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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 ΠR n and a, b ΠR , 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
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