9.2 The Directional Derivative 1)gradients 2)Gradient at a point 3)Vector differential operator.

9.2 The Directional Derivative

1) gradients

2) Gradient at a point

3) Vector differential operator

REV:Vectors in 2D & 3D

jiu 323,23

2

1332 22 u

jiu

u

13

3

13

2

Unit vector =

length =

Dot product =

22342)24()32( xxjijivu

Example1 gradients

Comput for),( yxf235),( yxyyxf

jy

fix

fyxf

),( = del f = grad f

Gradient

Example2 If

find at (2,-1,4)),,( zyxF

322 3),,( zxxyzyxF

Gradient at a point

Vector Differential Operator

yj

xi

zk

yj

xi

2D

3D

Directional Derivative.

Example3 Find the directional derivative of

. at (1,1) in the direction of

(A) the unit vector

(B) a unit vector in the direction of 3i+4j ….(C) a unit vector whose angle with the positive x-axis is

(D) The unit vector 0i+j

xyyxyxf 62),( 32

6

ji2

1

2

1

Directional Derivative.

Geometric representation (1D)

Generalization of Partial Diff

In this section we will introduce a type of derivative, called a directional derivative.

    Suppose that we wish to find the directional derivative of f at (x0,y0) in the direction of an arbitrary unit vector u = <a,b>

To do this we consider the surface S with equation z=f(x,y) And we let z0=f(x0,y0) then the point P(x0,y0,z0) lies on S.

The vertical plane that passes though P in the direction of u intersects S in a curve C.

The slope of the tangent line T to C at P is the directional derivative of f at (x0,y0) in the direction of u

See this link

See this

Directional Derivative.

Example5 Find the directional derivative of

. at (1,-1,2) in the direction of:222 4),,( zyxxyzyxF

kjiua2

10

2

1) kjiub 326)

Directional Derivative.

ezsurf('4*x^2+y^2',[0,4])

Example4

0 1 23 4

0

1

2

3

40

10

20

30

40

50

60

70

80

x

4 x2+y2

y

224),( yxyxf