Graphics Reference
In-Depth Information
Figure 4.6.
The support of a function.
y
f(x)
x
support of f
The
support
of a function f :
R
n
Definition.
Æ
R
is the closure of the set of points
where f is nonzero. (See Figure 4.6.)
4.3
Derivatives
Given a function f :
R
Æ
R
, the usual definition of the derivative for f at a point a is
(
)
-
()
fa h
+
fa
¢
()
=
fa
lim
,
(4.2)
h
h
Æ
0
if the limit exists. This works fine for functions of one variable, but just like the single
number “slope” cannot capture the direction of vectors in dimension larger than two,
we need a different definition of derivative in higher dimensions.
Let
U
be an open subset of
R
n
. A function f :
U
Æ
R
m
Definition.
is said to be
differentiable
at
p
Œ
U
if there is linear transformation T :
R
n
Æ
R
m
such that
(
)
-
()
-
()
=
f
ph
+
f
p
T
h
lim
0
.
(4.3)
h
h
Æ
0
In that case, T will be called the
derivative of f at
p
and will be denoted by Df(
p
).
See Figure 4.7. Note that if n = m = 1, then equation (4.3) is simply a rewrite of
the equation in (4.2) if we define T(h) = f¢(a)h. In other words, in arbitrary dimen-
sions, the derivative needs to be replaced by a linear transformation rather than
having it simply be a real number. The fact that a linear transformation from the reals
to the reals could be specified by a real number obscured what was really going on.
4.3.1. Proposition.
The linear transformation T in equation (4.3) is unique if it
exists.
Proof.
Suppose that S is another linear transformation satisfying equation (4.3)
where T is replaced by S. Then
