Math 200, Spring 2010
Handout
#29
Section 17-2 Stokes’ Theorem
Curl and Divergence
Suppose
1, 2, 3
F F F
=
F
is a vector field on
3
ℝ
with partial derivatives of
1,
2,
3
and
F F
F
all existing, and define
the del operator
,
,
x
y
z
∂
∂
∂
∇ =
∂
∂
∂
.
Then (1) the
curl
of
F
is the vector field defined by
( )
3
3
2
1
2
1
1
2
3
curl
,
,
F
F
F
F
F
F
x
y
z
y
z
z
x
x
y
F
F
F
∂
∂
∂
∂
∂
∂
∂
∂
∂
= ∇×
=
−
−
−
∂
∂
∂
∂
∂
∂
∂
∂
∂
i
j
k
F
F
=
and
(2) the
divergence
of
F
is the scalar field defined by
( )
3
1
2
1
2
3
div
,
,
,
,
F
F
F
F F F
x
y
z
x
y
z
∂
∂
∂
∂
∂
∂
= ∇
+
+
∂
∂
∂
∂
∂
∂
F
F =
=
i
i
Note: Suppose the vector field
F
represents the velocity field of the fluid.
In hydrodynamics, we are concerned
with two particular rates of change; these can be viewed as generalizations of the notion of the derivative
to vector fields:
1.
Circulation
, or rate of rotation, of a fluid around a point.
2.
"et flux
, or rate of flow, of a fluid out of a volume at a point.
The first is measured by
curl
and the second is measured by
divergence
.
1.
Find (a) the curl and (b) the divergence of the vector field
( )
3
5
z
x
y
y
e
=
−
+
+
F
i
j
k
2.
Compute the curl of (a)
( )
, ,
x y z
x
y
=
+
F
i
j
and
(b)
( )
, ,
x y z
y
x
= −
+
F
i
j
and interpret each graphically.
Graph of
,
x y
Graph of
,
y x
−
3.
Compute the divergence of (a)
( )
, ,
x y z
x
y
=
+
F
i
j
and
(b)
( )
, ,
x y z
y
x
= −
+
F
i
j
and interpret each graphically.