The Gradient
Directional Derivatives
Suppose you are given a topographical map and want to see how steep it is from a point
that is neither due West or due North. Recall that the slopes due north
and due west are the two partial derivatives. The slopes in other
directions will be called the directional derivatives. Formally, we
define
Definition
Let f(x,y)
be a differentiable function and let u
be a unit vector then the directional derivative of f
in the direction of u
is

Note that if u is i then the directional derivative is just
f_{x} and
if u is j the it is
f_{y}.
Just as there is a difficult and an easy way to compute partial derivatives,
there is a difficult way and an easy way to compute directional derivatives.
Theorem
Let f(x,y)
be a differentiable function, and
u be a unit vector with direction
q, then

Example:
Let
f(x,y) = 2x + 3y^{2}  xy
and
v = <3,2>
Find
D_{v}
f(x,y)
Solution
We have
f_{x} = 2  y
and
f_{y} = 6y 
x
and
Hence
D_{v} f(x,y) =
<2  y, 6y 
x> ^{. }<3/,
2/>
2
3
=
(2  y) +
(6y  x)
Exercise
Let
Find D_{v} f(x,y)
The Gradient
We define
Notice that
D_{u} f(x,y) = (grad f)
^{. }u 
The gradient has a special place among directional derivatives. The
theorem below states this relationship.
Theorem

If grad f(x,y) = 0
then for all
u,
D_{u}
f(x,y) = 0

The direction of
grad f(x,y) is the
direction with maximal directional derivative.

The direction of
grad f(x,y) is
the direction
with the minimal directional derivative.

Proof:

If
gradf(x,y) = 0
then
D_{u} f(x,y) =
grad f ^{.
}u = 0 ^{. }u = 0

D_{u} f(x,y) = grad f ^{. }u =
grad f  cos q
This is a maximum when q
= 0 and a minimum when
q = p. If q
= 0 then grad f and u point in the same direction. If
q
= p then u and
grad f point in opposite directions.
This proves 2 and 3.
Example:
Suppose that a hill has altitude
w(x,y) =
x^{2}  y
Find the direction that is the steepest uphill and the steepest downhill at the
point (2,3).
Solution
We find
grad w = <2x,
y> = <4, 3>
Hence the steepest uphill is in the direction
<4,3>
while the steepest downhill is in the direction
<4,3> = <4,3>
The Gradient and Level Curves
If f is differentiable at (a,b) and
grad f is
nonzero at (a,b) then grad f is
perpendicular to the level curve through (a,b).
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