For each of the following vector fields F , decide whether it is
conservative or not by computing the appropriate first order partial derivatives. Type in a potential function
f (that is, &nabla f = F) with f(0,0) = 0.

A. f (x, y) =

B. f (x, y) =

C. f (x, y) =

A)

B) Not Conservative

C)

The work

2.

Consider the vector field .

a) Find a function f such that F = &nabla f and
f(0,0,0) = 0.
f(x,y,z) =

b) Suppose C is any curve from (0, 0, 0 ) to (1, 1, 1 ).
Use part a) to compute the line integral .

A) 3zx +4yx +2zy
B) 9

The work

3.

Consider the vector field F (x, y, z) = xi + y j + zk

a) Find a function f such that F = &nabla f and f(0,0,0) = 0.
f(x,y,z) =

b) Use part a) to compute the work done by F
on a particle moving along the curve C given by .

A)
B) 17.5

The work

4.

Let and let C be the circle
, 0 &le t &le 2&pi

A. Compute

B. Compute

C. Compute

A)

B)

C) 2&pi

The work

5.

Consider the vector field .
Compute the line integrals and ,
where and
for 0 &le t &le 1.
=

=

= 38/15
= 38/15

The work

6.

Let be a vector field in the
plane, and C the path joining (0,0) to (1,2)
in the plane.