Deck 16: Vector Calculus

Question 1

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

Answer 1

The vector grad f is defined.It is the gradient (vector) field F of the scalar field So, which is not meaningful.

Question 2

Determine whether F is conservative.If so, find a function

Answer 2

Question 3

A thin wire in the shape of a quarter-circle has a linear mass density Find the mass and the location of the center of mass of the wire.

Answer 3

mass = 216;

Question 4

Find the curl of the vector field.

Question 5

Use Stoke's theorem to evaluate C is the curve of intersection of the plane and the cylinder

Question 6

Evaluate the line integral over the given curve C. where C is the line segment joining

Question 7

Find a vector representation for the surface.The plane that passes through the point and contains the vectors and

Question 8

Find the exact value of where C is the line segment from

Question 9

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.

Question 10

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.

Question 11

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

Question 12

Find the area of the surface S where S is the part of the sphere that lies to the right of the xz-plane and inside the cylinder

Question 13

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

Question 14

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

Question 15

Find parametric equations for C, if C is the curve of intersection of the hyperbolic paraboloid and the cylinder oriented counterclockwise as viewed from above.

Question 16

Find the curl of the vector field.

Question 17

Evaluate the line integral over the given curve C. where C is the line segment joining

Question 18

Determine whether F is conservative.If so, find a function

Question 19

Find the area of the part of paraboloid that lies inside the cylinder

Question 20

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

Question 21

Find a parametric representation for the part of the plane that lies inside the cylinder

Question 22

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

Question 23

Find the gradient vector field of

Question 24

Determine whether F is conservative.If so, find a function such that

Question 25

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

Question 26

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

Question 27

Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone if its density function is

Question 28

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 and then back to the origin along the y-axis.

Question 29

Find a function such that and use it to evaluate along the given curve C. C is the upper semicircle that starts at (1, 2) and ends at (5, 2).

Question 30

Evaluate the line integral over the given curve C. where C is the line segment joining

Question 31

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

Question 32

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

Question 33

Find a function such that and use it to evaluate along the given curve C.

Question 34

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.

Question 35

Find the curl of

Question 36

Calculate the work done by the force field when a particle moves under its influence around the edge of the part of the sphere that lies in the first octant, in a counterclockwise direction as viewed from above.

Question 37

Evaluate the line integral.

Question 38

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

Question 39

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

Question 40

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.

Question 41

Select the correct Answer for each question.
Find the work done by the force field on a particle that moves along the parabola

A)
B)
C)
D)
E)

Question 42

Select the correct Answer for each question.
Find the gradient vector field of the scalar function f.(That is, find the conservative vector field F for the potential function f of F.)

A)
B)
C)
D)

Question 43

Select the correct Answer for each question.
Find the gradient vector field of

A)
B)
C)
D)
E)

Question 44

Select the correct Answer for each question.
Evaluate the line integral where and C is the arc of the circle traversed counterclockwise from , 0) to Round yourAnswer to two decimal places.

A)
B)
C)
D)
E)

Question 45

Select the correct Answer for each question.
Below is given the plot of a vector field F in the xy-plane. By studying the plot, determine whether is positive, negative, or zero.

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

Question 46

Select the correct Answer for each question.
Evaluate S is the part of the cone between the planes and

A)
B)
C)
D)0

Question 47

Select the correct Answer for each question.
Find the curl of the vector field F.

A)
B)
C)
D)

Question 48

Select the correct Answer for each question.
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.

A)42
B)
C)19
D)

Question 49

Select the correct Answer for each question.
Suppose that where g is a function of one variable such that Evaluate where S is the sphere

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

Question 50

Select the correct Answer for each question.
Evaluate S is the part of the plane in the first octant.

A)
B)0
C)
D)

Question 51

Select the correct Answer for each question.
Evaluate where C is given by Round yourAnswer to two decimal place.

A)
B)
C)
D)
E)

Question 52

Select the correct Answer for each question.
Let S be the cube with vertices Approximate by using a Riemann sum as in Definition 1, taking the patches to be the squares that are the faces of the cube and the points to be the centers of the squares.

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

Question 53

Select the correct Answer for each question.
Find the exact mass of a thin wire in the shape of the helix if the density is 5.

A)
B)
C)
D)
E)

Question 54

Select the correct Answer for each question.
Evaluate the line integral over the given curve C.

A)28
B)
C)
D)

Question 55

Select the correct Answer for each question.
Evaluate the line integral over the given curve C.

A)
B)
C)
D)

Question 56

Select the correct Answer for each question.
A plane lamina with constant density occupies a region in the xy-plane bounded by a simple closed path C.Its moments of inertia about the axes are Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0), (4, 5) and

A)
B)
C)
D)
E)

Question 57

Select the correct Answer for each question.
Let D be a region bounded by a simple closed path C in the xy.Then the coordinates of the centroid where A is the area of D.Find the centroid of the triangle with vertices (0, 0), and (0,

A)
B)
C)
D)
E)

Question 58

Select the correct Answer for each question.
Evaluate S is the part of the torus with vector representation

A)0
B)
C)
D)

Question 59

Select the correct Answer for each question.
Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. where C is the cardioid

A)
B)
C)
D)

Question 60

Select the correct Answer for each question.
Show that F is conservative, and find a function f such that and use the result to evaluate where C is any curve from

A)0
B)-154
C)-107
D)-89

Question 61

Select the correct Answer for each question.
Show that F is conservative, and find a function f such that and use the result to evaluate where C is any curve from

A)-89
B)0
C)-154
D)-107

Question 62

Select the correct Answer for each question.
Use Green's Theorem to find the work done by the force in moving a particle in the positive direction once around the triangle with vertices

A)
B)
C)
D)

Question 63

Select the correct Answer for each question.
Use a computer algebra system to compute the flux of F s the surface of the cube cut from the first octant by the planes

A)
B)
C)
D)
E)

Question 64

Select the correct Answer for each question.
Find the area of the surface.The part of the paraboloid

A)
B)
C)
D)

Question 65

Select the correct Answer for each question.
Use Green's Theorem and/or a computer algebra system to evaluate where C is the circle with counterclockwise orientation.

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

Question 66

Select the correct Answer for each question.
Which plot illustrates the vector field

A)
B)
C)
D)

Question 67

Select the correct Answer for each question.
Show that F is conservative and find a function f such that and use this result to evaluate where C is any path from

A)
B)
C)
D)

Question 68

Select the correct Answer for each question.
Use Green's Theorem to evaluate the line integral along the positively oriented closed curve where C is the triangle with vertices

A)6
B)60
C)30
D)66

Question 69

Select the correct Answer for each question.
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

A)
B)0
C)
D)1

Question 70

Select the correct Answer for each question.
Find the correct identity, if f is a scalar field, F and G are vector fields.

A)
B)
C)
D)None of these

Question 71

Select the correct Answer for each question.
A plane lamina with constant density occupies a region in the xy-plane bounded by a simple closed path C.Its moments of inertia about the axes are Find the moments of inertia about the axes, if C is a rectangle with vertices (0, 0), (4, 0), (4, 5) and

A)
B)
C)
D)
E)

Question 72

Select the correct Answer for each question.
Find the exact mass of a thin wire in the shape of the helix if the density is 5.

A)
B)
C)
D)
E)

Question 73

Select the correct Answer for each question.
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.

A)8
B)6
C)5
D)7
E)0

Question 74

Select the correct Answer for each question.
Find a parametric representation for the part of the elliptic paraboloid that lies in front of the plane x = 0.

A)
B)
C)
D)
E)

Question 75

Select the correct Answer for each question.
Let

A)5
B)3
C)1
D)2
E)None of these

Question 76

Select the correct Answer for each question.
Evaluate where C is the right half of the circle

A)
B)
C)
D)
E)

Question 77

Select the correct Answer for each question.
Evaluate the surface integral.Round yourAnswer to four decimal places. S is surface

A)
B)
C)
D)
E)

Question 78

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

A)24
B)47
C)0
D)

Question 79

Select the correct Answer for each question.
Find the work done by the force field F on a particle that moves along the curve C.

A)17
B)19
C)34
D)

Question 80

Select the correct Answer for each question.
Let S be the cube with vertices Approximate by using a Riemann sum as in Definition 1, taking the patches to be the squares that are the faces of the cube and the points to be the centers of the squares.

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