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The Gradient and Applications This unit is based on Sections 9.5 and 9.6 , Chapter 9.

All assigned readings and exercises are from the textbook Objectives:Make certain that you can define, and use in context, the terms,

concepts and formulas listed below:1. find the gradient vector at a given point of a function.2. understand the physical interpretation of the gradient.3. find a multi-variable function, given its gradient4. find a unit vector in the direction in which the rate of change

is greatest and least, given a function and a point on the

function.5. the rate of change of a function in the direction of a vector.6. find normal vector and tangent vectors to a curve7. write equations for the tangent line and the normal line.

8. find an equation for the tangent plane to a surface Reading: Read Section 9.5, pages 474-482. Exercises: Complete problems 9.5 and 9.6 Prerequisites: Before starting this Section you should. . .

9 familiar with the concept of partial differentiation9 be familiar with vector functions

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Directional Derivatives: definition

We are interested in how Tchanges from one point to another.

T=20

T=25

T=15

A

B

Consider the temperature Tatvarious points of a heated metal

plate.

Some contours forTare shown inthe diagram.

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T=20

T=25

T=15

A

B

Directional Derivatives: definition

The rate of change ofTin the directionspecified byAB is given by:

(20-15)/AB = 5/h

an examples ofdirectional derivative

In general, for a given function T = T(x,y), the directionalderivative in the direction of a unit vectoru = < cos, sin> is

22

0

)()(

,),()sin,cos(

)(

yxh

h

yxThyhxTTD im

h

u

+

=

where

l

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Directional Derivatives: definition

A

the gradient vector at a pointA

magnitude = the largest directional derivative, and

pointing in the direction in which this largest directional

derivative occurs, is known as the gradient vector.

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The gradient vector: grad

kji

zyx +

+

=

A vector field, called thegradient, written:

grad F orFcan be associated with a scalar

field F. At every point the direction of

the vector field (F) isorthogonal to the scalar field

contour (C) and

in the direction of the

maximum rate of change of F.

called del

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The gradient vector: grad

zF

yF

xFzyxFFgrad

zyxFw

++==

kji

:Given

),,(

),,(

Gradient of a Function

FzxyF Find:Given ,/ 32Example:

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The gradient vector (Cont.)

|F| magnitude gives the rate of change (the slope) (rate of

change) of F, when moving along a certain direction.

Fr tounit vectotangentais,0 ttF =

Key points:

F direction is the normal vector to the surface

F is not constant in space. F is a scalar field while F is a vector field.

F points in the direction of most rapid increase of F.

- F points in the direction of most rapid decrease of F.

T

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The gradient vector (Cont.)

Example:

runit vecto,)( == uuFFDu

Generalization ofDirectional Derivative in u direction:

The maximum value of Du

(F) is ||F|| and occurs when u andFare in the same directions.

).1,2(8,6);/( < @(F)DFind uuyxxyF r

F

t

F(x,y)=C

u

The minimum value of Du(F) is -||F|| and occurs when u andFare in the opposite directions.

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The gradient vector: Some Applications

1. Electric Field (E) and Electric Potential (V):

),,(),,( zyxVzyxE =r

constant),,,(),,( == kzyxTkzyxH

r

),,(),,( zyxUzyxF r

Engineers use the gradient vector in many physical laws such as:

2. Heat Flow (H) and Temperature (T):

3. Force Field (F) and Potential Energy (U):

Example

HvectorflowheattheFind

:Given ,2100 2 yxT

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Insect Example!

Temperature Distribution:

fastet?theoffcooltogoshouldinsectthedirectionWhatcenter:locationCoolest

2225),( yxyxT +

(Assuming the insect is

familiar with vector analysis

and knows the temperature

distribution!!)

ji--

yxyxT

416

24),(

=

jid

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Equation of Tangent Plane

A vector normal to the surface F(x,y,z) = cat a point P (xo,yo,zo) is F, and can bedenoted by no.

Ifro is the position vector of the point P

relative to the origin, andr is the positionvector of any point on the tangent plane,the vector equation of the tangent plane is:

( ) Pat) oooo rFnrrn (,0 rrrr

0),,()(),,()(),,()( =ooozooooyooooxo zyxFzzzyxFyyzyxFxx The equivalent scalar equation of the tangent plane is:

Example: )1,2,2(,9:222 :PointSurface zyx

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Equation of Normal Line to a Surface

A vector normal to the surface F(x,y,z) = cat a point P (xo,yo,zo) is no = F.

Ifro is the position vector of the point P,andr is the position vector of any point on

the normal line, the vector equation of thenormal line to the surface is:

( ) Pat) oooo rFnrrn (,0 rrrr

),,(

),,,(

),,(

ooozo

oooyo

oooxo

zyxFtz

zyxFtyy

zyxFtxx

z

,

+++

The equivalent parametric equation of the normal line is:

Example:

r

)1,1,1(,42: 222 :PointSurface zyx