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Calc 3 13.5

 

<== Calculus 3
Number Question Answer
See the work
1. 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.
	extrm{div}(F) = 
abla \cdot F = .
	extrm{curl}(F) = 
abla 	imes F =  (( , ,)


B)
Consider the vector field F(x,y,z) = (-4x^2, -9(x+y)^2, -3(x+y+z)^2).
Find the divergence and curl of F.
	extrm{div}(F) = 
abla \cdot F = .
	extrm{curl}(F) = 
abla 	imes 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 \mathbf{F} = (18 xyz + 7\sin x, 9 x^2z, 9 x^2y). 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. \mathbf{F} \left( x, y, z 
ight) = -8 x^{2} \mathbf{i} +
4 y^{2} \mathbf{j} + 3 z^{2} \mathbf{k}
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 \int_{C} \mathbf{F}
\cdot d\mathbf{r} where C is the positively oriented circle x^{2} + y^{2} =
25.

628.319 The work
10. Let F = 4xi + 2yj and let n be the outward unit normal vector to the positively oriented circle x^{2} + y^{2} =
1. Compute the flux integral \int_{C} \mathbf{F \cdot n} \, ds.

18.85 The work
11. Apply the Laplace operator to the function h(x,y,z) = e^{-4x}\sin(-6y).

abla^2 h = .
The work