Question 1

Find the divergence of the field F.
-F = -3   i + 7   j + 6   k

Accepted Answer

A)  30 
B)  0 
C)  -9   + 21   + 18   
D)  -3   + 7   + 6   
A)  30 
B)  0 
C)  -9   + 21   + 18   
D)  -3   + 7   + 6

Question 2

Evaluate the line integral along the curve C.
-

Accepted Answer

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

Question 3

Evaluate the surface integral of G over the surface S.
-S is the parabolic cylinder y = 2   , 0 $\le$ x$\le$4 and 0 $\le$ z $\le$ 3; G(x, y, z) = 5x

Accepted Answer

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

Question 4

Calculate the area of the surface S.
-S is the lower portion of the sphere   +   +   = 16 cut by the cone z =   .

Accepted Answer

A)  16   
B)  8    $\pi$ 
C)  16    $\pi$ 
D)  32    $\pi$ 
A)  16   
B)  8    $\pi$ 
C)  16    $\pi$ 
D)  32    $\pi$

Question 5

Calculate the area of the surface S.
-S is the portion of the cylinder   +   = 36 that lies between z = 3 and z = 5.

Accepted Answer

A)  48 $\pi$ 
B)  12 $\pi$ 
C)  24 $\pi$ 
D)  96 $\pi$ 
A)  48 $\pi$ 
B)  12 $\pi$ 
C)  24 $\pi$ 
D)  96 $\pi$

Question 6

Find the flux of the curl of field F through the shell S.
-F = 5yi + 4xj + cos(z)k ; S: r(r,$\theta$) = 3 sin   cos $\theta$i + 3 sin   sin $\theta$j + 3 cos $\theta$k, 0 $\le$ $\theta$ $\le$ 2 $\pi$ and 0 $\le$   $\le$

Accepted Answer

A)  -18 $\pi$ 
B)  -9 
C)  -9 $\pi$ 
D)  18 $\pi$ 
A)  -18 $\pi$ 
B)  -9 
C)  -9 $\pi$ 
D)  18 $\pi$

Question 7

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-F = (y-x)i + (z-y)j + (z-x)k ; D: the region cut from the solid cylinder   +   $\le$ 49 by the planes z = 0 and

Accepted Answer

A)  0 
B)  -294 $\pi$ 
C)  294 $\pi$ 
D)  -294 
A)  0 
B)  -294 $\pi$ 
C)  294 $\pi$ 
D)  -294

Question 8

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-F =   i +   j + zk; D: the solid cube cut by the coordinate planes and the planes x = 3, y = 3, and z = 3

Accepted Answer

A)  81 
B)  189 
C)  164 
D)  108 
A)  81 
B)  189 
C)  164 
D)  108

Question 9

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 ;

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

Find the flux of the curl of field F through the shell S.
-F = (x-y)i + (x-z)j + (y-z)k; S is the portion of the cone z = 3   below the plane z = 5

Accepted Answer

A)  50/9  $\pi$ 
B)  - 100/9  $\pi$ 
C)  100/9 
D)  - 50/9  $\pi$ 
A)  50/9  $\pi$ 
B)  - 100/9  $\pi$ 
C)  100/9 
D)  - 50/9  $\pi$

Question 12

Find the flux of the vector field F across the surface S in the indicated direction.
-F = 5xi + 5yj + zk; S is portion of the plane x + y + z = 6 for which 0 $\le$ x $\le$ 1 and   direction is outward (away from origin)

Accepted Answer

A)  75 
B)  90 
C)  180 
D)  -30 
A)  75 
B)  90 
C)  180 
D)  -30

Question 13

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

Accepted Answer

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

Question 14

Evaluate the surface integral of the function g over the surface S.
-G(x, y, z) =  xS1U1P12S1S1P0 yS1U1P12S1S1P0 zS1U1P12S1S1P0  ; S is the surface of the rectangular prism formed from the coordinate planes and the planes x =  2,  y =  2, and z =  1

Accepted Answer

A)  32/9 
B)  16/9 
C)  128/3 
D)  128/9 
A)  32/9 
B)  16/9 
C)  128/3 
D)  128/9

Question 15

Find the flux of the vector field F across the surface S in the indicated direction.
-F(x, y, z) = 11xi + 11yj + 11zk , S is the surface of the sphere   +   +   = 1 in the first octant, direction away from the origin

Accepted Answer

A)  11/4 $\pi$ 
B)  0 
C)  11/3 $\pi$ 
D)  11/2 $\pi$ 
A)  11/4 $\pi$ 
B)  0 
C)  11/3 $\pi$ 
D)  11/2 $\pi$

Question 16

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction.
-F = 2yi + 7xj +   k; C: the counterclockwise path around the perimeter of the triangle in the x-y plane formed from the x-axis, y-axis , and the line y = 3 - 2x

Accepted Answer

A)  - 45/2 
B)  45/4 
C)  15/4 
D)  45/2 
A)  - 45/2 
B)  45/4 
C)  15/4 
D)  45/2

Question 17

Apply Green's Theorem to evaluate the integral.
-  
C:  Any simple closed curve in the plane for which Green's Theorem holds

Accepted Answer

A)  0 
B)  There is not enough information to evaluate the integral. 
C)  -2 
D)  2 
A)  0 
B)  There is not enough information to evaluate the integral. 
C)  -2 
D)  2

Question 18

Find the flux of the vector field F across the surface S in the indicated direction.
-  S is the portion of the parabolic cylinder z =  1 - yS1U1P12S1S1P0  for which  z ≥  0 and  2 ≤ x ≤  3; direction is outward (away from the x-y plane)

Accepted Answer

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

Question 19

Parametrize the surface S.
-S is the portion of the plane -8x + 8y + 2z = 4 that lies within the cylinder   .

Accepted Answer

Answers wi

Question 20

Find the gradient field F of the function f.        
-f(x, y, z) =     +

Accepted Answer

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

Find the divergence of the field F. -F = -3 i + 7 j + 6 k

Evaluate the line integral along the curve C. -

Evaluate the surface integral of G over the surface S. -S is the parabolic cylinder y = 2 $Evaluate the surface integral of G over the surface S. -S is the parabolic cylinder y = 2 , 0 \le x \le 4 and 0 \le z \le 3; G(x, y, z) = 5x$ , 0 $\le$ x $\le$ 4 and 0 $\le$ z $\le$ 3; G(x, y, z) = 5x

Calculate the area of the surface S. -S is the lower portion of the sphere + + = 16 cut by the cone z = .

Calculate the area of the surface S. -S is the portion of the cylinder + = 36 that lies between z = 3 and z = 5.

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = i + j + zk; D: the solid cube cut by the coordinate planes and the planes x = 3, y = 3, and z = 3

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 ;

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

Find the flux of the curl of field F through the shell S. -F = (x-y)i + (x-z)j + (y-z)k; S is the portion of the cone z = 3 below the plane z = 5

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

Evaluate the surface integral of the function g over the surface S. -G(x, y, z) = x² y² z² ; S is the surface of the rectangular prism formed from the coordinate planes and the planes x = 2, y = 2, and z = 1

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = 11xi + 11yj + 11zk , S is the surface of the sphere + + = 1 in the first octant, direction away from the origin

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. -F = 2yi + 7xj + k; C: the counterclockwise path around the perimeter of the triangle in the x-y plane formed from the x-axis, y-axis , and the line y = 3 - 2x

Apply Green's Theorem to evaluate the integral. - C: Any simple closed curve in the plane for which Green's Theorem holds

Find the flux of the vector field F across the surface S in the indicated direction. - S is the portion of the parabolic cylinder z = 1 - y² for which z ≥ 0 and 2 ≤ x ≤ 3; direction is outward (away from the x-y plane)

Parametrize the surface S. -S is the portion of the plane -8x + 8y + 2z = 4 that lies within the cylinder .

Find the gradient field F of the function f. -f(x, y, z) = +

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Exam 17: Vector Calculus