∂ f
f ( a + hv , b
)
, b)
( a
, b
) .
1
+ hv 2 − f ( a + hv 1
= hv 2
+ hv
+ αhv
∂y
1
2
By a similar argument, there exists a number 0 < β < 1 such that
∂ f
f ( a + hv , b)
( a
, b) .
1
− f ( a, b) = hv 1
+ βhv
∂x
1
Thus, by equation (2.11), we have
∂ f
∂ f
f ( a + hv , b
)
hv
( a
, b
)
( a
, b)
2
+ hv 1
+ αhv 2 + hv 1
+ βhv 1
1
+ hv 2 − f ( a, b)
∂ y
∂x
=
h
h
∂ f
∂ f
= v
( a
, b
)
( a
, b)
2
+ hv
+ αhv + v
+ βhv
∂y
1
2
1 ∂x
1
so by formula (2.9) we have
f ( a + hv , b + hv ) − f ( a, b)
D f ( a, b)
1
2
v
= lim
h→0
h
∂ f
∂ f
= lim v
( a
, b
)
( a
, b)
2
+ hv 1
+ αhv 2 + v 1
+ βhv 1
h→0
∂y
∂x
∂ f
∂ f
∂ f
∂ f
= v
( a, b)
( a, b) by the continuity of
and
, so
2
+ v
∂y
1 ∂x
∂x
∂y
∂ f
∂ f
D f ( a, b)
( a, b)
( a, b)
v
= v 1
+ v
∂x
2 ∂y
after reversing the order of summation.
QED
Note that D f ( a, b)
( a, b), ∂f ( a, b) . The second vector has a special name:
v
= v · ∂f
∂x
∂ y
80
CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES
Definition 2.6. For a real-valued function f ( x, y), the gradient of f , denoted by ∇ f , is the
vector
∂ f ∂ f
∇ f =
,
(2.12)
∂x ∂y
in R2. For a real-valued function f ( x, y, z), the gradient is the vector
∂ f ∂ f ∂ f
∇ f =
,
,
(2.13)
∂x ∂y ∂z
in R3. The symbol ∇ is pronounced “del” .5
Corollary 2.3. D f
v
= v · ∇ f
Example 2.15. Find the directional derivative of f ( x, y) = xy 2 + x 3 y at the point (1,2) in the direction of v = 1 , 1 .
2
2
Solution: We see that ∇ f = ( y 2 + 3 x 2 y,2 xy + x 3), so
D f (1, 2)
, 1
v
= v · ∇ f (1,2) =
1
· (22 + 3(1)2(2),2(1)(2) + 13) = 15
2
2
2
A real-valued function z = f ( x, y) whose partial derivatives ∂f and ∂f exist and are con-
∂x
∂ y
tinuous is called continuously differentiable. Assume that f ( x, y) is such a function and that
∇ f = 0. Let c be a real number in the range of f and let v be a unit vector in R2 which is
tangent to the level curve f ( x, y) = c (see Figure 2.4.1).
y
v
∇ f
f ( x, y) = c
x
0
Figure 2.4.1
5Sometimes the notation grad( f ) is used instead of ∇ f .
2.4 Directional Derivatives and the Gradient
81
The value of f ( x, y) is constant along a level curve, so since v is a tangent vector to this
curve, then the rate of change of f in the direction of v is 0, i.e. D f
v
= 0. But we know that
D f
v
= v · ∇ f = v
∇ f cos θ, where θ is the angle between v and ∇ f . So since v = 1 then
D f
f
v
= ∇ f cos θ. So since ∇ f = 0 then Dv = 0 ⇒ cos θ = 0 ⇒ θ = 90◦. In other words, ∇ f ⊥ v, which means that ∇ f is normal to the level curve.
In general, for any unit vector v in R2, we still have D f
v
= ∇ f cos θ, where θ is the angle
between v and ∇ f . At a fixed point ( x, y) the length ∇ f is fixed, and the value of D f then
v
varies as θ varies. The largest value that D f can take is when cos θ
v
= 1 ( θ = 0◦), while the
smallest value occurs when cos θ = −1 ( θ = 180◦). In other words, the value of the function
f increases the fastest in the direction of ∇ f (since θ = 0◦ in that case), and the value of
f decreases the fastest in the direction of −∇ f (since θ = 180◦ in that case). We have thus
proved the following theorem:
Theorem 2.4. Let f ( x, y) be a continuously differentiable real-valued function, with ∇ f = 0.
Then:
(a) The gradient ∇ f is normal to any level curve f ( x, y) = c.
(b) The value of f ( x, y) increases the fastest in the direction of ∇ f .
(c) The value of f ( x, y) decreases the fastest in the direction of −∇ f .
Example 2.16. In which direction does the function f ( x, y) = xy 2 + x 3 y increase the fastest from the point (1, 2)? In which direction does it decrease the fastest?
Solution: Since ∇ f = ( y 2 + 3 x 2 y,2 xy + x 3), then ∇ f (1,2) = (10,5) = 0. A unit vector in that direction is v = ∇ f
, 1 . Thus, f increases the fastest in the direction of
2 , 1 and
∇ f
= 25 5
5
5
decreases the fastest in the direction of −2 , −1 .
5
5
Though we proved Theorem 2.4 for functions of two variables, a similar argument can
be used to show that it also applies to functions of three or more variables. Likewise, the
directional derivative in the three-dimensional case can also be defined by the formula D f
v
=
v · ∇ f .
Example 2.17. The temperature T of a solid is given by the function T( x, y, z) = e− x + e−2 y +
e 4 z, where x, y, z are space coordinates relative to the center of the solid. In which direction
from the point (1, 1, 1) will the temperature decrease the fastest?
Solution: Since ∇ f = (− e− x,−2 e−2 y,4 e 4 z), then the temperature will decrease the fastest in the direction of −∇ f (1,1,1) = ( e−1,2 e−2,−4 e 4).
82
CHAPTER 2. FUNCTIONS OF SEVERAL VARIABLES
Exercises
A
For Exercises 1-10, compute the gradient ∇ f .
1
1. f ( x, y) = x 2 + y 2 − 1
2. f ( x, y) = x 2 + y 2
3. f ( x, y) =
x 2 + y 2 + 4
4. f ( x, y) = x 2 ey
5. f ( x, y) = ln( xy)
6. f ( x, y) = 2 x + 5 y
7. f ( x, y, z) = sin( xyz)
8. f ( x, y, z) = x 2 eyz
9. f ( x, y, z) = x 2 + y 2 + z 2
10. f ( x, y, z) =
x 2 + y 2 + z 2
For Exercises 11-14, find the directional derivative of f at the point P in the direction of
v = 1 , 1 .
2
2
1
11. f ( x, y) = x 2 + y 2 − 1, P = (1,1)
12. f ( x, y) =
, P = (1,1)
x 2 + y 2
13. f ( x, y) =
x 2 + y 2 + 4, P = (1,1)
14. f ( x, y) = x 2 ey, P = (1,1)
For Exercises 15-16, find the directional derivative of f at the point P in the direction of
v = 1 , 1 , 1 .
3
3
3
15. f ( x, y, z) = sin( xyz), P = (1,1,1)
16. f ( x, y, z) = x 2 eyz, P = (1,1,1)
17. Repeat Example 2.16 at the point (2, 3).
18. Repeat Example 2.17 at the point (3, 1, 2).
B
For Exercises 19-26, let f ( x, y) and g( x, y) be continuously differentiable real-valued func-
tions, let c be a constant, and let v be a unit vector in R2. Show that:
19. ∇( c f ) = c ∇ f
20. ∇( f + g) = ∇ f + ∇ g
g ∇ f − f ∇ g
21. ∇( f g) = f ∇ g + g ∇ f
22. ∇( f / g) =
if g( x, y) = 0
g 2
23. D f
f
24. D ( c f )
f
−v
= − Dv
v
= c Dv
25. D ( f
f
g
26. D ( f g)
g
f
v
+ g) = Dv + Dv
v
= f Dv + g Dv
27. The function r( x, y) =
x 2 + y 2 is the length of the position vector r = x i + yj for each
1
point ( x, y) in R2. Show that ∇ r = r when ( x, y) = (0,0), and that ∇( r 2) = 2r.
r
2.5 Maxima and Minima
83
2.5 Maxima and Minima
The gradient can be used to find extreme points of real-valued functions of several variables,
that is, points where the function has a local maximum or local minimum. We will consider
only functions of two variables; functions of three or more variables require methods using
linear algebra.
Definition 2.7. Let f ( x, y) be a real-valued function, and let ( a, b) be a point in the domain
of f . We say that f has a local maximum at ( a, b) if f ( x, y) ≤ f ( a, b) for all ( x, y) inside some disk of positive radius centered at ( a, b), i.e. there is some sufficiently small r > 0 such that
f ( x, y) ≤ f ( a, b) for all ( x, y) for which ( x − a)2 + ( y − b)2 < r 2.
Likewise, we say that f has a local minimum at ( a, b) if f ( x, y) ≥ f ( a, b) for all ( x, y) inside some disk of positive radius centered at ( a, b).
If f ( x, y) ≤ f ( a, b) for all ( x, y) in the domain of f , then f has a global maximum at ( a, b). If f ( x, y) ≥ f ( a, b) for all ( x, y) in the domain of f , then f has a global minimum at ( a, b).
Suppose that ( a, b) is a local maximum point for f ( x, y), and that the first-order partial
derivatives of f exist at ( a, b). We know that f ( a, b) is the largest value of f ( x, y) as ( x, y) goes in all directions from the point ( a, b), in some sufficiently small disk centered at ( a, b).
In particular, f ( a, b) is the largest value of f in the x direction (around the point ( a, b)), that is, the single-variable function g( x) = f ( x, b) has a local maximum at x = a. So we know that
g ′( a) = 0. Since g ′( x) = ∂f ( x, b), then ∂f ( a, b)
∂x
∂x
= 0. Similarly, f ( a, b) is the largest value of f
near ( a, b) in the y direction and so