Question 1

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.
curl (div F)

Accepted Answer

div F is a scalar fi

Question 2

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

Accepted Answer

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

Question 3

Find the work done by the force field   in moving an object along an arch of the cycloid

Accepted Answer

The answer of Find the work done by the force...

Question 4

Find an equation of the tangent plane to the parametric surface represented by r at the specified point.   ;

Accepted Answer

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

Question 5

Use Green's Theorem and/or a computer algebra system to evaluate   where C is the circle   with counterclockwise orientation.

Accepted Answer

A)    
B)    
C)    
D)    
E) None of these 
A)    
B)    
C)    
D)    
E) None of these

Question 6

Find the work done by the force field   on a particle that moves along the parabola

Accepted Answer

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

Question 7

Find the curl of the vector field.

Accepted Answer

A)    
B)    
C)    
D)    
E) None of these 
A)    
B)    
C)    
D)    
E) None of these

Question 8

Find the gradient vector field of

Accepted Answer

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

Question 9

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

Accepted Answer

The answer of Determine whether or not F is a...

Question 10

Use the Divergence Theorem to calculate the surface integral   ; that is, calculate the flux of   across   .   S is the surface of the box bounded by the coordinate planes and the planes   .

Accepted Answer

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

Question 11

Let f be a scalar field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.

Accepted Answer

The answer of Let f be a scalar field. Determine...

Question 12

Evaluate   where C is the right half of the circle

Accepted Answer

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

Question 13

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 14

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C.   , where C is the triangle with vertices   ,   , and   .

Accepted Answer

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

Question 15

Use Stoke's theorem to evaluate     C is the curve of intersection of the hyperbolic paraboloid   and the cylinder   oriented counterclockwise as viewed from above.

Accepted Answer

The answer of Use Stoke's theorem to evaluate  ...

Question 16

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

Accepted Answer

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

Question 17

Evaluate the line integral over the given curve C.   ;   ,

Accepted Answer

A) 1 
B)      
C) 3 
D)    
A) 1 
B)      
C) 3 
D)

Question 18

Evaluate the surface integral   for the given vector field F and the oriented surface S. In other words, find the flux of F across S.   in the first octant,
with orientation toward the origin.

Accepted Answer

The answer of Evaluate the surface integral   for...

Question 19

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C.   , where C is the cardioid   .

Accepted Answer

A)      
B)      
C) 270   
D)      
A)      
B)      
C) 270   
D)

Question 20

Find the area of the part of the surface   that lies between the planes x = 0, x = 4,   , and z = 1.

Accepted Answer

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

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field. curl (div F)

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

Find the work done by the force field in moving an object along an arch of the cycloid

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ;

Use Green's Theorem and/or a computer algebra system to evaluate where C is the circle with counterclockwise orientation.

Find the work done by the force field on a particle that moves along the parabola

Find the curl of the vector field.

Find the gradient vector field of

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

Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of across . S is the surface of the box bounded by the coordinate planes and the planes .

Let f be a scalar field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.

Evaluate where C is the right half of the circle

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 triangle with vertices , , and .

Use Stoke's theorem to evaluate C is the curve of intersection of the hyperbolic paraboloid and the cylinder oriented counterclockwise as viewed from above.

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

Evaluate the line integral over the given curve C. ; ,

Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S. in the first octant, with orientation toward the origin.

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the cardioid .

Find the area of the part of the surface that lies between the planes x = 0, x = 4, , and z = 1.

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

Exam 16: Vector Calculus

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field. curl (div F)

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

Find the work done by the force field in moving an object along an arch of the cycloid

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. ;

Use Green's Theorem and/or a computer algebra system to evaluate where C is the circle with counterclockwise orientation.

Find the work done by the force field on a particle that moves along the parabola

Find the curl of the vector field.

Find the gradient vector field of

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

Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of across . S is the surface of the box bounded by the coordinate planes and the planes .

Let f be a scalar field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field.

Evaluate where C is the right half of the circle

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 triangle with vertices , , and .

Use Stoke's theorem to evaluate C is the curve of intersection of the hyperbolic paraboloid and the cylinder oriented counterclockwise as viewed from above.

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

Evaluate the line integral over the given curve C. ; ,

Evaluate the surface integral for the given vector field F and the oriented surface S. In other words, find the flux of F across S. in the first octant, with orientation toward the origin.

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. , where C is the cardioid .

Find the area of the part of the surface that lies between the planes x = 0, x = 4, , and z = 1.

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