Deck 15: Vector Fields

Question 1

a piecewise smooth parametrization of the path C given in the following graph.

Answer 1

D

Question 2

Answer 2

C

Question 3

Sketch the vector field

Answer 3

A

Question 4

the conservative vector field for the potential function

Question 5

the work done by the force field

Question 6

Determine whether the vector field is conservative. If it is, find a potential function

Question 7

the divergence at

Question 8

a piecewise smooth parametrization of the path C given in the following graph.

Question 9

Determine whether the vector field is conservative. If it is, find a potential function

Question 10

Evaluate the integral

Question 11

Determine whether the vector field is conservative. If it is, find a potential function for the vector field.

Question 12

the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.)

Question 13

the divergence of the vector field F given by

Question 14

Evaluate

Question 15

Evaluate the line integral along the given path.

Question 16

the divergence of the vector field.

Question 17

Question 18

the curl for the vector field at the given point.

Question 19

the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.)

Question 20

the divergence of the vector field at the given point.

Question 21

Question 22

the vector field field is conservative.

Question 23

the work done by the force field

Question 24

Calculate the line integral along

Question 25

the value of the line integral

Question 26

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results.

Question 27

.

Question 28

Calculate the line integral along

Question 29

tractor engine has a steel component with a circular base modeled by the

Question 30

the area of the lateral surface over the curve C in the xy-plane and under the

Question 31

Evaluate the line integral

Question 32

the value of the line integral

Question 33

stone weighing 2 pounds is attached to the end of a four-foot string and is whirled horizontally with one end held fixed. It makes 1 revolution per second. Find the work done by the
Force F that keeps the stone moving in a circular path. [Hint: Use Force = (mass)(centripetal
Acceleration).] Round your answer to two decimal places, if required.

Question 34

the work done by a person weighing pounds walking exactly one revolution up a circular helical staircase of radius feet if the person rises feet.

Question 35

Question 36

Determine whether or not the vector field is conservative.

Question 37

Question 38

the moments of inertia for a wire that lies along

Question 39

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results.

Question 40

the value of the line integral

Question 41

Green's Theorem to calculate the work done by the force

Question 42

Verify Green's Theorem by evaluating both integrals

Question 43

up and evaluate a line integral to find the area of the region R bounded by the

Question 44

Identify the surface by eliminating the parameters from the vector-valued function

Question 45

Question 46

Green's Theorem to calculate the work done by the force moving counterclockwise around the closed path C.

Question 47

a computer algebra system and the result "The area of a plane region bounded by the simple closed path C given in polar coordinates is

Question 48

Green's Theorem to evaluate the integral

Question 49

Match the following vector-valued function with its graph.

Question 50

Green's Theorem to evaluate the integral

Question 51

Green's Theorem to evaluate the line integral

Question 52

15. Use a computer algebra system and the result "The area of a plane region bounded by

Question 53

the rectangular equation for the surface by eliminating parameters from the vector-valued function. Identify the surface.

Question 54

the rectangular equation for the surface by eliminating the parameters from the

Question 55

Match the following vector -valued function with its graph.

Question 56

Green's Theorem to evaluate the integral

Question 57

the maximum value of in the xy-plane, oriented counterclockwise.

Question 58

Green's Theorem to evaluate the line integral

Question 59

Green's Theorem to evaluate the integral

Question 60

a computer algebra system and the result "The centroid of the region

Question 61

a computer algebra system to evaluate

Question 62

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis.

Question 63

a vector-valued function for the hyperboloid

Question 64

Question 65

Write a set of parametric equations for the surface of revolution obtained by

Question 66

an equation of the tangent plane to the surface represented by the vector-valued function at the given point.

Question 67

surface of the dome on a new museum is given by

Question 68

a vector-valued function whose graph is the plane

Question 69

a computer algebra system to evaluate

Question 70

Question 71

Question 72

Determine the tangent plane for the hyperboloid

Question 73

the rectangular equation for the surface by eliminating the parameters from the

Question 74

the area of the surface of revolution

Question 75

the area of the surface over the given region. Use a computer algebra system to verify your results.
The part of the cone,

Question 76

Question 77

Question 78

Question 79

the area of the surface over the given region. Use a computer algebra system to verify your results.

Question 80

the area of the surface given by