del operator

Dr. Kent Chamberlin 1

The Del OperatorThe Del Operator

ˆ ˆ ˆ

The Del Operator ( ) has the properties of a vector, and a

differential operator:

in Cartesian coordinatesx y za a ax y z

The del operator is important to us since it providesThe del operator is important to us since it providesa number of indications as to how vector and scalara number of indications as to how vector and scalarfunctions vary with position. It shows up in the gradient,functions vary with position. It shows up in the gradient,curl, divergence, and Laplacian.curl, divergence, and Laplacian.

GradientGradientThe gradient of a scalar function of position results in aThe gradient of a scalar function of position results in avector that points in the direction of greatest increase forvector that points in the direction of greatest increase forthat function. The magnitude of the gradient indicates howthat function. The magnitude of the gradient indicates howquickly that function changes with position.quickly that function changes with position.

ExampleExample

If the temperature in a room is

in what direction is the temperature increasing most rapidly, and

what is the rate of change with distance

( , , ) (65 0.

in that direction?

Solut

1 0.25

ion:

. )0 5T x y z x y z C

T

ˆ ˆ /

Rate of cha

ˆ(0.1 0.25 0.5 )

0.5n /7ge

x y z C m(x, y,z) a a a

T(x, Cy,z) m

If is a scalar function of position in Cartesian Coordinates

then ˆ ˆ ˆx y z

Φ(x, y,z)

Φ(x, y,z) Φ(x, y,z) Φ(x, y,z)Φ(x, y,z) a a a

x y z


GradientGradient (2)(2)

The normal to any surface defined byThe normal to any surface defined by f(x,y,z)=constant isconstant isgiven by the gradient ofgiven by the gradient of f(x,y,z)

2 2 2

2 2 2

Example: find the unit normal to the sphere

in Cartesian coordinates.

9

( , , )

ˆ ˆ ˆ( , , ) 2 2 2

ˆ ˆ2 2 2( ,

In this case ,

so , and the unit normal is

, )

( , ,

:

)

x y z

x y

x y z

f x y z x y z

f x y z xa ya za

xa ya zf x y z

f x y z

2 2 2 2 2 2

2 2 2since on the spherical surfac

ˆ ˆ ˆ ˆ ˆ ˆ ˆ

3

9 e

4 4 4

z x y z x y za xa ya za xa ya za

x y z x y z

x y z

CurlCurly

x screen.theintopoints:rulehand

righttheObeysrotation.maximum

inresultwillthatnorientatioaxiswheel

-paddlethedefiningvectorainresults

thenvelocity,fluidrepresentsIf

F

FF

In Cartesian coordinatesIn Cartesian coordinates

zyx

zyx

FFFzyx

aaa

F

ˆˆˆ


How the Curl WorksHow the Curl Worksy

x

Consider only the z-component of curl, whichcorresponds to the paddlewheel oriented asshown. That wheel will turn only if the xcomponent of the vector changes with y, orthe y component changes with x. The zcomponent of the curl is given by:

y

F

x

F

FFFzyx

aaaxy

zyx

zyx

ˆˆˆ

Curl ExamplesCurl Examples

Definition of the curlDefinition of the curl

zayF ˆ2

xayF ˆ2

y

x

t

HE

E

)(tH

max where is the surface

enclosed by the closed contour and is the unit

normal to that surface. The curl is a

ˆl

measure of

ro

im

0ˆ

tation per unit area.

n

S

n

a A dl

A ss s

a


DivergenceDivergenceDivergence is a measure of compression or decompressionof a field. It indicates the field leaving a volume element.

Because the divergence is a differential operator, it tells ussomething about the vector field at a particular point in space.

If the divergence of a vector field is positive throughout aregion, it indicates that more field is leaving that region thanentering.

The divergence can only be performed on a vector

ˆ ˆ ˆ ˆ ˆ ˆis a scalar

= in Cartesian coordinates

x y z x x y y z z

y

F

x z

F a a a F a F a F ax y z

FF F

x y z

How Divergence WorksHow Divergence Works

yy

VV y

y

yV

y

Consider the volume of gas leaving each faceConsider the volume of gas leaving each face

msec

Consider the net volume leaving the differential volume element

shown if gas velocity ˆ ˆ ˆx x y y z z

V V a V a V a

m( ) ( ) ( )3m( )

c es ce sLeft Face: - m m

y yV x z V x z

3m( )sec

Right Face: yy

VV y x z

y

x

y

z


How Divergence WorksHow Divergence Works (2)(2)Rear Face: ( )

3msecx

V y z

Front Face: ( )3m

secx

x

VV x y z

x

Bottom Face: ( )3m

seczV x y

Top Face: ( )3m

secz

z

VV z x y

z

The net loss per unit volume is (summing the above)The net loss per unit volume is (summing the above)

1sec

yx z

yx z

VV Vx y z x y z x y z

x y zx y z

VV VV

x y z

Divergence ExamplesDivergence Examples

z

y

,3

ˆ ˆ ˆIf gas velocity

is the region in compression or

decompression?

Since decompression

is the same everywhere.

V

V xa ya zax y z

2 ˆ

2

If gas velocity is the region in

compression or decompression?

Since decompression

increases with

yV y a

V y

y


Using Divergence in an EquationUsing Divergence in an Equation

As air leaves a region, its pressure decreases

As heat leaves a region, it cools off

As current leaves a region, its chargedensity decreases

If a region contains an electric charge density,an electric field will emanate from that region

If is air velocity, is air pressure, and is a constant, thenV k

V kt

If is heat flow, is temperature, and is a constant, thenH T k

TH k

t

If is current density, and is charge density, thenv ev

evv

J

Jt

If is electric field, and is charge density, thenev

ev

E

E

Definition of DivergenceDefinition of Divergence

lim

0

V dsV

v v

A measure of compression or decompression of aA measure of compression or decompression of afield. Indicates the field leaving per unit volume.field. Indicates the field leaving per unit volume.

max

ˆlim

0

n

S

a A dl

As s

Now, reconsider the definition of the CurlNow, reconsider the definition of the Curl

del operator

Documents