Graphics Reference
In-Depth Information
Let f :
R
n
Æ
R
and let
p
Œ
R
n
.
Definition.
The limit
(
)
-
()
=
(
)
-
(
)
f
pe
+
h f
h
p
fp
,...,
p
,
p
+
hp
,
,...,
p
fp
,...,
p
i
1
i
-
1
i
i
+
1
n
1
n
lim
lim
,
h
h
Æ
0
h
Æ
0
if it exists, is called the
ith partial derivative of f of order 1 at
p
and denoted by D
i
f(
p
)
or ∂f/∂x
i
(
p
).
Note that the ith partial derivative is just the ordinary derivative h¢(0) of the
composite function h(t) = f(g(t)), where g(t) =
p
+ t
e
i
.
Since partial derivatives are again functions, one can take partial derivatives of
those to get the higher partial derivatives.
Definition.
If k > 1, then define the (
mixed
)
partial derivative
D
i
i
,..., i
k
f(
p
)
of order k
recursively by:
()
=
(
)( )
Df
p
DD f
p
.
i
,...,
i
i
i
,...,
i
1
k
k
1
k
-
1
Does it matter in which order the partial derivatives are taken? Usually not.
4.3.9. Theorem.
If D
i,j
f and D
j,i
f are continuous in an open neighborhood contain-
ing
p
, then
()
=
()
Df
p
Df
p
.
ij
,
ji
,
Proof.
See [Spiv65].
Some common notation for partial derivatives is:
∂
∂
f
x
∂
∂
f
y
∂
∂
f
z
or f
,
,
and
,
f
,
and f for D f D f and D f respectively
,
,
,
.
xy
z
1
2
3
∂
∂
f
x
df
dx
¢
()
If f
:
RR
Æ
,
then one does not usually write
but reverts to
or
fx
.
2
2
∂
∂
f
∂
∂∂
f
xy
or f
,
,
,
f
,...
for D
f D
,
f
,...
xx
xy
11
,
21
,
2
x
Finally, the following notation often comes in handy to simplify expressions.
Let f :
R
n
Æ
R
m
be differentiable. If f(
x
) = (f
1
(
x
),f
2
(
x
),...,f
m
(
x
)), then
Notation.
∂
∂
f
x
∂
∂
f
x
∂
∂
f
x
∂
∂
f
x
Ê
Ë
ˆ
¯
1
2
m
i
will denote
,
,...,
.
i
i
i
Definition.
Let
A
be an open set in
R
n
and let f :
A
Æ
R
. If f is continuous, it is said
to be of
class C
0
. Let k ≥ 1. The function f is said to be of
class C
k
if its partial deriv-
