8.2
Now that we are experts on the Chain Rule, we know at once how to compute such a
thing.
It is simply
D f
d
dt
f
t
f
t
u
a
a
u
u
( )
(
)
=
+
= ∇ ⋅
=
0
.
Example
The surface of a mountain is the graph of
f
x y
x
y
( ,
)
=


700
5
2
2
.
In other words, at
the point (
x
,
y
), the height is
f
(
x
,
y
).
The positive
y
axis points North, and, of course,
then the positive
x
axis points East.
You are on the mountain side above the point (2, 4)
and begin to walk Southeast.
What is the slope of the path at the starting point?
Are you
going uphill or downhill? (Which!?).
The answers to these questions call for the directional derivative.
We know we are at
the point
a
=
( , )
2 4
, but we need a unit vector
u
in the direction we are walking.
This is,
of course, just
u
=

1
2
1
1
( ,
)
.
Next we compute the gradient
∇
= 

f
x y
x
y
( ,
)
[
,
]
2
10
.
At
the
point
a
this
becomes
∇
= 

f
( , )
[
,
]
2 4
2
40
,
and
at
last
we
have
∇ ⋅
=

+
=
f
u
2
40
2
38
2
.
This gives us the slope of the path; it is positive so we are going
uphill.
Can you tell in which direction the path will be level?
Another Example
The temperature in space is given by
T x y z
x y
yz
( , , )
=
+
2
3
.
From the point (1,1,1), in
which direction does the temperature increase most rapidly?
We clearly need the direction in which the directional derivative is largest. The
directional derivative is simply
∇
⋅
=∇
T
T
u

cos
θ
, where
θ
is the angle between
∇
T
and
u
.
Anyone can see that this will be largest when
θ
= 0.
Thus
T
in creases most rapidly in