Show that the vector field
F(x,y,z) = (-8ycos(-x), -xsin(-8y), 0) is not a gradient vector field by computing its curl. How does this show what you intended?
curl(F) = &nabla × F = (
, ,).

( 0, 0, -sin(-8y) +8cos(-x))

The work

2.

Let F = 1xi + 5yj + 8z k. Compute the
divergence and the curl.

A. div F =

B. curl F = i + j + k

A) 14

B) 0i + 0j + 0k

The work

3.

A)
Consider the vector field
F(x,y,z) = (-2yz, -6xz, 3xy).
Find the divergence and curl of F.
.
(
, ,)

B)
Consider the vector field
.
Find the divergence and curl of F.
.
(
, ,)

A)
0

( 9x, -5y, -4z)
B)
-32x - 24y - 6z

( -6(x+y+z), 6(x+y+z), -18(x+y))

The work

4.

1.
Let F = 10xi + 3yj + 5zk. Compute the
divergence and the curl.

A. div F =

B. curl F = i + j + k

2.
LetF = (4xy, 3y, 6z).
The curl of F = (
,, )

1.
div F = 18
curl F = 0i + 0j + 0k

2.
curl of F = ( 0, 0, -4x)

The work

5.

Let F = (8yz)i + (7xz)j + (6xy)k. Compute the following:

A. div F =

B. curl F = i + j + k

C. div curl F =

A) 0

B) -xi + 2yj - zk

C) 0

The work

6.

Consider the vector field F(x,y,z) = (-7y, -7x, 3z). Show
that F is a gradient vector field F = &nabla V by
determining the function V which satisfies V(0,0,0) = 0.

V(x,y,z) =

The work

7.

Let .
Find a function f so that F = &nabla f, and
f(0,0,0) = 0.

The work

8.

For each of the following vector fields F , decide whether it is
conservative or not by computing curl F . Type in a potential function
f (that is, &nabla f = F).

A. F (x, y) = (-16x + 4y)i + (4x + 6y)j f (x, y) =

B. F (x, y) = -8yi - 7xj f (x, y) =

C. F ( x, y, z) = -8xi - 7yj + k f (x, y, z) =

D. F (x, y) = (-8sin(y))i + ( 8y - 8xcos(y) )j f (x, y) =

E. f (x, y, z) =

A)

B) Not Conservative

C)

D)

E)

The work

9.

Let F = -4yi + 4xj. Use the tangential vector
form of Green's Theorem to compute the circulation integral where C is the positively oriented circle .

628.319

The work

10.

Let F = 4xi + 2yj and let n be the
outward unit normal vector to the positively oriented circle . Compute the flux integral .