14.6

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14.6 Directional Derivatives and

the Gradient Vector

Directional Derivatives and the Gradient Vector

In this section we introduce a type of derivative, called a directional derivative, that enables us to find the rate of change of a function of two or more variables in any direction.

3 3

Directional Derivatives

Directional Derivatives

Recall that if z = f(x, y), then the partial derivatives f_x and f_y are defined as

and represent the rates of change of z in the

x- and y-directions, that is, in the directions of the unit vectors i and j.

5 5

Directional Derivatives

Suppose that we now wish to find the rate of change of z at (x₀, y₀) in the direction of an arbitrary unit vector u =

〈

〉

(See Figure 2.)

To do this we consider the surface S with the equation

z = f(x, y) (the graph of f) and we let z₀ = f(x₀, y₀). Then the point P(x₀, y₀, z₀) lies on S.

Figure 2

A unit vector u = 〈a, b〉 = 〈cos u, sin u 〉

Directional Derivatives

The vertical plane that passes through P in the direction of u intersects S in a curve C. (See Figure 3.)

Figure 3

7 7

Directional Derivatives

The slope of the tangent line T to C at the point P is the rate of change of z in the direction of u.

If Q(x, y, z) is another point on C and P′, Q′ are the

projections of P, Q onto the xy-plane, then the vector is parallel to u and so

= hu =

〈

〉

for some scalar h. Therefore x – x₀ = ha, y – y₀ = hb, so x = x₀ + ha, y = y₀ + hb, and

Directional Derivatives

If we take the limit as h → 0, we obtain the rate of change of z (with respect to distance) in the direction of u, which is

called the directional derivative of f in the direction of u.

9 9

Directional Derivatives

By comparing Definition 2 with Equations 1, we see that if u = i =

〈

〉

〈

〉

In other words, the partial derivatives of f with respect to x and y are just special cases of the directional derivative.

Example 1

Use the weather map in Figure 1 to estimate the value of the directional derivative of the temperature function at Reno in the southeasterly direction.

Figure 1

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Example 1 – Solution

The unit vector directed toward the southeast is

but we won’t need to use this expression.

We start by drawing a line through Reno toward the southeast (see Figure 4).

Figure 4

Example 1 – Solution

We approximate the directional derivative D_uT by the average rate of change of the temperature between the points where this line intersects the isothermals

T = 50 and T = 60.

The temperature at the point southeast of Reno is T = 60°F and the temperature at the point northwest of Reno is

T = 50°F.

The distance between these points looks to be about

75 miles. So the rate of change of the temperature in the southeasterly direction is

cont’d

13 13

Directional Derivatives

When we compute the directional derivative of a function defined by a formula, we generally use the following

theorem.

Directional Derivatives

If the unit vector u makes an angle θ with the positive x-axis (as in Figure 2), then we can write u =

〈

〉

and the formula in Theorem 3 becomes

D_uf(x, y) = f_x(x, y) cos θ + f_y(x, y) sin θ

Figure 2

A unit vector u = 〈a, b〉 = 〈cos u, sin u 〉

15 15

The Gradient Vector

The Gradient Vector

Notice from Theorem 3 that the directional derivative of a differentiable function can be written as the dot product of two vectors:

D_uf(x, y) = f_x(x, y)a + f_y(x, y)b

=

〈

〉

〈

〉

=

〈

〉

The first vector in this dot product occurs not only in computing directional derivatives but in many other contexts as well.

So we give it a special name (the gradient of f ) and a special notation (grad f or ∇f, which is read “del f ”).

17 17

The Gradient Vector

Example 3

If f(x, y) = sin x + e^xy, then

∇f(x, y) =

〈

〉

=

〈

〉

and ∇f(0, 1) =

〈

〉

19 19

The Gradient Vector

With the notation for the gradient vector, we can rewrite Equation 7 for the directional derivative of a differentiable function as

This expresses the directional derivative in the direction of a unit vector u as the scalar projection of the gradient

vector onto u.

Functions of Three Variables

21 21

Functions of Three Variables

For functions of three variables we can define directional derivatives in a similar manner.

Again D_uf(x, y, z) can be interpreted as the rate of change of the function in the direction of a unit vector u.

Functions of Three Variables

If we use vector notation, then we can write both

definitions (2 and 10) of the directional derivative in the compact form

where x₀ =

〈

〉

〈

〉

This is reasonable because the vector equation of the line through x₀ in the direction of the vector u is given by

x = x₀ + t u and so f(x₀ + hu) represents the value of f at a point on this line.

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Functions of Three Variables

If f(x, y, z) is differentiable and u =

〈

〉

D_uf(x, y, z) = f_x(x, y, z)a + f_y(x, y, z)b + f_z(x, y, z)c For a function f of three variables, the gradient vector, denoted by ∇f or grad f, is

∇f(x, y, z) =

〈

〉

or, for short,

Functions of Three Variables

Then, just as with functions of two variables, Formula 12 for the directional derivative can be rewritten as

25 25

Example 5

If f(x, y, z) = x sin yz, (a) find the gradient of f and (b) find the directional derivative of f at (1, 3, 0) in the direction of v = i + 2 j – k.

Solution:

(a) The gradient of f is

∇f(x, y, z) =

〈

〉

=

〈

〉

Example 5 – Solution

(b) At (1, 3, 0) we have ∇f(1, 3, 0) =

〈

〉

The unit vector in the direction of v = i + 2 j – k is

Therefore Equation 14 gives D_uf(1, 3, 0) = ∇f(1, 3, 0) ^ u

cont’d

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Maximizing the Directional

Derivatives

Maximizing the Directional Derivatives

Suppose we have a function f of two or three variables and we consider all possible directional derivatives of f at a

given point.

These give the rates of change of f in all possible directions.

We can then ask the questions: In which of these directions does f change fastest and what is the maximum rate of

change? The answers are provided by the following theorem.

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Example 6

(a) If f(x, y) = xe^y, find the rate of change of f at the point P(2, 0) in the direction from P to .

(b) In what direction does f have the maximum rate of change? What is this maximum rate of change?

Solution:

(a) We first compute the gradient vector:

∇f(x, y) =

〈

〉

〈

〉

∇f(2, 0) =

〈

〉

Example 6 – Solution

The unit vector in the direction of = is u = so the rate of change of f in the direction from P to Q is

D_uf(2, 0) = ∇f(2, 0) ^ u

(b) According to Theorem 15, f increases fastest in the direction of the gradient vector ∇f(2, 0) =

〈

〉

The maximum rate of change is

|∇f(2, 0)| = |

〈

〉

cont’d

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Tangent Planes to Level Surfaces

Tangent Planes to Level Surfaces

Suppose S is a surface with equation F(x, y, z) = k, that is, it is a level surface of a function F of three variables, and let P(x₀, y₀, z₀) be a point on S.

Let C be any curve that lies on the surface S and passes through the point P. Recall that the curve C is described by a continuous vector function r(t) =

〈

〉

Let t₀ be the parameter value corresponding to P; that is,

r(t₀) =

〈

〉

F(x(t), y(t), z(t)) = k

33 33

Tangent Planes to Level Surfaces

If x, y, and z are differentiable functions of t and F is also differentiable, then we can use the Chain Rule to

differentiate both sides of Equation 16 as follows:

But, since ∇F =

〈

〉

〈

〉

Equation 17 can be written in terms of a dot product as

∇F  r′(t) = 0

Tangent Planes to Level Surfaces

In particular, when t = t₀ we have r(t₀) =

〈

〉

∇F(x₀, y₀, z₀)  r′(t₀) = 0 Equation 18 says that the gradient vector at P,

∇F(x₀, y₀, z₀), is perpendicular to the tangent vector r′(t₀) to any curve C on S that passes through P. (See Figure 9.)

35 35

Tangent Planes to Level Surfaces

If ∇F(x₀, y₀, z₀) ≠ 0, it is therefore natural to define the tangent plane to the level surface F(x, y, z) = k at

P(x₀, y₀, z₀) as the plane that passes through P and has normal vector ∇F(x₀, y₀, z₀).

Using the standard equation of a plane, we can write the equation of this tangent plane as

Tangent Planes to Level Surfaces

The normal line to S at P is the line passing through P and perpendicular to the tangent plane.

The direction of the normal line is therefore given by the gradient vector ∇F(x₀, y₀, z₀) and so, its symmetric

equations are

37 37

Tangent Planes to Level Surfaces

In the special case in which the equation of a surface S is of the form z = f(x, y) (that is, S is the graph of a function f of two variables), we can rewrite the equation as

F(x, y, z) = f(x, y) – z = 0

and regard S as a level surface (with k = 0) of F. Then F_x(x₀, y₀, z₀) = f_x(x₀, y₀)

F_y(x₀, y₀, z₀) = f_y(x₀, y₀) F_z(x₀, y₀, z₀) = –1

so Equation 19 becomes

f_x(x₀, y₀)(x – x₀) + f_y(x₀, y₀)(y – y₀) – (z – z₀) = 0

Example 8

Find the equations of the tangent plane and normal line at the point (–2, 1, –3) to the ellipsoid

Solution:

The ellipsoid is the level surface (with k = 3) of the function

39 39

Example 8 – Solution

Therefore we have

F_x(x, y, z) = F_y(x, y, z) = 2y F_z(x, y, z) = F_x(–2, 1, –3) = –1 F_y(–2, 1, –3) = 2 F_z(–2, 1, –3) = Then Equation 19 gives the equation of the tangent plane at (–2, 1, –3) as

–1(x + 2) + 2(y – 1) – (z + 3) = 0 which simplifies to 3x – 6y + 2z + 18 = 0.

By Equation 20, symmetric equations of the normal line are

cont’d

Significance of the Gradient Vector

41 41

Significance of the Gradient Vector

We now summarize the ways in which the gradient vector is significant.

We first consider a function f of three variables and a point P(x₀, y₀, z₀) in its domain.

On the one hand, we know from Theorem 15 that the

gradient vector ∇f(x₀, y₀, z₀) gives the direction of fastest increase of f.

Significance of the Gradient Vector

On the other hand, we know that ∇f(x₀, y₀, z₀) is orthogonal to the level surface S of f through P. (Refer to Figure 9.)

These two properties are quite compatible intuitively

because as we move away from P on the level surface S, the value of f does not change at all.

Figure 9

43 43

Significance of the Gradient Vector

So it seems reasonable that if we move in the

perpendicular direction, we get the maximum increase.

In like manner we consider a function f of two variables and a point P(x₀, y₀) in its domain.

Again the gradient vector ∇f(x₀, y₀) gives the direction of fastest increase of f. Also, by considerations similar to our discussion of tangent planes, it can be shown that ∇f(x₀, y₀) is perpendicular to the level curve f(x, y) = k that passes

through P.

Significance of the Gradient Vector

Again this is intuitively plausible because the values of f remain constant as we move along the curve.

(See Figure 11.)

Figure 11

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Significance of the Gradient Vector

If we consider a topographical map of a hill and let f(x, y) represent the height above sea level at a point with

coordinates (x, y), then a curve of steepest ascent can be drawn as in Figure 12 by making it perpendicular to all of the contour lines.

Figure 12

Significance of the Gradient Vector

Computer algebra systems have commands that plot sample gradient vectors.

Each gradient vector ∇f(a, b) is plotted starting at the point (a, b). Figure 13 shows such a plot (called a gradient vector field ) for the function f(x, y) = x² – y² superimposed on a

contour map of f.

As expected, the gradient vectors point “uphill” and are perpendicular to the level curves.