Section 15.6

Directional Derivatives and the Gradient Vector

The directional derivative of f at (x0, y0) in the direction of a unit vector is

if this limit exists.

THE DIRECTIONAL DERIVATIVE

ba,u

h

yxfhbyhaxfyxfD

h

),(),(lim),( 00

000

u

COMMENTS ON THE DIRECTIONAL DERIVATIVE

If u = i , then Di f = fx.

If u = j , then Dj f = fy.

THE DIRECTIONAL DERIVATIVE AND PARTIAL DERIVATIVES

Theorem: If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector and

Du f (x, y) = fx(x, y) a + fy(x, y) b

ba,u

THE DIRECTIONAL DERIVATIVE AND ANGLES

If the unit vector u makes an angle θ with the positive x-axis, then we can write

and the formula for the directional derivative becomes

Du f (x, y) = fx(x, y) cos θ + fy(x, y) sin θ

sin,cosu

VECTOR NOTATION FOR THE DIRECTIONAL DERIVATIVE

u

u

),(),,(

,),(),,(

),(),(),(

yxfyxf

bayxfyxf

byxfayxfyxfD

yx

yx

yx

THE GRADIENT VECTOR

fIf f is a function of two variables x and y, then the gradient of f is the vector function defined by

jiy

f

x

fyxfyxfyxf yx

),(),,(),(

NOTATION: Another notation for the gradient is grad f.

THE DIRECTIONAL DERIVATIVE AND THE GRADIENT

The directional derivative can be expressed by using the gradient

uu ),(),( yxfyxfD

The directional derivative of f at (x0, y0, z0) in the direction of a unit vector is

if this limit exists.

THE DIRECTIONAL DERIVATIVE IN THREE VARIABLES

cba ,,u

h

zyxfhczhbyhaxfzyxfD

h

),,(),,(lim),,( 000000

0000

u

VECTOR FORM OF THE DEFINITION OF THE

DIRECTIONAL DERIVATIVE

The directional derivative of f at the vector x0 in the direction of the unit vector u is

h

fhffD

h

)()(lim)( 00

00

xuxxu

NOTE: This formula is valid for any number of dimensions: 2, 3, or more.

THE GRADIENT IN THREE VARIABLES

kjiz

f

y

f

x

fffff zyx

,,

If f is a function of three variables, the gradient vector is

The directional directive can be expressed in terms of the gradient as

uu ),,(),,( zyxfxyxfD

Theorem: Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative Du f (x) is and it occurs when u has the same direction as the gradient vector .

MAXIMIZING THE DIRECTIONAL DERIVATIVE

)(xf

)(xf

Let F be a function of three variables. The tangent plane to the level surface F(x, y, z) = k at P(x0, y0, z0) is the plane that passes through P and is tangent to F(x, y, z) = k. Its normal vector is , and its equation is

TANGENT PLANE AND NORMAL LINE TO A LEVEL SURFACE

),,( 000 zyxF

0)(),,()(),,()(),,( 000000000000 zzzyxFyyzyxFxxzyxF zyx

The normal line to the level surface F(x, y, z) = k at P is the line passing through P and perpendicular to the tangent plane. Its direction is given by the gradient, and its symmetric equations are

),,(),,(),,( 000

0

000

0

000

0

zyxF

zz

zyxF

yy

zyxF

xx

zyx