Question 1

Parametrize the surface S.
-S is the lower portion of the sphere   +   +   = 25 cut by the cone z =   .

Accepted Answer

Answers will vary. One possibility is r =  5 cos φ sin θi + 5 sin φ sin θj +  5 cos θk

Question 2

Parametrize the surface S.
-S is the portion of the cylinder   +   = 16 that lies between z = 2 and z = 7.

Accepted Answer

Answers will vary. One possibility is r =  4 cos θi + 4 sin θj + zk ,  2 ≤ z ≤  7, 0 ≤ θ ≤ 2π

Question 3

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction.
-F = 2yi + 3xj + 6   k ; C: the portion of the plane 3x + 3y + 5z = 6 in the first quadrant

Accepted Answer

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

Question 4

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)

Accepted Answer

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

Question 5

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)

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Find the mass of the wire that lies along the curve r and has density δ.
-r(t) = 7i + ( 9 - 4t)j + 3tk, 0 $\le$ t $\le$ 2 $\pi$ ;  = 5(1 + sin 7t)

Accepted Answer

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

Question 8

Find the flux of the vector field F across the surface S in the indicated direction.
-F(x, y, z) = 2xi + 2yj + 2k , S is the surface cut from the bottom of the paraboloid z =   +   by the plane   direction is outward

Accepted Answer

A)  180$\pi$ 
B)  12$\pi$ 
C)  -288 $\pi$ 
D)  144$\pi$ 
A)  180$\pi$ 
B)  12$\pi$ 
C)  -288 $\pi$ 
D)  144$\pi$

Question 9

Find the surface area of the surface S.
-S is the intersection of the plane 3x + 4y + 12z = 7 and the cylinder with sides y = 4   and y = 8 - 4   .

Accepted Answer

A)  104/3 
B)  13/18 
C)  104/9 
D)  13/9 
A)  104/3 
B)  13/18 
C)  104/9 
D)  13/9

Question 10

Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle  xS1U1P12S1S1P0 + yS1U1P12S1S1P0  = 4. 
-F =   i +   j

Accepted Answer

The answer of Sketch the vector field in the plane...

Question 11

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$\pi$

Accepted Answer

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

Question 12

Find the potential function f for the field F.   
-

Accepted Answer

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

Question 13

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-F = 6   i + 6   j + 6   k ; D: the thick sphere 4 $\le$  +   +   $\le$16

Accepted Answer

A)    $\pi$ 
B)  71,424$\pi$ 
C)  1008$\pi$ 
D)  1920$\pi$ 
A)    $\pi$ 
B)  71,424$\pi$ 
C)  1008$\pi$ 
D)  1920$\pi$

Question 14

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

Accepted Answer

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

Question 15

Parametrize the surface S.
-S is the portion of the cone   +   =   that lies between z = 1 and z = 9.

Accepted Answer

Answers will vary. O

Question 16

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

Accepted Answer

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

Question 17

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)

Accepted Answer

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

Question 18

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 planes x = ±  2,  y = ±  2, and   z = ± 1

Accepted Answer

A)  256/9 
B)  16/9 
C)  128/3 
D)  128/9 
A)  256/9 
B)  16/9 
C)  128/3 
D)  128/9

Question 19

Find the potential function f for the field F.   
-

Accepted Answer

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

Question 20

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D.
-F = x   i + y   j + z   k ; D: the thick cylinder 1 $\le$   +   $\le$ 3 ,

Accepted Answer

A)  104/3 $\pi$ 
B)  416/3$\pi$ 
C)  208/3$\pi$ 
D)  208$\pi$ 
A)  104/3 $\pi$ 
B)  416/3$\pi$ 
C)  208/3$\pi$ 
D)  208$\pi$

Parametrize the surface S. -S is the lower portion of the sphere + + = 25 cut by the cone z = .

Parametrize the surface S. -S is the portion of the cylinder + = 16 that lies between z = 2 and z = 7.

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. -F = 2yi + 3xj + 6 k ; C: the portion of the plane 3x + 3y + 5z = 6 in the first quadrant

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)

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)

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

Find the mass of the wire that lies along the curve r and has density δ. -r(t) = 7i + ( 9 - 4t)j + 3tk, 0 $\le$ t $\le$ 2 $\pi$ ; $Find the mass of the wire that lies along the curve r and has density δ. -r(t) = 7i + ( 9 - 4t)j + 3tk, 0 \le t \le 2 \pi ; = 5(1 + sin 7t)$ = 5(1 + sin 7t)

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = 2xi + 2yj + 2k , S is the surface cut from the bottom of the paraboloid z = + by the plane direction is outward

Find the surface area of the surface S. -S is the intersection of the plane 3x + 4y + 12z = 7 and the cylinder with sides y = 4 and y = 8 - 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 = i + j

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 $\pi$

Find the potential function f for the field F. -

Parametrize the surface S. -S is the portion of the cone + = that lies between z = 1 and z = 9.

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

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)

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 planes x = ± 2, y = ± 2, and z = ± 1

Find the potential function f for the field F. -

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Exam 17: Vector Calculus

Parametrize the surface S. -S is the lower portion of the sphere + + = 25 cut by the cone z = .

Parametrize the surface S. -S is the portion of the cylinder + = 16 that lies between z = 2 and z = 7.

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. -F = 2yi + 3xj + 6 k ; C: the portion of the plane 3x + 3y + 5z = 6 in the first quadrant

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)

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)

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

Find the mass of the wire that lies along the curve r and has density δ. -r(t) = 7i + ( 9 - 4t)j + 3tk, 0 ≤\le≤ t ≤\le≤ 2 π\piπ ; = 5(1 + sin 7t)

Find the flux of the vector field F across the surface S in the indicated direction. -F(x, y, z) = 2xi + 2yj + 2k , S is the surface cut from the bottom of the paraboloid z = + by the plane direction is outward

Find the surface area of the surface S. -S is the intersection of the plane 3x + 4y + 12z = 7 and the cylinder with sides y = 4 and y = 8 - 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 = i + j

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 π\piπ

Find the potential function f for the field F. -

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = 6 i + 6 j + 6 k ; D: the thick sphere 4 ≤\le≤ + + ≤\le≤ 16

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

Parametrize the surface S. -S is the portion of the cone + = that lies between z = 1 and z = 9.

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

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)

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 planes x = ± 2, y = ± 2, and z = ± 1

Find the potential function f for the field F. -

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = x i + y j + z k ; D: the thick cylinder 1 ≤\le≤ + ≤\le≤ 3 ,

Functions

Limits

Derivatives

Applications of the Derivative

Integration

Applications of Integration

Logarithmic, Exponential, and Hyperbolic Functions

Integration Techniques

Differential Equations

Sequences and Infinite Series

Power Series

Parametric and Polar Curves

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Filters

Find the mass of the wire that lies along the curve r and has density δ. -r(t) = 7i + ( 9 - 4t)j + 3tk, 0 $\le$ t $\le$ 2 $\pi$ ; $Find the mass of the wire that lies along the curve r and has density δ. -r(t) = 7i + ( 9 - 4t)j + 3tk, 0 \le t \le 2 \pi ; = 5(1 + sin 7t)$ = 5(1 + sin 7t)

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

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 $\pi$

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 planes x = ± 2, y = ± 2, and z = ± 1