Question 1

Let   where   . Which of the following equations does the line segment from   to   satisy?

Accepted Answer

A)    
B)    
C) none of these 
A)    
B)    
C) none of these

Question 2

Evaluate the surface integral where S is the surface with parametric equations   ,   .

Accepted Answer

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

Question 3

Let

Accepted Answer

The answer of Let    ...

Question 4

Use Stokes' Theorem to evaluate   .   ; S is the part of the paraboloid   lying below the plane   and oriented with normal pointing downward.

Accepted Answer

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

Question 5

Find (a) the divergence and (b) the curl of the vector field F.

Accepted Answer

The answer of Find (a) the divergence and (b) the...

Question 6

Find the mass of the surface S having the given mass density. S is part of the plane   in the first octant; the density at a point P on S is equal to the square of the distance between P and the xy-plane.

Accepted Answer

A)    
B) 49 
C)    
D) 20 
A)    
B) 49 
C)    
D) 20

Question 7

Find the divergence of the vector field F.

Accepted Answer

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

Question 8

Find the area of the part of the cone   that is cut off by the cylinder

Accepted Answer

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

Question 9

Use Stokes' Theorem to evaluate   .   ; C is the curve obtained by intersecting the cylinder   with the hyperbolic paraboloid   , oriented in a counterclockwise direction when viewed from above

Accepted Answer

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

Question 10

Set up, but do not evaluate, a double integral for the area of the surface with parametric equations

Accepted Answer

The answer of Set up, but do not evaluate, a...

Question 11

Find an equation of the tangent plane to the parametric surface represented by r at the specified point.   ; u = ln 9, v = 0

Accepted Answer

The answer of Find an equation of the tangent plane...

Question 12

Consider the vector field   If a particle starts at the point   in the velocity field given by F, find an equation of the path it follows.

Accepted Answer

The answer of Consider the vector field   If...

Question 13

Use Green's Theorem to find the work done by the force   in moving a particle from the origin along the x-axis to (1, 0) then along the line segment to
(0, 1) and then back to the origin along the y-axis.

Accepted Answer

The answer of Use Green's Theorem to find the work...

Question 14

The plot of a vector field is shown below. A particle is moved         . By inspection, determine whether the work done by F on the particle is positive, negative, or zero.

Accepted Answer

The answer of The plot of a vector field is...

Question 15

Which plot illustrates the vector field

Accepted Answer

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

Question 16

Find the area of the surface. The part of the plane   ;   ,

Accepted Answer

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

Question 17

Evaluate the surface integral.   S is the part of the plane   that lies in the first octant.

Accepted Answer

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

Question 18

Below is given the plot of a vector field F in the xy-plane. (The z-component of F is 0.) By studying the plot, determine whether div F is positive, negative, or zero.

Accepted Answer

A) cannot be determined 
B) positive 
C) negative 
D) zero 
A) cannot be determined 
B) positive 
C) negative 
D) zero

Question 19

Show that F is conservative, and find a function f such that   , and use the result to evaluate   , where C is any curve from   to   .   ;   and

Accepted Answer

A) -24 
B) -113 
C) -178 
D) -131 
A) -24 
B) -113 
C) -178 
D) -131

Question 20

Match the vector field with its plot.

Accepted Answer

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

Let where . Which of the following equations does the line segment from to satisy?

Evaluate the surface integral where S is the surface with parametric equations , .

Let

Use Stokes' Theorem to evaluate . ; S is the part of the paraboloid lying below the plane and oriented with normal pointing downward.

Find (a) the divergence and (b) the curl of the vector field F.

Find the mass of the surface S having the given mass density. S is part of the plane in the first octant; the density at a point P on S is equal to the square of the distance between P and the xy-plane.

Find the divergence of the vector field F.

Find the area of the part of the cone that is cut off by the cylinder

Use Stokes' Theorem to evaluate . ; C is the curve obtained by intersecting the cylinder with the hyperbolic paraboloid , oriented in a counterclockwise direction when viewed from above

Set up, but do not evaluate, a double integral for the area of the surface with parametric equations

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ; u = ln 9, v = 0

Consider the vector field If a particle starts at the point in the velocity field given by F, find an equation of the path it follows.

Use Green's Theorem to find the work done by the force in moving a particle from the origin along the x-axis to (1, 0) then along the line segment to (0, 1) and then back to the origin along the y-axis.

The plot of a vector field is shown below. A particle is moved . By inspection, determine whether the work done by F on the particle is positive, negative, or zero.

Which plot illustrates the vector field

Find the area of the surface. The part of the plane ; ,

Evaluate the surface integral. S is the part of the plane that lies in the first octant.

Below is given the plot of a vector field F in the xy-plane. (The z-component of F is 0.) By studying the plot, determine whether div F is positive, negative, or zero.

Show that F is conservative, and find a function f such that , and use the result to evaluate , where C is any curve from to . ; and

Match the vector field with its plot.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Second-Order Differential Equations

Final Exam

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Exam 16: Vector Calculus

Let where . Which of the following equations does the line segment from to satisy?

Evaluate the surface integral where S is the surface with parametric equations , .

Let

Use Stokes' Theorem to evaluate . ; S is the part of the paraboloid lying below the plane and oriented with normal pointing downward.

Find (a) the divergence and (b) the curl of the vector field F.

Find the mass of the surface S having the given mass density. S is part of the plane in the first octant; the density at a point P on S is equal to the square of the distance between P and the xy-plane.

Find the divergence of the vector field F.

Find the area of the part of the cone that is cut off by the cylinder

Use Stokes' Theorem to evaluate . ; C is the curve obtained by intersecting the cylinder with the hyperbolic paraboloid , oriented in a counterclockwise direction when viewed from above

Set up, but do not evaluate, a double integral for the area of the surface with parametric equations

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ; u = ln 9, v = 0

Consider the vector field If a particle starts at the point in the velocity field given by F, find an equation of the path it follows.

Use Green's Theorem to find the work done by the force in moving a particle from the origin along the x-axis to (1, 0) then along the line segment to (0, 1) and then back to the origin along the y-axis.

The plot of a vector field is shown below. A particle is moved . By inspection, determine whether the work done by F on the particle is positive, negative, or zero.

Which plot illustrates the vector field

Find the area of the surface. The part of the plane ; ,

Evaluate the surface integral. S is the part of the plane that lies in the first octant.

Below is given the plot of a vector field F in the xy-plane. (The z-component of F is 0.) By studying the plot, determine whether div F is positive, negative, or zero.

Show that F is conservative, and find a function f such that , and use the result to evaluate , where C is any curve from to . ; and

Match the vector field with its plot.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Second-Order Differential Equations

Final Exam

Filters