Question 1

Find the flux of F = xi + yj + zk upward through the hemisphere x2 + y2 + z2 = 1, z $\ge$ 0.

Accepted Answer

A)  2 $\pi$ 
B)     $\pi$ 
C)     $\pi$ 
D)  4 $\pi$ 
E)  0 
A)  2 $\pi$ 
B)     $\pi$ 
C)     $\pi$ 
D)  4 $\pi$ 
E)  0 A

Question 2

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 , -   ).

Accepted Answer

A)  ±   
B)  ± (- 1 , 0 ,0) 
C)  ±   
D)  ±   
E)  ±   
A)  ±   
B)  ± (- 1 , 0 ,0) 
C)  ±   
D)  ±   
E)  ±   C

Question 3

Find the flux of the field F = (x + y) i + (y + z) j + (x + z)k outward across the surface of the cube

Accepted Answer

A)  12 
B)  24 
C)  48 
D)  6 
E)  0 
A)  12 
B)  24 
C)  48 
D)  6 
E)  0 B

Question 4

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.

Accepted Answer

A) True 
 B)False

Question 5

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).

Accepted Answer

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

Question 6

Find   , where r = xi + yj + zk, over the entire surface of the cone with vertex at   and base given by   in the plane z = 5.

Accepted Answer

A)  6 $\pi$ 
B)  48 $\pi$ 
C)  12 $\pi$ 
D)  5 $\pi$ 
E)  45 $\pi$ 
A)  6 $\pi$ 
B)  48 $\pi$ 
C)  12 $\pi$ 
D)  5 $\pi$ 
E)  45 $\pi$

Question 7

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

Accepted Answer

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

Question 8

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   .

Accepted Answer

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

Question 9

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.

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

Let V(x,y) @#IMG-DLM&  hence ∇ V = x i +

Question 12

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

Accepted Answer

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

Question 13

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) =       .

Accepted Answer

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$\theta$) 
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 14

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.

Accepted Answer

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

Question 15

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

Accepted Answer

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 
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

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.

Accepted Answer

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

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

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).

Accepted Answer

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 
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

Find the flux of F = xi + yj + zk upward through the hemisphere x² + y²+ z² = 1, z $\ge$ 0.

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 , - ).

Find the flux of the field F = (x + y) i + (y + z) j + (x + z)k outward across the surface of the cube

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.

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).

Find , where r = xi + yj + zk, over the entire surface of the cone with vertex at and base given by in the plane z = 5.

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

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 .

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.

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

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

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

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) = .

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.

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

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.

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

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

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

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).

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Exam 16: Vector Fields

Find the flux of F = xi + yj + zk upward through the hemisphere x2 + y2 + z2 = 1, z ≥\ge≥ 0.

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 , - ).

Find the flux of the field F = (x + y) i + (y + z) j + (x + z)k outward across the surface of the cube

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.

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).

Find , where r = xi + yj + zk, over the entire surface of the cone with vertex at and base given by in the plane z = 5.

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

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 .

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.

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

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

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

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) = .

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.

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

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.

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

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

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

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).

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

Find the flux of F = xi + yj + zk upward through the hemisphere x² + y²+ z² = 1, z $\ge$ 0.

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 .

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.

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