsoid 4x² + 4y² + z² = 4 for which z $\ge$ 0.

Question 21

F (x, y, z) = 5xyz i + 5xyz j + 5xyz k. Find the outward flux of the vector field F across the unit cube in the first octant and including the origin as a vertex.

Question 22

Use the divergence theorem to evaluate where F(x, y, z) = 2e ^x i - 2ye ^x j + 6z k, n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the sphere by x² + y² + z² = 36.

A)
B)
C)
D)
E)

Question 23

Let and let $\sigma$ be a closed, orientable surface that surrounds the origin. Then the flux $\phi$ =

A) 7 $\pi$
B) 0
C) 100 $\pi$
D) 16 $\pi$
E) $\pi$

Question 24

Use the divergence theorem to evaluate where , n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the sphere by x² + y² + z² = 9.

A)
B)
C)
D)
E)

Question 25

Use the divergence theorem to evaluate where F(x, y, z) = 11yz i + 11xy j + 11xz k, n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface enclosed by the cylinder x² + z² = 1 and the planes y = -1 and y = 1.

Question 26

Use the divergence theorem to evaluate where F(x, y, z) = 2y²x i + 2yz² j + 2x²y² k, n is the outer unit normal to $\sigma$ , and $\sigma$ is the sphere x² + y² + z² = 4.

Question 27

Find the outward flux of the vector field across the sphere .

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

Question 28

Use the divergence theorem to evaluate where F(x, y, z) = (x³ + 3xy²)i + z³ k, n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the sphere of radius a centered at the origin.

Question 29

Use the divergence theorem to evaluate where , n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the sphere by x² + y² + z² = 36.

A)
B)
C)
D)
E)

Question 30

F (x, y, z) = 8x³ i + 16y² j + 24z² k. Find the outward flux of the vector field F across the unit cube in the first octant and including the origin as a vertex.

Question 31

Find the outward flux of F(x, y, z) = 4x i + 4y j + 4(z - 2)k across the rectangular box with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (3, 0, 0), (3, 1, 0), (3, 0, 1), and (3, 1, 1).

A) 12
B) 36
C) 0
D) 1
E) 4

Question 32

Let Find the outward flux of the vector field F across the surface of the region bounded above by the sphere by x² + y² + z² = 4 and below by the plane z = 0.

A)
B)
C)
D)
E)

Question 33

Use the divergence theorem to evaluate where , n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the sphere by x² + y² + z² = 49.

A)
B)
C)
D)
E)

Question 34

Find the outward flux of F(x, y, z) = 5(x - 1)i + 5(y - 3)j + 5z k across the rectangular box with vertices (0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (4, 0, 0), (4, 1, 0), (4, 0, 1), and (4, 1, 1).

A) 15
B) 60
C) 0
D) 1
E) 6

Question 35

F (x, y, z) = 12xyz i + 12xyz j + 12xyz k. Find the outward flux of the vector field F across the cube with vertices (0, 0, 0), (0, 0, 2), (0, 2, 2), (2, 2, 2), (0, 2, 0), (2, 0, 0), (2, 2, 0), and (2, 0, 2).

Question 36

Determine whether the flow field F(x, y, z) = 10x² i + 10y² j + 10x² k is free of all sources and sinks. If it is not, find the location of all sources and sinks.

Question 37

Let F(x, y, z) = 8x² i + 6y j + 3z k . Find the outward flux of the vector field F across the unit cube in the first octant and including the origin as a vertex.

A) 17
B) 18
C) 16
D) 27
E) 37

Question 38

Find the outward flux of F(x, y, z) = 5x i + (y + 3)j + 8z² k across the unit cube in the first octant that has a vertex at the origin.

A) 1
B) 0
C) 8
D) 9
E) 10

Question 39

Use the divergence theorem to evaluate where F(x, y, z) = 4x² i + 4y² j + 4z² k, n is the outer unit normal to $\sigma$ , and $\sigma$ is the surface of the cube enclosed by the planes 0 $\le$ x $\le$ 1, 0 $\le$ y $\le$ 1, and 0 $\le$ z $\le$ 1.

Question 40

Determine whether the flow field F(x, y, z) = 5x³ i + 5y³ j + 5z³ k is free of all sources and sinks. If it is not, find the location of all sources and sinks.

Question 41

Evaluate where F(x, y, z) = 2 i - z j + y k and $\sigma$ is that portion of the paraboloid x = y² + z² between x = 0 and x = 4. The surface is oriented by forward unit normals.

Question 42

Evaluate where F(x, y, z) = 2x i + 2y j + 4z k and $\sigma$ is that portion of the surface z = 4 - x² - y² above the xy-plane oriented by upward unit normals.

Question 43

Evaluate where F(x, y, z) = -x i - 2x j + (z - 1)k and $\sigma$ is the surface enclosed by that portion of the paraboloid z = 4 - y² which lies in the first octant and is bounded by the coordinate planes and the plane y = x. The surface is oriented by upward unit normals.

Question 44

If F(x, y, z) = 3y j + 3z k, the magnitude of the flux through the portion of the surface $\sigma$ that lies right of the yz-plane, where $\sigma$ is defined by x = 1 - y² - z², is

A)
B)
C) 3 $\pi$
D) 0
E) $\pi$

Question 45

Find the surface area of (x - 7)² + (y + 1)² + (z - 4)² = 4 that lies below z = 6.

A) 8 $\pi$
B) 32 $\pi$
C) 16 $\pi$
D) 64 $\pi$
E) 128 $\pi$

Question 46

Evaluate where F(x, y, z) = 2y i + 2z j + 2y k and $\sigma$ is that portion of the cone which lies in the first octant between x = 1 and x = 3. The surface is oriented by forward unit normals.

Question 47

Evaluate where F(x, y, z) = 18 i - 2z j + 2y k and $\sigma$ is that portion of the paraboloid x = 4 - y² - z² to the right of x = 0 oriented by forward unit normals.

A)
B)
C) 0
D)
E)

Question 48

Evaluate where F(x, y, z) = 9y i - 9x j - 36z² k and $\sigma$ is that portion of the cone which lies above the square in the xy-plane with vertices (0, 0), (1, 0), (1, 1), and (0, 1), and oriented by downward unit normals.

A) 72
B) 18
C) 0
D) 24
E) 45

Question 49

Find the flux of the vector field F(x, y, z) = 5z k across the sphere 5x² + 5y² + 5z² = 20 oriented outward.

A)
B)
C) 0
D)
E)

Question 50

If F(x, y, z) = 7y j + 7z k, the magnitude of the flux through the portion of the surface $\sigma$ that lies in front of the xz-plane, where $\sigma$ is defined by y = 1 - x² - z², is

A)
B)
C) 7 $\pi$
D) 0
E) $\pi$

Question 51

Evaluate where F(x, y, z) = 4y i + 8x j + 4xy k and $\sigma$ is that portion of the cylinder x² + y² = 9 in the first octant between z = 1 and z = 4. The surface is oriented by right unit normals.

A) 0
B) 4
C) 162
D) 8
E) -32

Question 52

Evaluate where F(x, y, z) = 4x i + 4y j + 4z k and $\sigma$ is that portion of the plane 2x + 3y + 4z = 12 which lies in the first octant and is oriented by upward unit normals.

A) 144
B) 16
C) 36
D) 72
E) 0

Question 53

Find the surface area of (x - 1)² + (y + 1)² + (z - 4)² = 4 that lies below z = 4.

A) 8 $\pi$
B) 16 $\pi$
C) 32 $\pi$
D) 64 $\pi$
E) 128 $\pi$

Question 54

Evaluate where F(x, y, z) = 12x i + 24j + 24x² k and $\sigma$ is that portion of the paraboloid z = x² + y² which lies above the xy-plane enclosed by the parabolas y = 1 - x² and y = x² - 1. The surface is oriented by downward unit normals.

Question 55

Let F(x, y, z) = 3y i. The flux outward between the planes z = 0 and z = 2 is

A) 0
B)
C) 9
D) 3
E)

Question 56

Evaluate where F(x, y, z) = 3x i + 3y j - 6z k and $\sigma$ is that portion of the sphere x² + y² + z² = 9 which lies above the xy-plane and is oriented by upward unit normals.

A) 12
B) 12 $\pi$
C) 9
D) 0
E) 6

Question 57

Let F(x, y, z) = 10x i + 10y j + 10z k and $\sigma$ be the portion of the surface z = 5 - x² - y² that lies above the xy-plane. Find the magnitude of the flux of the vector field across $\sigma$ .

A)
B)
C)
D) 0
E)

Question 58

Find the flux of the vector field F(x, y, z) = 3z k across the sphere x² + y² + z² = 9 oriented outward.

A) $\pi$
B) 3 $\pi$
C) 0
D) 108 $\pi$
E) $\pi$

Question 59

Find the surface area of the cone that lies between the planes z = 6 and z = 7.

A) 7 $\pi$
B)
C) 8 $\pi$
D)
E)

Question 60

F(x, y, z) = 4xyz i + 4xyz j + 4xyz k. Find the outward flux of the vector field F across the sphere x² + y² + z² = 25.

Question 61

Evaluate the surface integral , where $\sigma$ is the portion of the cone r(u, v) = u cos v i + u sin v j + u k for which 1 $\le$ u $\le$ 2, .

Question 62

Evaluate the surface integral where $\sigma$ is that portion of the plane x + y + z = 1 which lies in the first octant.

Question 63

Evaluate the surface integral where is the part of the plane in the first octant.

A) 3
B) 192
C)
D) 400
E) 800 $\pi$

Question 64

Evaluate the surface integral where $\sigma$ is the surface enclosed by y = x², 0 $\le$ x $\le$ 2, and -1 $\le$ z $\le$ 2.

A)
B)
C)
D)
E) 0

Question 65

Evaluate the surface integral where $\sigma$ is the surface enclosed by z = x³, 0 $\le$ x $\le$ 2, and 0 $\le$ y $\le$ $\pi$ .

A)
B)
C)
D) 0
E)

Question 66

Evaluate the surface integral over the sphere x² + y² + z² = 25.

A) 8 $\pi$
B) 800 $\pi$
C) $\pi$
D) 400 $\pi$

Question 67

Evaluate the surface integral where $\sigma$ is the portion of the cone for 0 $\le$ x $\le$ 3.

Question 68

Use Green's Theorem to evaluate , where C is x² + y² = 16.

A) 6 $\pi$
B) 9 $\pi$
C) 18 $\pi$
D) 0
E) 12 $\pi$

Question 69

Use Green's Theorem to evaluate , where C is the circle .

A) $\pi$
B) 7
C) 49 $\pi$
D) 7 $\pi$
E) 0

Question 70

Evaluate the surface integral where $\sigma$ is the portion of the cone for 0 $\le$ z $\le$ 3.

Question 71

Evaluate the surface integral where $\sigma$ is that portion of the plane x + 2y + 3z = 6 which lies in the first octant.

Question 72

Evaluate the surface integral where $\sigma$ is that portion of the cylinder x² + z² = 1 that lies above the xy-plane enclosed by 0 $\le$ y $\le$ 5.

Question 73

Use Green's Theorem to evaluate , where C is the square bounded by x = y = 0, and x = y = 1.

A) 1/2
B) -5
C) 2
D) 0
E) 3

Question 74

Use Green's Theorem to evaluate , where C is the square bounded by x = y = 0, and x = y = 1.

A) 1/2
B) 2
C) 6
D) 0
E) 4

Question 75

Use Green's Theorem to evaluate , where C is the square bounded by x = y = 0, and x = y = 1.

A) 1/2
B) 1
C) 2
D) 0
E) 3

Question 76

Evaluate the surface integral where $\sigma$ is that portion of 3x + 3y + 3z = 3 which lies in the first octant.

A)
B) 0
C)
D)
E)

Question 77

Evaluate the surface integral where $\sigma$ is that portion of the plane x + y + z = 1 that lies in the first octant.

Question 78

Use Green's Theorem to evaluate , where C is the circle .

A) 0
B) 7 $\pi$
C) 49
D) 49 $\pi$
E) $\pi$

Question 79

Evaluate the surface integral where $\sigma$ is that portion of the cylinder y² + z² = 1 that lies above the xy-plane enclosed by 0 $\le$ x $\le$ 5 and -1 $\le$ y $\le$ 1.

A)
B)
C)
D)
E) 0

Question 80

Evaluate the surface integral where $\sigma$ is that portion of the paraboloid z = x² + y² enclosed by 1 $\le$ z $\le$ 9.

A)
B)
C) 0
D)
E)