Question 1

Compute   for the vector field given by   .

Accepted Answer

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

Question 2

Verify Green's Theorem by evaluating both integrals   for the path   defined as the boundary of the region lying between the graphs of   and   . ​

Accepted Answer

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

Question 3

Let   and let S be the cylinder   ,   Verify the Divergence Theorem by evaluating   as a surface integral and as a triple integral. ​   ​

Accepted Answer

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

Question 4

Use Divergence Theorem to evaluate   and find the outward flux of   through the surface S of the solid bounded by the planes   and   . ​

Accepted Answer

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

Question 5

Evaluate   where   and   . ​

Accepted Answer

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

Question 6

Use a computer algebra system to evaluate   where S is     . Round your answer to two decimal places. ​

Accepted Answer

A)  -1,115.02 
B)  -1,168.00 
C)  -491.00 
D)  -2,283.38 
E)  -99.93 
A)  -1,115.02 
B)  -1,168.00 
C)  -491.00 
D)  -2,283.38 
E)  -99.93

Question 7

Let   and let S be the graph of   . Verify Stokes's Theorem by evaluating   as a line integral and as a double integral.
​

Accepted Answer

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

Question 8

Evaluate   , where   is the unit circle given by   . ​

Accepted Answer

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

Question 9

Use Green's Theorem to calculate the work done by the force   on a particle that is moving counterclockwise around the closed path   .

Accepted Answer

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

Question 10

Find the value of the line integral   on the closed path consisting of line segments from   to   , from   to   , and then from   to   , where   .
(Hint: If F is conservative, the integration may be easier on an alternate path.)

Accepted Answer

A)  4,224 
B)  72 
C)  576 
D)  520 
E)  0 
A)  4,224 
B)  72 
C)  576 
D)  520 
E)  0

Question 11

Evaluate   , where   ​   . ​

Accepted Answer

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

Question 12

Find the flux   of through S,   , where   is the upward unit normal vector to S. ​   ​   ​

Accepted Answer

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

Question 13

Find the work done by the force field   on a particle moving along the given path. ​   ,   from   to   .
​

Accepted Answer

A)  0 
B)  4 
C)    
D)  2 
E)    
A)  0 
B)  4 
C)    
D)  2 
E)

Question 14

Let   and let S be the surface bounded by   and   . Verify the Divergence Theorem by evaluating   as a surface integral and as a triple integral. Round your answer to two decimal places. ​

Accepted Answer

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

Question 15

Let   and let S be the plane   in the first octant. Verify Stokes's Theorem by evaluating   as a line integral and as a double integral. ​

Accepted Answer

A)  24 
B)  8 
C)  16 
D)  0 
E)  1 
A)  24 
B)  8 
C)  16 
D)  0 
E)  1

Question 16

Evaluate the integral   along the path C, defined as   from   to   .

Accepted Answer

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

Question 17

Evaluate   along the path C, defined as y-axis from   to   .

Accepted Answer

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

Question 18

Let   and let S be the surface bounded by the planes   and   and the coordinate planes. Verify the Divergence Theorem by evaluating   as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable. ​   ​

Accepted Answer

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

Question 19

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field   . Find   , where S is the upper surface of the cylindrical container. ​

Accepted Answer

A)  -3 
B)  -7 
C)  1 
D)  0 
E)  3 
A)  -3 
B)  -7 
C)  1 
D)  0 
E)  3

Question 20

Find   .

Accepted Answer

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

Compute for the vector field given by .

Verify Green's Theorem by evaluating both integrals for the path defined as the boundary of the region lying between the graphs of and .

Let and let S be the cylinder , Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral.

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the planes and .

Evaluate where and .

Use a computer algebra system to evaluate where S is . Round your answer to two decimal places.

Let and let S be the graph of . Verify Stokes's Theorem by evaluating as a line integral and as a double integral.

Evaluate , where is the unit circle given by .

Use Green's Theorem to calculate the work done by the force on a particle that is moving counterclockwise around the closed path .

Find the value of the line integral on the closed path consisting of line segments from to , from to , and then from to , where . (Hint: If F is conservative, the integration may be easier on an alternate path.)

Evaluate , where .

Find the flux of through S, , where is the upward unit normal vector to S.

Find the work done by the force field on a particle moving along the given path. , from to .

Let and let S be the surface bounded by and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places.

Let and let S be the plane in the first octant. Verify Stokes's Theorem by evaluating as a line integral and as a double integral.

Evaluate the integral along the path C, defined as from to .

Evaluate along the path C, defined as y-axis from to .

Let and let S be the surface bounded by the planes and and the coordinate planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable.

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field . Find , where S is the upper surface of the cylindrical container.

Find .

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

Compute for the vector field given by .

Verify Green's Theorem by evaluating both integrals for the path defined as the boundary of the region lying between the graphs of and . ​

Let and let S be the cylinder , Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. ​ ​

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the planes and . ​

Evaluate where and . ​

Use a computer algebra system to evaluate where S is . Round your answer to two decimal places. ​

Let and let S be the graph of . Verify Stokes's Theorem by evaluating as a line integral and as a double integral. ​

Evaluate , where is the unit circle given by . ​

Use Green's Theorem to calculate the work done by the force on a particle that is moving counterclockwise around the closed path .

Find the value of the line integral on the closed path consisting of line segments from to , from to , and then from to , where . (Hint: If F is conservative, the integration may be easier on an alternate path.)

Evaluate , where ​ . ​

Find the flux of through S, , where is the upward unit normal vector to S. ​ ​ ​

Find the work done by the force field on a particle moving along the given path. ​ , from to . ​

Let and let S be the surface bounded by and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places. ​

Let and let S be the plane in the first octant. Verify Stokes's Theorem by evaluating as a line integral and as a double integral. ​

Evaluate the integral along the path C, defined as from to .

Evaluate along the path C, defined as y-axis from to .

Let and let S be the surface bounded by the planes and and the coordinate planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable. ​ ​

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field . Find , where S is the upper surface of the cylindrical container. ​

Find .

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

Verify Green's Theorem by evaluating both integrals for the path defined as the boundary of the region lying between the graphs of and .

Let and let S be the cylinder , Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral.

Use Divergence Theorem to evaluate and find the outward flux of through the surface S of the solid bounded by the planes and .

Evaluate where and .

Use a computer algebra system to evaluate where S is . Round your answer to two decimal places.

Let and let S be the graph of . Verify Stokes's Theorem by evaluating as a line integral and as a double integral.

Evaluate , where is the unit circle given by .

Evaluate , where .

Find the flux of through S, , where is the upward unit normal vector to S.

Find the work done by the force field on a particle moving along the given path. , from to .

Let and let S be the surface bounded by and . Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places.

Let and let S be the plane in the first octant. Verify Stokes's Theorem by evaluating as a line integral and as a double integral.

Let and let S be the surface bounded by the planes and and the coordinate planes. Verify the Divergence Theorem by evaluating as a surface integral and as a triple integral. Round your answer to two decimal places wherever applicable.

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field . Find , where S is the upper surface of the cylindrical container.