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Divergence and Curl of a Vector Function

This unit is based on Section 9.7 , 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 divergence and curl of a vector field.

2. understand the physical interpretations of the Divergence andCurl.

3. solve practical problems using the curl and divergence. Reading: Read Section 9.7, pages 483-487.

Exercises: Complete problems

Prerequisites: Before starting this Section you should . . .

9 be familiar with the concept of partial differentiation

9 be familiar with vector dot and cross multiplications

9 be familiar with 3D coordinate system

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Differentiation of vector fields

There are two kinds of differentiation of a vector field F(x,y,z):

1. divergence (div F = . F) and2. curl (curl F = x F)

Example of a vector field: Suppose fluid moves down apipe, a river flows, or the air circulates in a certain pattern.The velocity can be different at different points and may beat different time.

The velocity vector F gives the direction of flow and speed offlow at every point.

Applications of Vector Fields:

Mechanics

Electric and Magnetic fields

Fluids motions

Heat transfer

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Examples of Vector Fields

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The Divergence of a Vector Field

Consider the vector fields

kjiF

jiF

),,(),,(),,(),,(:variableith threefunction wVector

),(),(),(

:variableith twofunction wVector

zyxRzyxQzyxPzyx

yxQyxPyx

++=

+=

r

r

We define the divergence ofF

z

R

y

Q

x

PFDiv

+

+

=

r

In terms of the differential operator, the divergence ofF

z

R

y

Q

x

PFFDiv

+

+

==

rr

A key point: F is a vector and the divergence ofF is a scalar.

Example: F.kyzjyzxixyFrr Find,)(3)22(4 222

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Divergence

Divergence is the outflow of flux from asmall closed surface area (per unit volume)as volume shrinks to zero.

Air leaving a punctured tire: Divergence is positive,as closed surface (tire) exhibits net outflow

The divergence measures sources and

drains of flow:

sinkorsourcenoF

sinkF

sourceF

0)(@

0)(@

0)(@

P

P

P

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Physical Interpretation of the Divergence

The divergence also enters electrical

engineering topics such as electric and

magnetic fields: For a magnetic field: B = 0, that is

there are no sources or sinks of magnetic

field, a solenoidal filed.

For an electric field: E = /, that is

there are sources of electric field..

Consider a vector field F that represents a fluid velocity:

The divergence ofF at a point in a fluid is a measure of the rateat which the fluid is flowing away from or towards that point.

A positive divergence is indicating a flow away from the point.

Physically divergence means that either the fluid is expanding or

that fluid is being supplied by a source external to the field.

The lines of flow diverge from a source and converge to a sink.

If there is no gain or loss of fluid anywhere then div F = 0. Such a

vector field is said to be solenoidal.

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The Curl of a Vector Field

Consider the vector fields

kjiF ),,(),,(),,(),,( zyxRzyxQzyxPzyx +r The curl ofF is another vector field defined as:

In terms of the differential operator, the curl ofF

RQP

zyx

kji

=

Fcurlr

A key point: F is a vector and the curl ofF is a vector.

kjiFFCurl )()()(y

P

x

Q

x

R

z

P

z

Q

y

R

+

+

=rr

Example: FkyzjyzxixyFrr Find,)(3)22(4 222

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Physical Interpretation of the Curl Consider a vector field F that represents a fluid velocity:

The curl ofF at a point in a fluid is a measure of the rotation ofthe fluid. If there is no rotation of fluid anywhere then x F = 0. Such a

vector field is said to be irrotational orconservative.

For a 2D flow with F represents the fluid velocity, x F isperpendicularto the motion and represents the direction of axis ofrotation.

The curl also enters electrical engineering topics such as electric

and magnetic fields:

A magnetic field (denoted by H) has the property x H = J.

An electrostatic field (denoted by E) has the property x E = 0,

an irrotational (conservative) field. Related Course: Elec 251/351

Related Course:

ENGR361

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Further properties of the vector differential operator

[ ]222

;),,(1

z

f

y

f

x

f

ffzyxf

+

+=

222

2

graddiv)

[ ][ ][ ][ ] )()()()()5 )()()()4)()()3

)()()2

GFFGrGrF

FfFfrFrf

fFFfrFrf

gffgrgrf

rrrrrr

rrr

rrr

0))(7

0)()(6

==

frf

FrF

(]curl[grad)

]div[curl)rr

Verification Examples: >< 222322 ,, yzzxyyxFzyxf r;

2 is called theLaplacian operator

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Vector Calculus and Heat Transfer

Consider a solid material with density , heat capacity c , thetemperature distribution T(x,y,z,t) and heat flux vector q.

conservation of heat energy

In many cases the heat flux is given by Ficks law

Which results in heat equation:

Related Course: MECH352

0cTt =+

q)(

Tk=q

,2Tt

T =

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Vector Calculus and Fluid Mechanics

Conservation of Mass:Let

be the fluid density and

vbe the fluid velocity.Conservation of mass in a volume gives

Which can be written as0t =+

)(

v

0t

=++

vv

Related Course: ENGR361

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Vector Calculus and Electromagnetics

Maxwell equations in free space

Maxwell Equations describe the transmission

of information ( internet data, TV/radio

program, phone,) using wireless

communication.

t

tv

/,0

/,/

00

0

EJBB

BEE

Solutions of this equations are essential for the analysis, designand advancement of wireless devices and system, high-speed

electronics, microwave imaging, remote sensing, etc.

Related Courses: ELEC251, ELEC351, ELEC353, ELEC453,ELEC 456, ELEC 457

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Magneto-static Field Example

cabletheinside,

cabletheoutside,

0

0

=

=

B

B

JB

Magneto-static Field is an example of rotational field