Graphics Reference
In-Depth Information
The different versions of the product rule for differentiation, in tensor
notation, all just fall out of the regular single-variable calculus rule. For
example,
(
fg
)=
∂f
∂x
i
∂g
∂x
i
∂
∂x
i
g
+
f
,
∂
∂x
i
(
fu
i
)=
∂f
∂x
i
u
i
+
f
∂u
i
∂x
i
.
The integral identities also simplify. For example,
=
∂
Ω
∂u
i
∂x
i
u
i
n
i
,
Ω
=
∂
Ω
∂f
∂x
i
fn
i
,
Ω
g
=
∂
Ω
∂f
∂x
i
∂g
∂x
i
fgn
i
−
f
.
Ω
Ω
A.2 Numerical Methods
This topic concentrate on methods based on finite differences, which them-
selves boil down simply to applications of Taylor series.
Assuming a function
f
has at least
k
smooth derivatives, then
∂
2
f
∂
3
f
∂x
3
(
x
)Δ
x
3
+
f
(
x
+Δ
x
)=
f
(
x
)+
∂f
∂x
(
x
)Δ
x
+
1
∂x
2
(
x
)Δ
x
2
+
1
···
2
6
∂
k−
1
f
∂x
k−
1
(
x
)Δ
x
k−
1
+
R
k
.
1
+
(
k
−
1)!
The remainder term
R
k
can be expressed in several ways, for example,
R
k
=
x
+Δ
x
x
∂
k
f
∂x
k
(
s
)
s
k−
1
ds,
1
k
!
∂
k
f
∂x
k
(
s
)Δ
x
k
1
k
!
R
k
=
for some
s
∈
[
x, x
+Δ
x
]
,
R
k
=
O
(Δ
x
k
)
.
Note that Δ
x
could be negative, in which case the second form of the
remainder uses the interval [
x
+Δ
x, x
]. We'll generally stick with the last
form, using the simple
O
() notation, but do remember that the hidden
