Graphics Reference
In-Depth Information
A.1.4
Laplacian
The Laplacian is usually formed as the divergence of the gradient (as it
repeatedly appears in fluid dynamics). Sometimes it is written as
2
or
Δ, but since these symbols are occasionally used for other purposes, I will
stick to writing it as
∇
∇·∇
. In two dimensions,
∂
2
f
∂x
2
+
∂
2
f
∂y
2
∇·∇
f
=
and in three dimensions,
∂
2
f
∂x
2
+
∂
2
f
∂y
2
+
∂
2
f
∇·∇
f
=
∂z
2
.
The Laplacian can also be applied to vector or even matrix fields, and the
result is simply the Laplacian of each component.
Incidentally, the partial differential equation
f
= 0 is called
Laplace's equation, and if the right-hand side is replaced by something
non-zero,
∇·∇
f
=
q
we call it the Poisson equation. More generally, you
can multiply the gradient by a scalar field
a
(such as 1
/ρ
), like
∇·∇
∇·
(
a
∇
f
)=
q
and still call it a Poisson problem.
A.1.5 Differential Identities
There are several identities based on the fact that changing the order of
mixed partial derivatives doesn't change the result (assuming reasonable
smoothness), e.g.,
∂
∂x
∂
∂y
f
=
∂
∂y
∂
∂x
f.
Armed with this, it's simple to show that for any smooth function,
∇·
(
∇×
u
)
≡
0
,
∇×
(
∇
f
)
≡
0
.
Another identity that shows up in vorticity calculations is
∇×
(
∇×
u
)
≡∇
(
∇·
u
)
−∇·∇
u.
The Helmholtz or Hodge decomposition is the result that any smooth vector
field
u
can be written as the sum of a divergence-free part and a curl-free
part. In fact, referring back to the first two identities above, the divergence-
free part can be written as the curl of something and the curl-free part can
