Page 15-5
Curl
The curl of a vector field
F
(x,y,z) = f(x,y,z)
i
+g(x,y,z)
j
+h(x,y,z)
k
, is defined by
a “cross-product” of the del operator with the vector field, i.e.,
[ ]
[ ]
[ ]
)
,
,
(
)
,
,
(
)
,
,
(
z
y
x
h
z
y
x
g
z
y
x
f
z
y
x
curl
∂
∂
∂
∂
∂
∂
=
×
∇
=
k
j
i
F
F
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
+
∂
∂
−
∂
∂
=
z
g
y
h
x
h
z
f
z
g
y
h
k
j
i
The curl of vector field can be calculated with function CURL. For example,
for the function
F
(X,Y,Z) = [XY,X
2
+Y
2
+Z
2
,YZ], the curl is calculated as follows:
Irrotational fields and potential function
In an earlier section in this chapter we introduced function POTENTIAL to
calculate the potential function
φ
(x,y,z) for a vector field,
F
(x,y,z) = f(x,y,z)
i
+
g(x,y,z)
j
+ h(x,y,z)
k
, such that
F
= grad
φ
=
∇φ
. We also indicated that the
conditions for the existence of
φ
, were:
∂
f/
∂
y =
∂
g/
∂
x,
∂
f/
∂
z =
∂
h/
∂
x, and
∂
g/
∂
z =
∂
h/
∂
y. These conditions are equivalent to the vector expression
curl
F
=
∇×
F
= 0.
A vector field
F
(x,y,z), with zero curl, is known as an irrotational field. Thus,
we conclude that a potential function
φ
(x,y,z) always exists for an irrotational
field
F
(x,y,z).