Graphics Reference
In-Depth Information
Other useful symbols for tensor expressions are the Kronecker delta
δ
ij
and the Levi-Civita symbol
ijk
. The Kronecker delta is
δ
ij
,whichis
actually just the identity matrix in disguise:
δ
ij
x
j
=
x
i
. The Levi-Civita
symbol has three indices, making it a third-order tensor (kind of like a
three-dimensional version of a matrix!). It is zero if any of the indices
are repeated, +1 if (
i, j, k
) is just a rotation of (1
,
2
,
3), and -1 if (
i, j, k
)
is a rotation of (3
,
2
,
1). What this boils down to is that we can write a
cross-product using it:
r
×
u
is just
ijk
r
j
u
k
,
which is a vector with free index
i
.
Putting all this together, we can translate the definitions, identities
and theorems from before into very compact notation. Furthermore, just
by keeping our indices straight, we won't have to puzzle over what an
expression like
∇∇u · n ×∇f
might actually mean. Here are some of the
translations that you can check:
∂f
∂x
i
(
∇
f
)
i
=
,
∂
∂x
i
∇
=
,
f
(
x
i
)+
∂f
∂x
i
f
(
x
i
+Δ
x
i
)
≈
Δ
x
i
,
∂f
∂n
=
∂f
∂x
i
n
i
,
∂f
i
∂x
j
f
)
ij
=
(
∇
,
∂u
i
∂x
i
∇·
u
=
,
∂u
k
∂x
j
(
∇×
u
)
i
=
ijk
,
∂
2
f
∂x
i
∂x
i
∇·∇
f
=
,
∂
∂x
i
∂u
k
∂x
j
ijk
=0
,
∂
∂x
j
∂f
∂x
k
ijk
=0
.
