Graphics Reference
In-Depth Information
be written as the gradient of something else. In three dimensions,
ψ
u
=
∇×
−∇
φ,
where
ψ
is a vector-valued potential function and
φ
is a scalar potential
function. In two dimensions this reduces to
ψ
being a scalar potential
function as well:
u
=
∇×
ψ
−∇
φ.
This decomposition is highly relevant to incompressible fluid flow, since
we can interpret the pressure projection step as decomposing the inter-
mediate velocity field
u
n
+1
into a divergence-free part and something else
which we throw away, just keeping the divergence-free part. When we ex-
press a divergence-free velocity field as the curl of a potential
ψ
,wecall
ψ
the
streamfunction
.
Some more useful identities are generalizations of the product rule:
∇
(
fg
)=(
∇
f
)
g
+
f
∇
g,
∇·
(
fu
)=(
∇
f
)
·
u
+
f
∇·
u.
A.1.6 Integral Identities
The Fundamental Theorem of Calculus (that the integral of a derivative is
the original function evaluated at the limits) can be generalized to multiple
dimensions in a variety of ways.
The most common generalization is the divergence theorem discovered
by Gauss:
Ω
∇·
u
=
∂
Ω
u
·
n.
That is, the volume integral of the divergence of a vector field
u
is the
boundary integral of
u
dotted with the unit outward normal
n
.Thisac-
tually is true in any dimension (replacing volume with area or length or
hypervolume as appropriate). This provides our intuition of the divergence
measuring how fast a velocity field is expanding or compressing: the bound-
ary integral above measures the net speed of fluid entering or exiting the
volume.
Stokes' Theorem applies to the integral of a curl. Suppose we have
a bounded surface
S
with normal
n
and with boundary curve Γ whose
