1.10: 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: Directional Derivatives
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
f (x + tu1 , y + tu2 ) − f (x, y)
Du f (x, y) = lim
.
t
t→0
Note that if u is ^i then the directional derivative is just f and if u is ^i the it is f . 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.
x
y
^ , then
Let f (x, y) be a differentiable function, and u be a unit vector with direction q
Du f (x, y) = ⟨fx , fy ⟩ ⋅ ⟨cos θ, sin θ⟩ .
 Example 1.10.1
Let
f (x, y) = 2x + 3 y
2
− xy
and
v = ⟨3, 2⟩ .
Find
Dv f (x, y).
Solution
We have
fx = 2 − y
and
fy = 6y − x
and
−
−
−−−
−
||v|| = √ 9 + 4 = √13.
Hence
3
Dv f (x, y)
= ⟨2 − y, 6y − x⟩ ⋅ ⟨
2
=
−
−
√13
2
,
−
−
√13
⟩
3
−
−
√13
(2 − y) +
−
−
√13
(6y − x).
 Exercise 1.10.1
Let
f (x, y) = e
Find D
v
xy
2
and
v = ⟨2, −5⟩.
f (x, y)
1.10.1
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The Gradient
We define
∇f = ⟨fx , fy ⟩.
Notice that
Du f (x, y) = (∇f ) ⋅ u.
The gradient has a special place among directional derivatives. The theorem below states this relationship.
 Theorem
1. If ∇f (x, y) = 0 then for all u, D f (x, y) = 0.
2. The direction of ∇f (x, y) is the direction with maximal directional derivative.
3. The direction of ∇f (x, y) is the direction with the minimal directional derivative.
u
 Proof
1. If
∇f (x, y) = 0
then
Du f (x, y) = ∇f ⋅ u = 0 ⋅ u = 0.
2.
Du f (x, y) = ∇f ⋅ u = ||∇f || cos q.
This is a maximum when q = 0 and a minimum when q = p . If q = 0 then ∇f and u point in the same direction. If q = p then
u and ∇f point in opposite directions. This proves 2 and 3.
 Example 1.10.2
Suppose that a hill has altitude
2
w(x, y) = x
− y.
Find the direction that is the steepest uphill and the steepest downhill at the point (2, 3).
Solution
We find
∇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 ∇f is nonzero at (a, b) then ∇ is perpendicular to the level curve through (a, b).
1.10.2
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This page titled 1.10: The Gradient is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Larry Green.
1.10.3
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