EEL 3472 Review of Vector Analysis. EEL 3472 2 Review of Vector Analysis Vector analysis is a...

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Review of Review of Vector AnalysisVector Analysis

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Review of Vector Analysis

Vector analysis is a mathematical tool with which electromagnetic (EM) concepts are most conveniently expressed and best comprehended.

A quantity is called a scalar if it has only magnitude (e.g.,mass, temperature, electric potential, population).

A quantity is called a vector if it has both magnitude anddirection (e.g., velocity, force, electric field intensity).

The magnitude of a vector is a scalar written as A or

AA A

Review of Vector AnalysisReview of Vector Analysis

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A unit vector along is defined as a vector whose magnitude is unity (that is,1) and its direction is along

AA

A

AeA )e( A 1

Thus

Ae

which completely specifies in terms of A and its direction Ae

A

AeAA


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A vector in Cartesian (or rectangular) coordinates may be represented as

or

where AX, Ay, and AZ are called the components of in the

x, y, and z directions, respectively; , , and are unit vectors in the x, y and z directions, respectively.

zzyyxx eAeAeA )A,A,A( zyx

A

A


xe

ze

ye

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Suppose a certain vector is given

by

The magnitude or absolute value of

the vector is

(from the Pythagorean theorem)

zyx e4e3e2V V

385.5432V 222

V


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The Radius Vector

A point P in Cartesian coordinates may be represented by specifying (x, y, z). The radius vector (or position vector) of point P is defined as the directed distance from the origin O to P; that is,

The unit vector in the direction of r is

zyx ezeyexr

rr

zyx

ezeyexe zyx

r

222


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Vector Algebra

Two vectors and can be added together to giveanother vector ; that is ,

Vectors are added by adding their individual components.Thus, if and

A BC

BAC

zzyyxx eAeAeA zzyyxx eBeBeBB

zzzyyyxxx e)BA(e)BA(e)BA(C


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Parallelogram Head to rule tail rule

Vector subtraction is similarly carried out as

zzzyyyxxx e)BA(e)BA(e)BA(D

)B(ABAD


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The three basic laws of algebra obeyed by any given vector A, B, and C, are summarized as follows:

Law Addition Multiplication Commutative

Associative

Distributive

where k and l are scalars

ABBA

C)BA()CB(A

kAAk

A)kl()Al(k

BkAk)BA(k


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When two vectors and are multiplied, the result iseither a scalar or a vector depending on how they aremultiplied. There are two types of vector multiplication:

1. Scalar (or dot) product:

2.Vector (or cross) product:

The dot product of the two vectors and is definedgeometrically as the product of the magnitude of and theprojection of onto (or vice versa):

where is the smaller angle between and

A

ABcosABBA

BA

B

AB

A

BA

A BB

B

A B


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If and then

which is obtained by multiplying and component bycomponent

),A,A,A(A ZYX )B,B,B(B ZYX

ZZYYXX BABABABA

A B

ABBA

CABACBA )(

A A A2A2

eX ex eyey eZ ez 1

eX ey eyez eZ ex 0


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The cross product of two vectors and is defined as

where is a unit vector normal to the plane containing and . The direction of is determined using the right-hand rule or the right-handed screw rule.

A

A

nABesinABBA

B

B

ne

ne

BA Direction of and using (a) right-hand rule,(b) right-handed screw rule

ne


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If and then

zyx

zyx

zyx

BBB

AAA

eee

BA

),A,A,A(A ZYX )B,B,B(B ZYX

zxyyxyzxxzxyzzy e)BABA(e)BABA(e)BABA(


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Note that the cross product has the following basicproperties:(i) It is not commutative:

It is anticommutative:

(ii) It is not associative:

(iii) It is distributive:

(iv)

ABBA

ABBA

C)BA()CB(A

CABACBA )(

0AA )0(sin


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Also note that

which are obtained in cyclic permutation and illustrated below.

yxz

xzy

zyx

eee

eee

eee

Cross product using cyclic permutation: (a) moving clockwise leads to positive results;

(b) moving counterclockwise leads to negative results


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Scalar and Vector Fields

A field can be defined as a function that specifies a particularquantity everywhere in a region (e.g., temperature distribution in a building), or as a spatial distribution of a quantity, which may or may not be a function of time.

Scalar quantity scalar function of position scalar fieldVector quantity vector function of position vector field


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Line Integrals

A line integral of a vector field can be calculated whenever apath has been specified through the field.

The line integral of the field along the path P is defined asV

2

1

P

PP

dl cos Vdl V


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Example. The vector is given by where Vo

is a constant. Find the line integral

where the path P is the closed path below.

It is convenient to break the path P up into the four parts P1,

P2, P3 , and P4.

dl VIP

V xoeVV


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For segment P1, Thus

For segment P2, and

xedxdl

o o

1

xx

0x

x

0ooooxxoxxo

P

xV)0x(Vdx)ee(V)edx()eV(dl V

yedydl

)0e (since 0)()(dl x

02

y

yy

y

yxo

P

eedyeVVo


V Voe x

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For segment P3,

dl dxe x (the differential length dl points to the left)

oo

xx

xxxo

P

xV- )edx()eV(dl Vo

03

04

dl VP

field) ive(conservat 00xV0xV I ooooP P PP 2 3 41


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Example. Let the vector field be given by . Find the line integral of over the semicircular path shownbelow

xoeVV V

V

Consider the contribution of the path segment located at the angle

dl dl cose x

dl sine y

Since - 90cos cos( - 90 ) sinsin sin( - 90 ) cos

dl dl sine x

dl cose y

addl

{ (sine x cose y )


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o

oo

yxxxo

yxxo

aV

aVdaV

deeeeaV

adeeeVI

2

)0cos180cos(sin

])(cos)([sin

)cos(sin)(

11

180

0

0

180

0 1

180

0


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Surface Integrals

Surface integration amounts to adding up normal components of a vector field over a given surface S.

We break the surface S into small surface elements and assign to each element a vector

is equal to the area of the surface elementis the unit vector normal (perpendicular) to the surface

element

ne dsds

neds

The flux of a vector field A through surface S


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(If S is a closed surface, is by convention directed outward)Then we take the dot product of the vector field at the position of the surface element with vector . The result isa differential scalar. The sum of these scalars over all thesurface elements is the surface integral.

is the component of in the direction of (normal to the surface). Therefore, the surface integral can beviewed as the flow (or flux) of the vector field through thesurface S(the net outward flux in the case of a closed surface).

ds

ds

ds

V

cosV

SS

cos ds VdsV

V


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Example. Let be the radius vector

The surface S is defined by

The normal to the surface is directed in the +z direction

Find

V

dyd

dxd

cz

S

dsV

zyx ezeyexV


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V is not perpendicular to S, except at one point on the Z axis

Surface S


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SS

cosdsVdsV

c4d(-d)]-2dc[d

dx)]d(d[cdydxcyx

ccyxdsV

cyx

ccos dxdyds cyxV

2

dx

dx

dscos

222

dx

dx

dy

dy

V

222

S

222

222


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Introduction to Differential Operators

An operator acts on a vector field at a point to producesome function of the vector field. It is like a function of afunction.If O is an operator acting on a function f(x) of the single variable X , the result is written O[f(x)]; and means that first f acts on X and then O acts on f.

Example. f(x) = x2 and the operator O is (d/dx+2) O[f(x)]=d/dx(x2 ) + 2(x2 ) = 2x +2(x2 ) = 2x(1+x)


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An operator acting on a vector field can produceeither a scalar or a vector.

Example. (the length operator), Evaluate at the point x=1, y=2, z=-2

Thus, O is a scalar operator acting on a vector field.

Example. , , x=1, y=2, z=-2

Thus, O is a vector operator acting on a vector field.

)]z,y,x(V[O

O(A ) A A yx ezey3V )V(O

scalar32.640zy9VV)V(O 22

A2AAA)A(O yx ezey3V

vectore65.16e49.95

e4e1240)e2e(6

ez2ey6zy9)ezey3()V(O

yx

yxyx

yx22

yx


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Vector fields are often specified in terms of their rectangular components:

where , , and are three scalar features functions ofposition. Operators can then be specified in terms of , , and .

The divergence operator is defined as

zzyyxx e)z,y,x(Ve)x,y,x(Ve)z,y,x(V)z,y,x(V

xV yV zV

zyx Vz

Vy

Vx

V

xV

yV zV


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Example . Evaluate at thepoint x=1, y=-1, z=2.

zyx2 e)x2(eyexV V

0Vz

1Vy

x2Vx

x2VyVxV

zyx

zy2

x

31x2V

Clearly the divergence operator is a scalar operator.


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1. - gradient, acts on a scalar to produce a vector

2. - divergence, acts on a vector to produce a scalar

3. - curl, acts on a vector to produce a vector

4. -Laplacian, acts on a scalar to produce a scalar

Each of these will be defined in detail in the subsequentsections.

V

V

V

V2


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Coordinate Systems

In order to define the position of a point in space, an appropriate coordinate system is needed. A considerableamount of work and time may be saved by choosing a coordinate system that best fits a given problem. A hard problem in one coordinate system may turn out to be easyin another system.

We will consider the Cartesian, the circular cylindrical, andthe spherical coordinate systems. All three are orthogonal(the coordinates are mutually perpendicular).


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Cartesian coordinates (x,y,z)The ranges of the coordinate variables are

A vector in Cartesian coordinates can be written as

The intersection of three orthogonal infinite places

(x=const, y= const, and z = const)

defines point P.

z

y

x

zzyyxxzyx eAeAeAor )A,A,A(

A

Constant x, y and z surfaces


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zyx edzedyedxdl

Differential elements in the right handed Cartesian coordinate system

dxdydzd


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z

y

x

adxdy

adxdz

adydzdS


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Cylindrical Coordinates .

- the radial distance from the z – axis- the azimuthal angle, measured from the

x- axis in the xy – plane- the same as in the Cartesian system.

A vector in cylindrical coordinates can be written as

Cylindrical coordinates amount to a combination ofrectangular coordinates and polar coordinates.

)z,,(

z

20

0

2/12z

22

zzz

)AAA(A

eAeAeAor )AA,A(


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Positions in the x-y plane are determined by the values of

Relationship between (x,y,z) and )z,,(

and

zz xy

tan yx 122


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eee

eee

eee

z

z

z

0eeeeee

1eeeeee

z

zz

Point P and unit vectors in the cylindrical coordinate system


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z and ,

semi-infinite plane with its edge along the z - axis

Constant surfaces


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Differential elements in cylindrical coordinates

Metric coefficient

zp adzadaddl

dzdddv


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Planar surface ( = const)

Cylindrical surface

( =const)

dS ddza ddza dda z

Planar surface ( z =const)


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Spherical coordinates .

- the distance from the origin to the point P- the angle between the z-axis and the

radius vector of P- the same as the azimuthal angle in cylindrical coordinates

),,r( Review of Vector AnalysisReview of Vector Analysis

0 r 0

Colatitude( polar angle)

0 2

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2/1222r

rrr

)AAA(A

eAeAeAor )AA,A(

eee

eee

eee

r

r

r

0eeeeee

1eeeeee

rr

rr

A vector A in spherical coordinates may be written as

Point P and unit vectors in spherical coordinates


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cosrz

sinsinry

cossinrx

22

11-22

1222

yx

xcos

xy

tan z

yxtan zyxr

rz

cosz

tan 11

Relationships between space variables )z,,( and ),,,r(),z,y,x(


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and ,,rConstant surfaces


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Differential elements in the spherical coordinate system

adsinrardadrdl r

ddrdsinrdv 2


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a d rdr

a d rd sinr

a d d sinrdS r2


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1.

2.

3.

POINTS TO REMEMBER


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4.

5.

6.

7.


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