1_vectors & coordinate systems

8/11/2019 1_Vectors & Coordinate Systems

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Vectors & Scalars

Vectors

Magnitude

Direction

Eg: velocity, force etc

Scalars

Magnitude only

Eg: speed, distance

ECE 320 Electromagnetics & WavePropagation

2

332211 aAaAaAA

AAA

aAaAaAa

A2

3

2

2

2

1

332211

Unit Vector: have magnitude unity, denoted by symbol awith

subscript. Use the right-handed system throughout.


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DOT or Scalar Product

Scalar

AB = AB cos

A cosscalarprojection of Aonto B


3

332211 aAaAaAA 332211 aBaBaBB

BBBAAA

BABABA

A

A

2

3

2

2

2

1

2

3

2

2

2

1

332211cos

B

B

A

B

A cos


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DOT Product - properties

1. Commutative -AB = BA

2. Distributive -A(B+C) = AB+ AC

3. Bilinear -A(rB+C) = r(AB)+ AC4. When multiplied by a scalar value, dot product

satisfies: k1A+ k2 B=k1k2(AB )

5. two non-zero vectors are orthogonal if & onlyif :AB = 0


4


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Cross Product

Vector

AB = AB sin

Useful for finding unitvector perpendicular totwo vectors.


5

332211

aAaAaAA

332211 aBaBaBB

B

B

AAa

n


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6

321

321

321

BBB

AAA

aaa

BA

Geometric meaning: crossproduct can be interpreted as thearea of a parallelogram with sides

a& b


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Cross Product - properties

1. Anti-Commutative -AB = -B A

2. Distributive -A (B+C) = A B+ A C

3. two non-zero vectors are parallel if & only if :A B = 0


7

Vector triple product)()()( BACCAB CBA

Back cab rule


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Scalar triple product


8

321

321

321

CCC

BBB

AAA

BACACBCBA

Geometric meaning: interpretedas the volume of a parallelepiped

A

B C


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Differential change

Length (dli) Surface(ds)

Volume(dv)

Differential Length (dli)

1. Differential change in duiis converted to lengthusing metric coefficient hi, function of u1, u2

& u3


10

dli= hidui


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11

u2u1u3

u1u3u2

u3u2u1

aaa

aaa

aaa

0aaaaaa u1u3u3u2u2u1

1aaaaaau3u3u2u2u1u1

332211 uuu aAaAaAA

AAA uuuA 2

3

2

2

2

1


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dl = au1(h1du1)+ au2(h2du2)+ au3(h3du3)

A directed differential length in an arbitrary direction

dl= [(h1du1)2+ (h2du2)

2+ (h3du3)2] 1/2

Differential area (ds)

ds= ands

annormal to ds


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ds1= dl2dl3ds1= h2h3du2du3

ds2= h1h3du1du3

ds3= h1h2du1du2

Differential volume (dv)

dv= h1h2h3du1du2du3


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Cartesian Coordinates

(u1, u2,u3) = (x,y,z)

3 orthogonal planes

x= 0,yzplane

y= 0, xzplane

z= 0, xyplane

Right handed system

Base vectors ax, ay, az


14

yxz

xzy

zyx

aaaaaa

aaa


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15

111 zayaxaOP zyx

x

yx

y

z

Oaz z

az

ay

ayax

ax

P

1

1

1


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zzyyxx aAaAaAA

zzyyxx BABABAA B

zyx

zyx

zyx

BBB

AAA

aaa

BA

h1= h2= h3= 1

dl = axdx+ aydy+azdz


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dsx= dydz dsy= dxdz dsz= dxdy

dv= dxdydz


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Cylindrical Coordinates

(u1, u2,u3) = (r,,z)

3 orthogonal planes Circular cylindrical surface r

Half plane containing z-axismaking an angle with xz plane

Plane parallel to xyplane at z

measured from + x axis

Base vectors ar, a, az (atangential to the cylindricalaxis)


18

aaaaaa

aaa

rz

rz

zr


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zzrr aAaAaAA h1= h3= 1, h2= r

dl = ardr + a r d+azdz

dv= r drddz

dsr= r ddz

ds= drdz

dsz= r drd


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Conversion


21

z

r

z

y

x

A

A

A

A

A

A

100

0sin

0sin-

cos

cos

zz

ry

rx

sin

cos

zzx

y

yxr

1

22

tan


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Spherical Coordinates

(u1, u2,u3) = (R,, )

3 orthogonal planes Spherical surface centered at

origin with a radius R Right circular cone with apex at

the origin, its axis coincidingwith +z axis & having a halfangle

Half plane containing z-axis &making angle with the xz plane

Base vectors aR, a

, a


22

aaa

aaa

aaa

R

R

R


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aAaAaAA RR h1= 1,h2= R,h3=Rsin

dl = aRdR + aR d+a R sind

dsR= R2sindd

ds= RsindR d

ds= R dRd

dv= R2sindR dd


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Conversion


25

cos

sinsin

cossin

Rz

Ry

Rx

x

y

z

yx

zyxR

1

22

1

222

tan

tan

z

y

x

A

A

A

0cos

sin

sinsincoscoscos

cossinsincossin

A

A

AR


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Integrals

Line integralintegral of tangentialcomponent of a vector along the curve


26

L dlA

dlAdlA

b

aL

cos

abcaalong

aroundofncirculatio LAdlAL


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Integrals

Surface integralvector through the surface


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S

ndSaA

S

n

S

dSaAdSA cos

S

dSA

S

dSA


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Fields

Variation of a physical quantity from one pointto other in a region

Scalar field: constant magnitude contour

Vector field: constant magnitude contour &direction

Static field: no variation with time

Dynamic field: varies with time


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Gradient of a Scalar Field

Gradient of a scalarfield represent boththe magnitude &

the direction of themaximum space rateof increase of the

field


29


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dn

dVaVVgradn

33

3

22

2

11

1uh

Va

uh

Va

uh

VaV

uuu

33

3

22

2

11

1uh

auh

auh

auuu

dlVdVdl

alongderivativeldirectiona


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31

za

ya

xa

zyx

,coordinaterRectangula

za

Ra

ra

zr

l,Cylindrica

sinSpherical,

Ra

Ra

RaR


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Divergence of a Vector Field


32


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33

z

A

y

A

x

A

A zyx

,coordinaterRectangula

3213

2312

1321321

1

AhhuAhhuAhhuhhhA

v

dSA

A Sv

0limA,div


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z

AA

rrA

rrA z

r

11l,Cylindrica

A

RAR

RAR

RRA

R

sin

1sin

sin

11

Spherical,

22

2

FieldSolenoidal,0 A


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max0

1

limA,c

CnS dlAaSAurl

332211

321

332211

321

1

AhAhAh

uuu

hahaha

hhhA

uuu

zyx

zyx

AAA

zyx

aaa

A

Cartesian,

Curl of a Vector Field


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zr

zr

ArAA zr

aaa

r

AlCylindrica

1,

ARRAA

R

aaa

RASpherical

r

R

sinsin

1,2

FieldveconservatioralIrrotation,0 A


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ECE 320 Electromagnetics & Wave 37

V S

dSAdVA

Divergence TheoremThe volume integral of the divergence of a vector field

= total outward flux of the vector through the surfacethat bounds the volume

V S

dSAdVA

Stokes TheoremThe surface integral of the Curl of a vector field overan open surface= the closed line integral of the vector

along the contour bounding the surface.

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