Question 1

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

Accepted Answer

A)  4 
B)  2 
C)  10 
D)    
E)  3 
A)  4 
B)  2 
C)  10 
D)    
E)  3 D

Question 2

Let S be the surface   oriented upwards. Use a computer algebra system to find the rate of mass flow of a fluid of density   through   if the velocity field is given by   . ​

Accepted Answer

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

Question 3

Evaluate   , where   and S is given by   . ​

Accepted Answer

A)  8 
B)  40 
C)  16 
D)  120 
E)  640 
A)  8 
B)  40 
C)  16 
D)  120 
E)  640 E

Question 4

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

Accepted Answer

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

Question 5

Find a vector-valued function whose graph is the part of the paraboloid   that lies inside the cylinder   . ​

Accepted Answer

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

Question 6

Find the divergence at   for the vector field   .

Accepted Answer

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

Question 7

Use Green's Theorem to evaluate the line integral   where   is the boundary of the region lying between the graphs of the circle   and the ellipse   . ​

Accepted Answer

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

Question 8

Find the curl of the vector field   ​

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

Evaluate   where   is represented by   .     ,   ​

Accepted Answer

A)  49 
B)  0 
C)  7 
D)  36 
E)  12 
A)  49 
B)  0 
C)  7 
D)  36 
E)  12

Question 11

Find the area of the surface given by   , where   and   . ​

Accepted Answer

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

Question 12

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

Accepted Answer

A)  4,798.52 
B)  10.80 
C)  2,399.26 
D)  20,349.51 
E)  3,280.50 
A)  4,798.52 
B)  10.80 
C)  2,399.26 
D)  20,349.51 
E)  3,280.50

Question 13

Sketch several representative vectors in the vector field given by   . ​

Accepted Answer

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

Question 14

Find the work done by the force field   in moving an object from P to Q. ​   ​

Accepted Answer

A)  4,664 
B)  9,329 
C)  13,994 
D)  6,997 
E)  11,662 
A)  4,664 
B)  9,329 
C)  13,994 
D)  6,997 
E)  11,662

Question 15

Determine whether the vector field is conservative. If it is, find a potential function for the vector field.

Accepted Answer

A)  conservative with potential function   
B)  conservative with potential function   
C)  conservative with potential function   
D)  conservative with potential function   
E)  not conservative 
A)  conservative with potential function   
B)  conservative with potential function   
C)  conservative with potential function   
D)  conservative with potential function   
E)  not conservative

Question 16

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   . Round your answer to two decimal places. ​

Accepted Answer

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

Question 17

Find the conservative vector field for the potential function   by finding its gradient.

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

Evaluate   , where   and S is given by   . ​

Accepted Answer

A)  1,296 
B)    
C)    
D)    
E)    
A)  1,296 
B)    
C)    
D)    
E)

Question 20

Find the moments of inertia for a wire that lies along   ,   with density   . ​

Accepted Answer

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

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

Let S be the surface oriented upwards. Use a computer algebra system to find the rate of mass flow of a fluid of density through if the velocity field is given by .

Evaluate , where and S is given by .

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

Find a vector-valued function whose graph is the part of the paraboloid that lies inside the cylinder .

Find the divergence at for the vector field .

Use Green's Theorem to evaluate the line integral where is the boundary of the region lying between the graphs of the circle and the ellipse .

Find the curl of the vector field

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

Evaluate where is represented by . ,

Find the area of the surface given by , where and .

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

Sketch several representative vectors in the vector field given by .

Find the work done by the force field in moving an object from P to Q.

Determine whether the vector field is conservative. If it is, find a potential function for the vector field.

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 . Round your answer to two decimal places.

Find the conservative vector field for the potential function by finding its gradient.

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

Evaluate , where and S is given by .

Find the moments of inertia for a wire that lies along , with density .

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

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

Let S be the surface oriented upwards. Use a computer algebra system to find the rate of mass flow of a fluid of density through if the velocity field is given by . ​

Evaluate , where and S is given by . ​

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

Find a vector-valued function whose graph is the part of the paraboloid that lies inside the cylinder . ​

Find the divergence at for the vector field .

Use Green's Theorem to evaluate the line integral where is the boundary of the region lying between the graphs of the circle and the ellipse . ​

Find the curl of the vector field ​

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

Evaluate where is represented by . , ​

Find the area of the surface given by , where and . ​

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

Sketch several representative vectors in the vector field given by . ​

Find the work done by the force field in moving an object from P to Q. ​ ​

Determine whether the vector field is conservative. If it is, find a potential function for the vector field.

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 . Round your answer to two decimal places. ​

Find the conservative vector field for the potential function by finding its gradient.

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

Evaluate , where and S is given by . ​

Find the moments of inertia for a wire that lies along , with density . ​

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

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

Let S be the surface oriented upwards. Use a computer algebra system to find the rate of mass flow of a fluid of density through if the velocity field is given by .

Evaluate , where and S is given by .

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

Find a vector-valued function whose graph is the part of the paraboloid that lies inside the cylinder .

Use Green's Theorem to evaluate the line integral where is the boundary of the region lying between the graphs of the circle and the ellipse .

Find the curl of the vector field

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

Evaluate where is represented by . ,

Find the area of the surface given by , where and .

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

Sketch several representative vectors in the vector field given by .

Find the work done by the force field in moving an object from P to Q.

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 . Round your answer to two decimal places.

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

Evaluate , where and S is given by .

Find the moments of inertia for a wire that lies along , with density .