Ch 14.6 : Directional Derivatives and Gradient vector

Ch 14.6 : Directional Derivatives and Gradient vector

In this section, we will define a directional derivative define the gradient of a function look at examples

fx and fy

Recall that if z = f (x, y ), then fx and fy at (x0, y0) represent the rates of change of z in the x and y directions at the point. That is, in the direction of the unit vectors i =< 1, 0 > and j =< 0, 1 >.

Goal

We would like to find the rate of change of z = f (x, y ) at P(x0, y0) in the direction of an arbitrary unit vector u =< a, b >.

Note that (x0, y0) moves to in the direction of u to (x0 + ha, y0 + ha).

Directional derivatives

Definition

The directional derivative of f at (x0, y0) in the direction of a unit vector u =< a, b > is

Duf

(x0,

y0)

=

lim

h0

f

(x0

+

ha,

y0

+ hb) h

-

f

(x0,

y0)

Note: This is the general version which includes the cases fx and fy .

fx is the directional derivative of f at (x0, y0) in the direction of u =< 1, 0 >.

fy is the directional derivative of f at (x0, y0) in the direction of u =< 0, 1 >.

Theorem

Instead of using the definition, we generally use the following theorem to compute the directional derivative of a function defined by a formula.

Theorem

If f is a differentiable function of x and y , then f has a directional derivative in the direction of any unit vector u =< a, b > and

Duf (x0, y0) = fx (x0, y0)a + fy (x0, y0)b

Notice that the above theorem can be written as follows: Duf (x0, y0) =< fx (x0, y0), fy (x0, y0) > ? < a, b > := f (x0, y0) ? < a, b >

f (x, y ) is called the gradient of f . If u =< a, b > makes the angle with the positive axis, then < a, b >=< cos , sin >.

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