Question 1

Sketch the vector field   . ​

Accepted Answer

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

Question 2

Use Stokes's Theorem to evaluate   where   and S is   . Use a computer algebra system to verify your result. ​

Accepted Answer

A)    
B)    
C)    
D)    
E)  0 
A)    
B)    
C)    
D)    
E)  0

Question 3

Find   .

Accepted Answer

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

Question 4

Use Green's Theorem to evaluate the integral   for the path   defined as   . ​

Accepted Answer

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

Question 5

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function   about the x-axis. ​

Accepted Answer

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

Question 6

Use Green's Theorem to evaluate the integral ​   for the path C: boundary of the region lying between the graphs of   and   .​

Accepted Answer

A)  0 
B)    ​ 
C)    ​ 
D)    ​ 
E)  13 
A)  0 
B)    ​ 
C)    ​ 
D)    ​ 
E)  13

Question 7

Use the Divergence Theorem to evaluate   and find the outward flux of   through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.

Accepted Answer

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

Question 8

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. ​   ​
C: a smooth curve from   to

Accepted Answer

A)  60 
B)  39 
C)  19 
D)  224 
E)  361 
A)  60 
B)  39 
C)  19 
D)  224 
E)  361

Question 9

Find the divergence of the vector field.

Accepted Answer

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

Question 10

Use Stokes's Theorem to evaluate   where   and S is the first-octant portion of   over   . Use a computer algebra system to verify your result. ​

Accepted Answer

A)    ​ 
B)  0​ 
C)    ​ 
D)    ​ 
E)    ​ 
A)    ​ 
B)  0​ 
C)    ​ 
D)    ​ 
E)    ​

Question 11

Evaluate   , where ​   .
​

Accepted Answer

A)    
B)    
C)  0 
D)    
E)    
A)    
B)    
C)  0 
D)    
E)

Question 12

Find a vector-valued function whose graph is the ellipsoid   . ​

Accepted Answer

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

Question 13

Find the area of the surface of revolution   , where   and   . ​

Accepted Answer

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

Question 14

Evaluate the line integral   using the Fundamental Theorem of Line Integrals, where C is the line segment from   to   . ​

Accepted Answer

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

Question 15

Find a vector-valued function for the hyperboloid   . ​

Accepted Answer

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

Question 16

Evaluate   , where S is   , first octant. ​

Accepted Answer

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

Question 17

Let   be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere   and its circular base in the xy-plane. ​

Accepted Answer

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

Question 18

Find the flux of   over the closed surface (let   be the outward unit normal vector of the surface). ​   ​
S: cube bounded by   .
​

Accepted Answer

A)  4,063 
B)  16,250 
C)  8,125 
D)  20,313 
E)  12,188 
A)  4,063 
B)  16,250 
C)  8,125 
D)  20,313 
E)  12,188

Question 19

Verify Green's Theorem by setting up and evaluating both integrals   for the path C: square with vertices (0,0), (5,0), (5,5), (0,5).

Accepted Answer

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

Question 20

Use a computer algebra system and the result "The area of a plane region bounded by the simple closed path   given in polar coordinates is   " to find the area of the region bounded by the graphs of the polar equation   . ​

Accepted Answer

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

Sketch the vector field .

Use Stokes's Theorem to evaluate where and S is . Use a computer algebra system to verify your result.

Find .

Use Green's Theorem to evaluate the integral for the path defined as .

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the x-axis.

Use Green's Theorem to evaluate the integral for the path C: boundary of the region lying between the graphs of and .

Use the Divergence Theorem to evaluate and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. C: a smooth curve from to

Find the divergence of the vector field.

Use Stokes's Theorem to evaluate where and S is the first-octant portion of over . Use a computer algebra system to verify your result.

Evaluate , where .

Find a vector-valued function whose graph is the ellipsoid .

Find the area of the surface of revolution , where and .

Evaluate the line integral using the Fundamental Theorem of Line Integrals, where C is the line segment from to .

Find a vector-valued function for the hyperboloid .

Evaluate , where S is , first octant.

Let be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere and its circular base in the xy-plane.

Find the flux of over the closed surface (let be the outward unit normal vector of the surface). S: cube bounded by .

Verify Green's Theorem by setting up and evaluating both integrals for the path C: square with vertices (0,0), (5,0), (5,5), (0,5).

Use a computer algebra system and the result "The area of a plane region bounded by the simple closed path given in polar coordinates is " to find the area of the region bounded by the graphs of the polar equation .

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Filters

Exam 15: Vector Anal

Sketch the vector field . ​

Use Stokes's Theorem to evaluate where and S is . Use a computer algebra system to verify your result. ​

Find .

Use Green's Theorem to evaluate the integral for the path defined as . ​

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the x-axis. ​

Use Green's Theorem to evaluate the integral ​ for the path C: boundary of the region lying between the graphs of and .​

Use the Divergence Theorem to evaluate and find the outward flux of through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. ​ ​ C: a smooth curve from to

Find the divergence of the vector field.

Use Stokes's Theorem to evaluate where and S is the first-octant portion of over . Use a computer algebra system to verify your result. ​

Evaluate , where ​ . ​

Find a vector-valued function whose graph is the ellipsoid . ​

Find the area of the surface of revolution , where and . ​

Evaluate the line integral using the Fundamental Theorem of Line Integrals, where C is the line segment from to . ​

Find a vector-valued function for the hyperboloid . ​

Evaluate , where S is , first octant. ​

Let be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere and its circular base in the xy-plane. ​

Find the flux of over the closed surface (let be the outward unit normal vector of the surface). ​ ​ S: cube bounded by . ​

Verify Green's Theorem by setting up and evaluating both integrals for the path C: square with vertices (0,0), (5,0), (5,5), (0,5).

Use a computer algebra system and the result "The area of a plane region bounded by the simple closed path given in polar coordinates is " to find the area of the region bounded by the graphs of the polar equation . ​

Preparation for Calculus

Limits and Their Properties

Differentiation

Applications of Differentiation

Integration

Differential Equations

Applications of Integration

Integration Techniques and Improper Integrals

Infinite Series

Conics, Parametric Equations, and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables

Multiple Integration

Filters

Sketch the vector field .

Use Stokes's Theorem to evaluate where and S is . Use a computer algebra system to verify your result.

Use Green's Theorem to evaluate the integral for the path defined as .

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the x-axis.

Use Green's Theorem to evaluate the integral for the path C: boundary of the region lying between the graphs of and .

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. C: a smooth curve from to

Use Stokes's Theorem to evaluate where and S is the first-octant portion of over . Use a computer algebra system to verify your result.

Evaluate , where .

Find a vector-valued function whose graph is the ellipsoid .

Find the area of the surface of revolution , where and .

Evaluate the line integral using the Fundamental Theorem of Line Integrals, where C is the line segment from to .

Find a vector-valued function for the hyperboloid .

Evaluate , where S is , first octant.

Let be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere and its circular base in the xy-plane.

Find the flux of over the closed surface (let be the outward unit normal vector of the surface). S: cube bounded by .

Use a computer algebra system and the result "The area of a plane region bounded by the simple closed path given in polar coordinates is " to find the area of the region bounded by the graphs of the polar equation .