16.5 Curl and Divergence

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Chapter 16 – Vector Calculus
16.5 Curl and Divergence
Objectives:
 Understand the operations
of curl and divergence
 Use curl and divergence to
obtain vector forms of
Green’s Theorem
16.5 Curl and Divergence
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Vector Calculus

Here, we define two operations that:
◦ Can be performed on vector fields.
◦ Play a basic role in the applications of vector
calculus to fluid flow, electricity, and
magnetism.

Each operation resembles differentiation.

However, one produces a vector field whereas the
other produces a scalar field.
16.5 Curl and Divergence
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Definition - Curl

Suppose F = P i + Q j + R k is a vector field on 3
and the partial derivatives of P, Q, and R all exist,
then the curl of F is the vector field defined by:
 R Q   P R   Q P 
curl F  

i


k
 j 


 y z   z x   x y 
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Curl

As a memory aid, let’s rewrite Equation 1
using operator notation.
◦ We introduce the vector differential
operator (“del”) as:



  i  j k
x
y
z
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Curl
i

F 
x
P
j
k

y
Q

z
R
 R Q   P R   Q P 


i


k
 j


 y z   z x   x y 
 curl F
16.5 Curl and Divergence
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Curl

Thus, the easiest way to remember
Definition 1 is by means of the
symbolic expression
curl F    F
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Theorem 3

If f is a function of three variables
that has continuous second-order
partial derivatives, then
curl f   0
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Conservative Vector Field

A conservative vector field is one for which
F  f

So, Theorem 3 can be rephrased as:
If F is conservative, then curl F = 0.
◦ This gives us a way of verifying that a vector
field is not conservative.
16.5 Curl and Divergence
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Theorem 4

If F is a vector field defined on all of 3 whose
component functions have continuous partial
derivatives and curl F = 0, then F is a
conservative vector field.

NOTE: Theorem 4 is the 3-D version of Theorem 6
in Section 16.3

NOTE: This theorem says that it is true if the
domain is simply-connected—that is, “has no
hole.”
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Curl

The reason for the name curl is that the curl
vector is associated with rotations.
◦ One connection is explained in Exercise 37.
◦ Another occurs when F represents the velocity
field in fluid flow (Example 3 in Section 16.1).
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Curl

Particles near (x, y, z) in the fluid tend to rotate
about the axis that points in the direction of curl
F(x, y, z).
◦ The length of
this curl vector is
a measure of
how quickly
the particles move
around the axis.
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Irrotational Curl

If curl F = 0 at a point P, the fluid is
free from rotations at P.

F is called irrotational at P.
◦ That is, there is no whirlpool or eddy at
P.
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
If curl F = 0, a tiny paddle wheel
moves with the fluid but doesn’t
rotate about its axis.

If curl F ≠ 0, the paddle wheel
rotates about its axis.
◦ We give a more detailed explanation in Section
16.8 as a consequence of Stokes’ Theorem.
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Example 1 – pg. 1068 # 21

Show that any vector field of the
form
F( x, y, z)  f ( x) i  g ( y) j  h( z) k
where f, g, and h are differentiable
functions, is irrotational.
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Definition - Divergence

If F = P i + Q j + R k is a vector field on 3 and
∂P/∂x, ∂Q/∂y, and ∂R/∂z exist, the divergence of
F is the function of three variables defined by:
P Q R
div F 


x y z
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Divergence

In terms of the gradient operator
      

   i    j  k
 x   y   z 
the divergence of F can be written
symbolically as the dot product of del and
F:
div F    F
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Curl versus Divergence

Observe that:
◦ Curl F is a vector field.
◦ Div F is a scalar field.
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Theorem 11

If F = P i + Q j + R k is a vector field on 3
and P, Q, and R have continuous secondorder partial derivatives, then
div curl F = 0
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Divergence

Again, the reason for the name divergence can be
understood in the context of fluid flow.
◦ If F(x, y, z) is the velocity of a fluid (or gas),
div F(x, y, z) represents the net rate of change
(with respect to time) of the mass of fluid (or
gas) flowing from the point (x, y, z) per unit
volume.
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Incompressible Divergence

In other words, div F(x, y, z) measures the
tendency of the fluid to diverge from the point
(x, y, z).

If div F = 0, F is said to be incompressible.
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Example 2 – pg. 1068 # 22

Show that any vector of the form
F( x, y, z )  f ( y, z ) i  g ( x, z ) j  h( x, y) k
is incompressible.
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Example 3 – pg. 1068

Find the following for the given
vector field:
a) The curl
b) The divergence
2. F( x, y, z )  x yzi  xy zj  xyz k
2
2
8. F( x, y, z )  e , e , e
x
xy
2
xyz
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Gradient Vector Fields

Another differential operator occurs when
we compute the divergence of a gradient vector
field f.
◦ If f is a function of three variables, we have:
div  f      f 
 f  f  f
 2  2  2
x
y
z
2
2
2
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Laplace Operator

This expression occurs so often that
we abbreviate it as 2f.

The operator 2    is called
the Laplace operator due to its
relation to Laplace’s equation
2
2
2

f

f

f
2
 f  2  2  2 0
x
y
z
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Green’s Theorem – Vector Form

Hence, we can now rewrite the equation
in Green’s Theorem in the vector form
using curl as equation 12:
 F  dr    curl F   k dA
C
D
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Green’s Theorem – Vector Form

Equation 12 expresses the line integral of the
tangential component of F along C as the double
integral of the vertical component of curl F over
the region D enclosed by C.
◦ We now write a similar formula involving
the normal component of F and the divergence.
(see book for proof)
16.5 Curl and Divergence
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Green’s Theorem – Vector Form
F

n
ds

div
F
x
,
y
dA




C

D
This version says that the line integral of the
normal component of F along C is equal to the
double integral of the divergence of F over the
region D enclosed by C.
16.5 Curl and Divergence
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Example 4 – pg. 1068

Determine whether or not the vector field is
conservative. If it is conservative, find a function
f such that F = f.
14. F ( x, y, z )  xyz 2 i  x 2 yz 2 j  x 2 y 2 zk
16. F ( x, y, z )  e z i  j  xe z k
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