Graphics Reference
In-Depth Information
x
n+1
Figure 4.9.
The directional derivative.
f(p)
f(p+hv)
v
p
R
n
p+hv
exists, then that limit is called the
directional derivative of f in the direction
v
and is
denoted by D
v
f(
p
).
The directional derivative at a point
p
is essentially the derivative of the curve
obtained by intersecting the graph of f with a vertical plane through
p
. See Figure 4.9.
Some topics require that
v
be a unit vector. Although this is a natural requirement in
some application (see Examples 4.3.19 and 4.3.20 below), there is no reason to assume
this in the definition. In fact, part (2) of the next proposition shows a useful linear
relationship in the general case.
4.3.18. Proposition
∂
∂
f
x
()
=
()
=
( )
(1) D
v
f(
p
) =—f(
p
)•
v
. In particular,
Df
p
Df
p
p
.
e
i
i
i
(2) The directional derivative is a linear function of its direction vector, that is, if
v
,
w
Œ
R
n
and a, b Œ
R
, then
Df
=
aDf
+
Df
.
a
vw
+
v
w
Proof.
To prove part (1), define functions t and g by
()
=+
()
=
(
()
)
=+
(
)
t
h
pv
h
and
g h
f
t
h
f
pv
.
h
The chain rule implies that
¢
()
=—
(
()
)
∑
()
=—
(
()
)
∑
gh
f h
t
t
h
f h
t
v
.
But D
v
f(p) is just g¢(0) and so we are done. Part (2) follows easily from (1) because
(
)
=—∑ +—∑ =
D
f
=— ∑
f
a
vw
+
b
a
f
v
b
f
w
aD f
+
bD f
.
a
vw
+
v
w