Graphics Reference
In-Depth Information
is explained by thinking of it as symbolically taking
a dot-product between the gradient operator and the vector field, e.g., in
three dimensions,
The notation
∇·
∇·u
=
∂
∂
∂y
,
∂
∂z
∂x
,
·
(
u, v, w
)
∂
∂x
u
+
∂
∂y
v
+
∂
∂z
w.
=
A.1.3 Curl
The curl operator measures how much a vector field is rotating around any
point. In three dimensions this is a vector:
(
u, v, w
)=
∂w
.
∂v
∂z
,
∂u
∂z
−
∂w
∂x
,
∂v
∂x
−
∂u
∂y
∇×
u
=
∇×
∂y
−
We can reduce this formula to two dimensions in two ways. The curl of
a two-dimensional vector field results in a scalar, the third component of
the expression above, as if we were looking at the three-dimensional vector
field (
u, v,
0):
(
u, v
)=
∂v
∂u
∂y
.
∇×
u
=
∇×
∂x
−
The curl of a two-dimensional scalar field results in a vector field, as if we
were looking at the three-dimensional field (0
,
0
,w
):
w
=
∂w
.
∂w
∂x
∇×
∂y
,
−
The simple way to remember these formulas is that the curl is taking a
symbolic cross-product between the gradient operator and the function.
For example, in three dimensions,
u
=
∂
∂
∂y
,
∂
∂z
∇×
∂x
,
×
(
u, v, w
)
∂y
u
.
The curl is a way of measuring how fast (and in three dimensions along
what axis) a vector field is rotating locally. If you imagine putting a little
paddle wheel in the flow and letting it be spun, then the curl is twice the
angular velocity of the wheel. You can check this by taking the curl of the
velocity field representing a rigid rotation.
A vector field whose curl is zero is called curl-free, or
irrotational
for
obvious reasons.
=
∂
∂
∂z
v,
∂
∂x
w,
∂
∂y
w
−
pdzu
−
pdxv
−
