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tangent vector is
τ
. Then,
n
=
(
∇×
u
)
·
u
·
τ.
S
Γ
This can obviously be restricted to two dimensions with
n
=(0
,
0
,
1). The
curve integral is called the
circulation
in the context of a fluid velocity field.
We can also integrate a gradient:
Ω
∇
f
=
∂
Ω
f n.
Some of the most useful identities of all are ones called
integration by parts
,
which is what we get when we combine integration identities based on the
Fundamental Theorem of Calculus with the product rule for derivatives.
They essentially let us move a differential operator from one factor in a
product to the other. Here are some of the most useful:
f
)
g
=
∂
Ω
(
∇
fgn
−
f
(
∇
g
)
,
Ω
Ω
u
=
∂
Ω
f
∇·
fu
·
n
−
(
∇
f
)
·
u,
Ω
Ω
u
=
∂
Ω
∇
f
)
·
fu
·
n
−
f
∇·
u.
(
Ω
Ω
Replacing
u
by
∇
g
in the last equation gives us one of Green's identities:
g
)=
∂
Ω
∇
f
)
·
∇
f
∇
g
·
n
−
f
∇·∇
g
(
(
Ω
Ω
=
∂
Ω
g
∇
f
·
n
−
g
∇·∇
f.
Ω
A.1.7 Basic Tensor Notation
When you get into two or three derivatives in multiple dimensions, it can get
very confusing if you stick to using the
symbols. An alternative is to use
tensor notation, which looks a little less friendly but makes it trivial to keep
everything straight. Advanced differential geometry is almost impossible to
do without this notation. We'll present a simplified version that is adequate
for most of fluid dynamics.
The basic idea is to label the separate components of a vector with
subscript indices 1, 2, and in three dimensions, 3. Usually we'll use variables
i
,
j
,
k
, etc., for these indices.
∇
Note that this can get very confusing if
