Graphics Reference
In-Depth Information
you also are thinking of discretizing on a grid—if you want to avoid that
confusion, it's often a good idea to only use Greek letters for your tensor
indices, e.g.,
α
,
β
,
γ
instead.
The gradient of a function is (
∂f/∂x
1
,∂f/∂x
2
,∂f/∂x
3
). This is still a
bit longwinded, so we instead use the generic
∂f/∂x
i
without specifying
what
i
is: it's a “free” index.
We could then write the divergence, for example, as
∂u
i
∂x
i
.
i
This brings us to the
Einstein summation convention
. It's tedious to have
to write the sum symbol Σ again and again. Thus we just won't bother
writing it: instead, we will assume that in any expression that contains the
index
i
twice, there is an implicit sum over
i
in front of it. If we don't want
a sum, we use different indices, like
i
and
j
. For example, the dot-product
of two vectors
u
and
n
can be written very succinctly as
u
i
n
i
.
Note that by expression I mean a single term or a product—it does not
include addition. So this
u
i
+
r
i
is a vector,
u
+
r
,notascalarsum.
Einstein notation makes it very simple to write a matrix-vector product,
such as
Ax
:
A
ij
x
j
.
Note that the free index in this expression is
j
: this is telling us the
j
th
component of the result. This is also an introduction to second-order ten-
sors, which really are a fancy name for matrices: they have two indices
instead of the one for a vector (which canbecalledafirst-ordertensor).
We can write matrix multiplication just as easily: the product
AB
is
A
ij
B
jk
with free indices
i
and
k
: this is the
i, k
entry of the result. Similarly, the
outer-product matrix of vectors
u
and
n
is
u
i
n
j
.
