Deck 13: Vector Calculus

Question 1

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

Answer 1

Question 2

Use Stokes' Theorem to evaluate where is the curve of intersection of the plane and the cylinder is oriented counterclockwise as viewed from above.

A)
B)
C)
D)
E)

Answer 2

Question 3

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

Answer 3

Question 4

Use Stoke's theorem to evaluate C is the curve of intersection of the plane z = x + 9 and the cylinder

Question 5

The temperature at the point in a substance with conductivity is Find the rate of heat flow inward across the cylindrical

A)
B)
C)
D)
E)

Question 6

Use a computer algebra system to compute the flux of F across S. S is the surface of the cube cut from the first octant by the planes

A)4
B)1
C)3
D)
E)0.67

Question 7

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

Question 8

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 9

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.

Question 10

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

Question 11

Use Gauss's Law to find the charge contained in the solid hemisphere , if the electric field is

Question 12

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.

Question 13

Use Stokes' Theorem to evaluate where and is the triangle with vertices is oriented counterclockwise as viewed from above.

A)
B)
C)
D)
E)

Question 14

Use Stokes' Theorem to evaluate is the part of the paraboloid that lies inside the cylinder oriented upword.

A)2
B)
C)4
D)3
E)1

Question 15

Use Stokes' Theorem to evaluate where is the circle . is oriented counterclockwise as viewed from above.

A)
B)
C)
D)
E)

Question 16

Evaluate the surface integral. Round your answer to four decimal places. S is surface

A)
B)
C)
D)
E)

Question 17

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

A)
B)
C)
D)
E)

Question 18

A fluid with density flows with velocity Find the rate of flow upward through the paraboloid

Question 19

Match the equation with one of the graphs below.

A)
B)
C)
D)

Question 20

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 .

A)
B)
C)
D)
E)

Question 21

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), ( , 0) and (0, ).

A)
B)
C)
D)
E)

Question 22

Let

A)45
B)9
C)18
D)27
E)None of these

Question 23

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

Question 24

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 25

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

A)
B)not conservative
C)
D)

Question 26

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

A)
B)
C)none of these

Question 27

Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and

Question 28

Find the curl of the vector field.

Question 29

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

A)
B)
C)
D)

Question 30

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.curl f

Question 31

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

Question 32

Find the divergence of the vector field.

Question 33

Let

Question 34

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 .

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

Question 35

Find the curl of the vector field.

Question 36

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.

Question 37

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

A)0
B)
C)
D)
E)

Question 38

Find the curl of the vector field F.

A)
B)
C)
D)

Question 39

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.

Question 40

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

Question 41

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

Question 42

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

Question 43

Evaluate the line integral over the given curve C. , where C is the line segment joining (-2, -1) to (4, 5)

Question 44

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

A)
B)
C)
D)

Question 45

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 46

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

A)
B)
C)
D)
E)

Question 47

A particle is moving in a velocity field At time t = 1 the particle is located at the point (1, 5, 5).a). What is the velocity of the particle at t = 1?
b). What is the approximate location of the particle at t = 1.01?

Question 48

Evaluate where C is the right half of the circle

A)
B)
C)
D)
E)

Question 49

A thin wire is bent into the shape of a semicircle If the linear density is , find the exact mass of the wire.

A)
B)
C)
D)
E)

Question 50

Suppose that F is an inverse square force field, that is, where Find the work done by F in moving an object from a point along a path to a point in terms of the distances and from these points to the origin.

Question 51

The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines. The flow lines of the vector field satisfy the differential equations and Solve these differential equations to find the equations of the family of flow lines.

Question 52

Which plot illustrates the vector field

A)
B)
C)
D)

Question 53

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

Question 54

Find the gradient vector field of