Download Curl, and Divergence Calculus 3 – Section 14.5 • Let F = 〈M,N,P

Curl, and Divergence
Calculus 3 – Section 14.5
• Let F = hM, N, P i and let ∇ = h∂/∂x, ∂/∂y, ∂/∂zi
– Curl:
i
j
∇ × F = ∂/∂x ∂/∂y
M
N
k ∂/∂z = h
P ,
– Divergence:
∇·F =
• Examples in 3D: Interpret each solution in terms of spread and rotation.
1. Calculate Divergence and Curl of the radial vector field F = hx, y, zi
2
0
-2
2
0
-2
-2
0
2
2. Calculate Divergence and Curl of the rotation vector field F = h−y, z, xi
2
0
-2
2
0
-2
-2
0
2
3. Calculate Divergence and Curl of the spiral vector field F = h−y, x, zi
2
0
-2
2
0
-2
-2
0
2
1
,
i
• Divergence and flux from a plot:
Let F = x, x + y 2 .
(a) without computing the divergence, does the graph suggest that the divergence is positive or negative at
P (−1, 1) and Q(1, 1)?
(b) Compute the divergence and confirm your conjecture in part (a).
(c) On what part of C is the flux outward? Inward?
(d) Is the net outward flux across C positive or negative? (compute the line integral to check your conjecture
. . . if possible)
2
1
æ
æ
0
-1
-2
-2
-1
0
1
2
2
• General Rotation Vector Field:
Generate a vector field by F = a × r where a = ha1 , a2 , a3 i and r = hx, y, zi.
Questions:
1. F =
2. ∇ · F =
3. ∇ × F =
4. |∇ × F| =
5. (∇ × F) · n =
Hint: let θ be the angle between ∇ × F and n
6. For what value of θ is the rotation of the vector field fastest?
• Properties of divergence and curl:
1. ∇ · (F + G) =
2. ∇ · (cF) =
3. ∇ × (F + G) =
4. ∇ × (cF) =
5. Let F be a conservative vector field. That is, F = ∇ϕ where ϕ is a potential function with continuous
second partial derivatives. Then
∇ × F = ∇ × ∇ϕ =
6. Suppose that F = hM, N, P i where M, N, and P have continuous second partial derivatives. Then the
divergence of the curl is
∇ · (∇ × F) =
7. Let u be a scalar-valued function that is twice differentiable on a region D. The divergence of the gradient
is known as Laplace’s Equation.
∇ · ∇u =
8. Let u be a scalar-valued function that is differentiable on a region D and let F be a vector field that is
differentiable on D. Then the product rule for divergence can be stated as
∇ · (uF) =
3
• Properties of Conservative Vector Fields:
Let F be a conservative vector field whose components have continuous second partial derivatives on an open
connected region D in R3 . Then F has the following equivalent properties:
1. There exists a potential function ϕ such that
.
R
for all points A and B in D and all smooth oriented curves C
2. C F · dr =
from A to B.
H
3. C F · dr =
4. ∇ × F =
on all simple smooth closed oriented curves C in D
at all points of D
4