Question 1

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) -109 
B) 45 
C) -62 
D) -44 
A) -109 
B) 45 
C) -62 
D) -44

Question 2

Find the divergence of the vector field.

Accepted Answer

The answer of Find the divergence of the vector field....

Question 3

A particle starts at the point   , moves along the x-axis to (3, 0) and then along the semicircle   to the starting point. Use Green's Theorem to find the work done on this particle by the force field

Accepted Answer

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

Question 4

Evaluate   , that is, find the flux of F across S.   ; S is the hemisphere   ; n points upward.

Accepted Answer

A) 162   
B) 162 
C)      
D)      
A) 162   
B) 162 
C)      
D)

Question 5

Use Stokes' Theorem to evaluate   .   ;
S is the part of the ellipsoid   lying above the xy-plane and oriented with normal pointing upward.

Accepted Answer

The answer of Use Stokes' Theorem to evaluate  ...

Question 6

Evaluate   , that is, find the flux of F across S.   ; S is the part of the paraboloid   between the planes z = 0 and z = 5; n points upward.

Accepted Answer

The answer of Evaluate   , that is, find...

Question 7

Match the equation with one of the graphs below.

Accepted Answer

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

Question 8

Find the area of the surface S where S is the part of the surface   that lies inside the cylinder

Accepted Answer

The answer of Find the area of the surface S...

Question 9

Evaluate   for the vector field F and the path C. (Hint: Show that F is conservative, and pick a simpler path.)   C:

Accepted Answer

A) 54 
B) 0 
C)    
D) 19 
A) 54 
B) 0 
C)    
D) 19

Question 10

Find the mass of the surface S having the given mass density. S is the hemisphere   ,   ; the density at a point P on S is equal to the distance between P and the xy-plane.

Accepted Answer

A)    
B)    
C) 9 
D)    
A)    
B)    
C) 9 
D)

Question 11

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 12

Determine whether or not vector field is conservative. If it is conservative, find a function f such that

Accepted Answer

The answer of Determine whether or not vector field is...

Question 13

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C.   , where C is the boundary of the region bounded by the parabolas   and   .

Accepted Answer

A)    
B)    + e 
C)    + e 
D)    
A)    
B)    + e 
C)    + e 
D)

Question 14

Find the work done by the force field F on a particle that moves along the curve C.   ;   ,

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

Evaluate the line integral.

Accepted Answer

The answer of Evaluate the line integral.  ...

Question 17

Use Stokes' Theorem to evaluate   .   ;
C is the boundary of the triangle with vertices   ,   , and   oriented in a counterclockwise direction when viewed from above

Accepted Answer

The answer of Use Stokes' Theorem to evaluate  ...

Question 18

Find an equation in rectangular coordinates, and then identify the surface.

Accepted Answer

The answer of Find an equation in rectangular coordinates, and...

Question 19

Find the div F if   .

Accepted Answer

The answer of Find the div F if  ...

Question 20

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find   , where a is the constant vector.

Accepted Answer

A) 5 
B) 3 
C) 8 
D) 7 
E) 6 
A) 5 
B) 3 
C) 8 
D) 7 
E) 6

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

Find the divergence of the vector field.

A particle starts at the point , moves along the x-axis to (3, 0) and then along the semicircle to the starting point. Use Green's Theorem to find the work done on this particle by the force field

Evaluate , that is, find the flux of F across S. ; S is the hemisphere ; n points upward.

Use Stokes' Theorem to evaluate . ; S is the part of the ellipsoid lying above the xy-plane and oriented with normal pointing upward.

Evaluate , that is, find the flux of F across S. ; S is the part of the paraboloid between the planes z = 0 and z = 5; n points upward.

Match the equation with one of the graphs below.

Find the area of the surface S where S is the part of the surface that lies inside the cylinder

Evaluate for the vector field F and the path C. (Hint: Show that F is conservative, and pick a simpler path.) C:

Find the mass of the surface S having the given mass density. S is the hemisphere , ; the density at a point P on S is equal to the distance between P and the xy-plane.

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.

Determine whether or not vector field is conservative. If it is conservative, find a function f such that

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the boundary of the region bounded by the parabolas and .

Find the work done by the force field F on a particle that moves along the curve C. ; ,

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

Evaluate the line integral.

Use Stokes' Theorem to evaluate . ; C is the boundary of the triangle with vertices , , and oriented in a counterclockwise direction when viewed from above

Find an equation in rectangular coordinates, and then identify the surface.

Find the div F if .

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find , where a is the constant vector.

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

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

Find the divergence of the vector field.

A particle starts at the point , moves along the x-axis to (3, 0) and then along the semicircle to the starting point. Use Green's Theorem to find the work done on this particle by the force field

Evaluate , that is, find the flux of F across S. ; S is the hemisphere ; n points upward.

Use Stokes' Theorem to evaluate . ; S is the part of the ellipsoid lying above the xy-plane and oriented with normal pointing upward.

Evaluate , that is, find the flux of F across S. ; S is the part of the paraboloid between the planes z = 0 and z = 5; n points upward.

Match the equation with one of the graphs below.

Find the area of the surface S where S is the part of the surface that lies inside the cylinder

Evaluate for the vector field F and the path C. (Hint: Show that F is conservative, and pick a simpler path.) C:

Find the mass of the surface S having the given mass density. S is the hemisphere , ; the density at a point P on S is equal to the distance between P and the xy-plane.

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.

Determine whether or not vector field is conservative. If it is conservative, find a function f such that

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the boundary of the region bounded by the parabolas and .

Find the work done by the force field F on a particle that moves along the curve C. ; ,

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

Evaluate the line integral.

Use Stokes' Theorem to evaluate . ; C is the boundary of the triangle with vertices , , and oriented in a counterclockwise direction when viewed from above

Find an equation in rectangular coordinates, and then identify the surface.

Find the div F if .

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find , where a is the constant vector.

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