Question 1

Use Gauss' Law for electricity and the relationship   to find the total charge in the hemisphere   where   .

Accepted Answer

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

Question 2

Determine whether   is conservative, and if so, find a potential function   of the vector field.

Accepted Answer

A)    is conservative with   
B)    is conservative with   
C)    is conservative with   
D)    is not conservative 
A)    is conservative with   
B)    is conservative with   
C)    is conservative with   
D)    is not conservative D

Question 3

Determine if   is conservative, and if so, find a potential function   of the vector field.

Accepted Answer

A)    is conservative with   
B)    is conservative with   
C)    is conservative with   
D)    is not conservative 
A)    is conservative with   
B)    is conservative with   
C)    is conservative with   
D)    is not conservative B

Question 4

Find a parametric representation of the surface   .

Accepted Answer

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

Question 5

Evaluate   , where C is the portion of   in the plane   from   to   .

Accepted Answer

A)  -16 
B)  -32 
C)  2 
D)  -80 
A)  -16 
B)  -32 
C)  2 
D)  -80

Question 6

Use Green's Theorem to evaluate   where C is the rectangle from   to   to   to   .

Accepted Answer

A)  -1080 
B)  -135 
C)  -45 
D)  0 
A)  -1080 
B)  -135 
C)  -45 
D)  0

Question 7

Use Guass' Law for electricity and the relationship   to find the total charge in the hemisphere   for the electric field   .

Accepted Answer

Gauss' Law: @#IMG-DLM& @#IMG-DLM& Theref

Question 8

Express the line integral   where C is the portion of   from   to   in parametric form. Estimate the integral.

Accepted Answer

The answer of Express the line integral   where...

Question 9

Find the surface area of the cone   below the plane   .

Accepted Answer

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

Question 10

Determine if   is conservative, and if so, find a potential function   of the vector field.

Accepted Answer

A)    is conservative with   
B)    is conservative with   
C)    is conservative with   
D)    is not conservative 
A)    is conservative with   
B)    is conservative with   
C)    is conservative with   
D)    is not conservative

Question 11

Find the surface area of the portion of the paraboloid   inside the cylinder   .

Accepted Answer

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

Question 12

Use Green's Theorem to evaluate   where   and C is formed by   and   Assume the curve is oriented positively.

Accepted Answer

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

Question 13

Evaluate   , where C is the quarter circle   from   to   . [Hint:   ]

Accepted Answer

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

Question 14

Use the Divergence Theorem to compute   for   and Q bounded by   and the coordinate planes.

Accepted Answer

A)    
B)  0 
C)    
D)  4 
A)    
B)  0 
C)    
D)  4

Question 15

Compute   using the easiest method available. Q is bounded by   and   and

Accepted Answer

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

Question 16

Find the outward flux of F over   where Q is bounded by   and   , and   .

Accepted Answer

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

Question 17

Set up a double integral that equals   where S is the portion of the sphere   that lies inside the cylinder   and above the xy-plane.

Accepted Answer

The answer of Set up a double integral that equals...

Question 18

Find the gradient field corresponding to

Accepted Answer

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

Question 19

Find the divergence of   .

Accepted Answer

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

Question 20

Compute the work done by the force field   along the curve C, where C is the line segment from (2, 1) to (3, 5).

Accepted Answer

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

Use Gauss' Law for electricity and the relationship to find the total charge in the hemisphere where .

Determine whether is conservative, and if so, find a potential function of the vector field.

Determine if is conservative, and if so, find a potential function of the vector field.

Find a parametric representation of the surface .

Evaluate , where C is the portion of in the plane from to .

Use Green's Theorem to evaluate where C is the rectangle from to to to .

Use Guass' Law for electricity and the relationship to find the total charge in the hemisphere for the electric field .

Express the line integral where C is the portion of from to in parametric form. Estimate the integral.

Find the surface area of the cone below the plane .

Determine if is conservative, and if so, find a potential function of the vector field.

Find the surface area of the portion of the paraboloid inside the cylinder .

Use Green's Theorem to evaluate where and C is formed by and Assume the curve is oriented positively.

Evaluate , where C is the quarter circle from to . [Hint: ]

Use the Divergence Theorem to compute for and Q bounded by and the coordinate planes.

Compute using the easiest method available. Q is bounded by and and

Find the outward flux of F over where Q is bounded by and , and .

Set up a double integral that equals where S is the portion of the sphere that lies inside the cylinder and above the xy-plane.

Find the gradient field corresponding to

Find the divergence of .

Compute the work done by the force field along the curve C, where C is the line segment from (2, 1) to (3, 5).

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Second Order Differential Equations

Filters

Exam 16: Vector Calculus

Use Gauss' Law for electricity and the relationship to find the total charge in the hemisphere where .

Determine whether is conservative, and if so, find a potential function of the vector field.

Determine if is conservative, and if so, find a potential function of the vector field.

Find a parametric representation of the surface .

Evaluate , where C is the portion of in the plane from to .

Use Green's Theorem to evaluate where C is the rectangle from to to to .

Use Guass' Law for electricity and the relationship to find the total charge in the hemisphere for the electric field .

Express the line integral where C is the portion of from to in parametric form. Estimate the integral.

Find the surface area of the cone below the plane .

Determine if is conservative, and if so, find a potential function of the vector field.

Find the surface area of the portion of the paraboloid inside the cylinder .

Use Green's Theorem to evaluate where and C is formed by and Assume the curve is oriented positively.

Evaluate , where C is the quarter circle from to . [Hint: ]

Use the Divergence Theorem to compute for and Q bounded by and the coordinate planes.

Compute using the easiest method available. Q is bounded by and and

Find the outward flux of F over where Q is bounded by and , and .

Set up a double integral that equals where S is the portion of the sphere that lies inside the cylinder and above the xy-plane.

Find the gradient field corresponding to

Find the divergence of .

Compute the work done by the force field along the curve C, where C is the line segment from (2, 1) to (3, 5).

Preliminaries

Limits and Continuity

Differentiation

Applications of the Derivative

Integration

Applications of the Definite Integral

Exponentials, Logarithms and Other Transcendental Functions

Integration Techniques

First-Order Differential Equations

Infinite Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions

Functions of Several Variables and Partial Differentiation

Multiple Integrals

Second Order Differential Equations

Filters