1.1 Vector Algebra

1.2 Differential Calculus

1.3 Integral Calculus

1.4 Curvilinear Coordinate

1.5 The Dirac Delta Function

1.6 The Theory of Vector Fields

Chapter 1 Vector Analysis

1.1 Vector Algebra

1.1.1 Scalar , Vector , Tensor

1.1.2 Vector Operation 1.1.3 Triple Products

1.1.4 Vector Transform

1.1.1 Scalar , Vector , Tensor

Scalar:Vector:Function :

3A

A�

magnitude , 0 directionmagnitude , 1 directionmagnitude , 2 direction

Tensor

all quantities are tensor .

Scalar :

Vector :

function :

ˆ ˆ

i j ij

i je e

i j

1

0

1.1.1

i

n

iieAA ˆ

1

i

n

ii faf

1

iaan

ii

1

i

ie

base of number system

base of coordinates

if base of functions

Any component of the base is independent to rest of the base

orthogonal

that is ,

1.1.1

Tensor :

magnitude , 0 direction , 0 rank tensor scale

[ no dimension ]

magnitude , 1 direction , 1st rank tensor vector

[in N dimention space : N components ]

magnitude , 2 direction , 2nd rank tensor 2nd rank tensor

[in N dimention space : N2 component ]

magnitude , 3 direction , 3rd rank tensor 3rd rank tensor

[in N dimention space : N3 component]. . .

. . .

. . .

1.1.2 Vector operation

n

ii aaaa

1321

•Addition

642111 bac

Ex.jiA ˆ3ˆ2

jiB ˆ5ˆ4

BAjcicC

ˆˆ21

642111 bac 853222 bac

; ;

;

i i ic a b

n

iii

n

i

n

i

n

iiiiiii ecebaebeaBAC

11 1 1

ˆˆˆˆ

Vector addition=sum of components

i

1.1.2

ji

n

i

n

jji

n

jjj

n

iii eebaebeaBAc ˆˆˆˆ

1 111

cosn n n

i j ij i ii j i

a b a b A B

1 1 1

=

•Inner product

Ex.

jiA ˆ3ˆ2

jiB ˆ5ˆ4

;

jijiBAC ˆ5ˆ4ˆ3ˆ2

jjijjiii ˆ5ˆ3ˆ4ˆ3ˆ5ˆ2ˆ4ˆ2 231585342

sinBABAC

ijiikkijkjjkkji

kji

kji

kji

babaebabaebabae

bbb

aaa

eee

BAC ˆˆˆ

ˆˆˆ

n

kji

kjiijkkjiijk baebae

111

ˆˆ =

1.1.2

sinBABAC

Area of

•Cross product

kjiijk baeBA ˆ

ijk= 1 clockwise = -1 counterclockwise= 0

kji

kji ji kj ik or or

1

23 1-1

1.1.2

Define :

kjiijkkjijkiii cbacbaCBaCBA

ABC

ccc

bbb

aaa

321

321

321

(volume enclosed by vectors , ,and )

1.1.3 triple products

A

B

C

Figure 1.12

1.1.4 Vector transform

ˆ ˆx yA A x A y

ˆ ˆx yA A x A y

2

1

2221

1211

2

1

A

A

aa

aa

A

A

cossin

sincosR�

y

x

y

x

B

B

B

B

cossin

sincos

BRB�

y

x

y

x

A

A

A

A

cossin

sincos

ARA�

Vector transform jiji ARA

1.2 Differential calculus

1.2.1 Differential Calculus for Rotation

1.2.2 Ordinary Derivative

1.2.3 Gradient

1.2.4 Divergence

1.2.5 The Curl

1.2.6 Product Rules

1.2.7 Second Derivatives

yyxx BABABABA cos

yyxx BABABABA cos

yxx ˆsinˆcosˆ yxy ˆcosˆsinˆ

y

x

y

y

x

y

y

x

x

x

y

x

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

yxy

xy

x

xxx

ˆsinˆcos

ˆ

ˆˆ

ˆ

ˆˆˆ

yxy

yy

x

yxy

ˆcosˆsin

ˆ

ˆˆ

ˆ

ˆˆˆ

1.2.1 Differential Calculus for Rotation

1.2.2 Ordinary Derivative

dxdx

dfdf )(

Geometrical Interpretation: The derivative is the slope of the graph of f versus x dxdf

ld

=

ldfldfee yyxx

)ˆˆ(

Ex. f =xy f=xy2

yx

f

x

y

f

;

2yx

f

; xyy

f2

1.2.3 Gradient

For a function ),( yxf

ye

xe yx

ˆˆ

Define i

ey

ex

e iyx

ˆˆˆ

f

: gradient of f

ld

dffldfldfdf cos

ˆ ˆ ˆ ˆ( ) ( )df df f f f fx y x ydx dy x y x ydf dx dy dx dy e e dxe dye

1.2.4 Divergence

y

V

x

Viiii

yxvvv

iiijjijijijjii vveevveev

ˆˆˆ)ˆ(

[a scalars]

dx

xVdxxV

xdxx

xVdxxV

x

Vx

)(

>0 : blow out

<0 : blow in

dyy y xV dxxV

dxx x

1.2.5 The Curl

ˆ ˆ

ˆ( )y x

ijk i j k ijk i j k

V Vx y

v v e e v

z

)(ˆ)(ˆ)(ˆ

ˆˆˆ

y

v

x

vk

x

v

z

vj

z

v

y

vi

vvvzyx

kji

v xyzxyz

zyx

0x

Vy

0y

Vx

0)(ˆ

y

V

x

VxyzV

1.2.6 Product Rules

The similar relations between calculation of derivatives and vector derivatives (1)Sum rules

dx

dfkkf

dx

d)(

)()(

)()(

)(

AkAk

AkAk

fkkf

(2)The rule for multiplication by a constant

( )d df dg

f gdx dx dx

( )

( ) ( ) ( )

( ) ( ) ( )

f g f g

A B A B

A B A B

1.2.6 (2)

(produce of two scalar functions) (dot product of two vectors) BA

gf

dx

dfg

dx

dgffg

dx

d)(

(3) Product rules there are six product rules:two each for gradient, divergence and curl. Product rule for divergence:

ABBA

ABBABA

)()(

)()()(

fggffg )(

two product rules for gradient

1.2.6 (3)

)()()(

)()()(

BAABBA

fAAffA

two product rules for divergences

Af

(scalar times vector)(cross product of two vector functions))( BA

two product rules for curls

( ) ( ) ( )

( ) ( ) ( ) + ( ) ( )

fA f A A f

A B B A A B A B B A

1.2.6 (4)

(4)The quotient rule

2

2

2

)()()(

)()()(

)(

g

gAAg

g

A

g

gAAg

g

A

g

gffg

g

f

2)(g

dx

dgf

dx

dfg

g

f

dx

d

The quotient rule for derivative:

The quotient rules for gradient,divergence ,and curl

1.2.7 Second Derivatives

1. Divergence of gradient:

TTTee

TeeTeT

ijiijjiji

jjiijj

2)ˆˆ(

)ˆ()ˆ()ˆ()(

jjii eVVeV ˆ)()ˆ( 222

TTTT zyx2222

so

2 is called Laplacian,T is a scalar; is a vectorV

iie ˆ a derivative vector

(inner product of same vector; ) A A A 2

1.2.7 (2)

0

)(ˆ)(ˆ)(ˆ

)(ˆ

)(ˆ)(ˆ

ˆ

)ˆ()ˆ()(

122133113223321

12321213123

1323131213223132321231

TTeTTeTTe

TTe

TTeTTe

Te

eTeT

jikkij

jjii

2. Curl of gradient :

a

(cross product of same vector, )0AA

1.2.7 (3)

)()(2 VVV

)( V

3. gradient of divergence:

4. divergence of curl

3

1,

ˆ ˆ( ) ( ) ( )

( ) 0

l l kij k i j kij lk l i j

ijk k i j k i i k jj cw

i j k

V e e V V

V V

(similar to ) 0)( VAA

1.2.7 (4)

2

2

2

2

ˆ( ) ( )

ˆ

ˆ ( )

ˆ ˆ

ˆ ˆ( ) ( )

ˆ ˆ( ) ( )

( )

ijk i j k

l lmk m ijk i j

l li mj lj mi m i j

i j i j j i j

i i j j i j j

i i j j

V V e

e V

e V

e V e V

e V V e

e V V e

V V

VVV 2)()(

5.Curl of curl

Poof:

(similar to ) )( VAA

1.3 Integral Calculus

1.3.1 Line, Surface, and Volume Integrals

1.3.2 The Fundamental Theorem of Calculus

1.3.3 The Fundamental Theorem of Gradients

1.3.4 The Fundamental Theorem of Divergences

1.3.5 The Fundamental Theorem for Curls

1.3.6 Relations Among the Fundamental Theorems

1.3.7 Integration by parts

1.3.1 Line,Surface,and Volume Integrals

(a) Line integrals

(b)Surface Integrals

b

aPldV

P:the path(e.g.(1) or (2) )

s

adV

S:the surface of integral

1.3.1 (2)

(c)Volume Integral

VTd dzdydxd

dVzdVydVxdzVyVxVdV zyxzyx ˆˆˆ)ˆˆˆ(

Example1.8

?2 TdzyxT

8

3)

12

1)(9(

2

1)1(

2

1

}][{

1

0

23

0

2

1

0

1

0

3

0

2

dyyydzz

dzdydxxyzdTy

Solution:

Suppose f(x) is a function of one variable. The fundamental theorem of calculus:

)()()(

)()()(

afbfdxxF

afbfdxdx

df

b

a

b

a

dx

dfxF )(

1.3.2 The Fundamental Theorem of Calculus

Figure 1.25

Suppose we have a scalar function of three variables T(x,y,z) We start at point ,and make the journey to point

(1)path independent(2) ,a closed loop(a=b)

),,( zyx aaa ),,( zyx bbb

0)( dlT

A line is bounded by two points

1.3.3 The Fundamental Theorem of Gradients

Figure 1.26

( )

( ) ( ) ( )b

a

dT T dl

T dl T b T a

1.3.4 The Fundamental Theorem of Divergences

ss

kj

s

ikjii

v

i

volumn

adVidaiVddVdddVdV ˆ)ˆ()(

Proof:

is called the flux of through the surface.V

( )volumn surface

V d V da

surface

V da

1.3.4 (2)

Example1.102 2ˆ ˆ ˆ(2 ) (2 )

sV y x xy z y yz z V d Vda

Solution:

2

dV

)(2 yxV

1

0

1

0

1

0)(2)(2 dzdydxyxdyx

ydxyx 1

0 2

1)( 1)(,

1

0 2

1 dyy

1

01 1dz

3

11

0

1

0

2 dzdyyadV

3

11

0

1

0

2 dzdyyadV

3

4)2(

1

0

1

0

2 dxdzzxadV

1

0

1

0

2

3

1dxdzzadV

121

0

1

0 ydxdyadV

1

0

1

000dxdyadV

2013

1

3

4

3

1

3

1 s adV

(2)

(1)

1.3.5 The Fundamental Theorem for Curls

line

k

line

kkjk

s

jikji

s

ijk

surface

ldVdVddVdaeVeadV

)ˆ()ˆ()(

Proof:

adVsurface

)(

0)( adVsurface

(1) dependents only on the boundary line, not on the particular surface used.

(2) for any closed surface.

A surface is enclosed by a closed line

Figure 1.31

( )surface boundary

line

V da V dl

1.3.5 (2)

Examples 1.11

( )s p

V da V dl

zyzyyxzV ˆ)4(ˆ)32( 22

xdzdyadandzzxxzV ˆˆ2ˆ)24( 2

3

44)(

1

0

1

0

2 dzdyzadVs

,13,3,001

0

22 dyyldVdyyldVzx

,4,4,103

41

0

22 dzzldVdzzldVyx

,13,3,100

1

22 dyyldVdyyldVzx

,00,0,000

1 dzldVldVyx

3

4

3

4011 dlV

Solution:(1)

(2)

1.3.6 Relations Among the Fundamental Theorems

(1)Gradient :

lineboundarysurface

ldVadV

)(

•combine (1)and (3)

•combine (3)and(2)

0)(0)]([

0)(

TadT

ldT

surface

line

(2)Divergence :

(3)Curl :

( ) ( ) ( )b

aT dl T b T a

( )volumn surface

V d V da

( ) 0

[ ( )] 0 ( ) 0

surface

volumn

V da

V d V

1.3.7 Integration by parts

b

a

b

a

b

a

b

a

b

a

b

a

b

a

fgdxdx

dfgdx

dx

dgfor

dxdx

dfgdx

dx

dgffgdxfg

dx

ddx

dfg

dx

dgffg

dx

d

)()(

,)()()(

)()()(

adAfdfAdAfor

adAfdfAdAfdAf

fAAfAf

s

)()(

,)()()(

)()()(

1.4 Curvilinear Coordinates

1.4.1 General Coordinates

1.4.2 Gradient

1.4.3 Divergence

1.4.4 Curl

1.4 Curvilinear Coordinates

Spherical Polar Coordinate and Cylindrical Coordinate),,( r

cossinsincossin rzryrx

ˆsinˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

drdrrdr

dldlrdlkdzjdyidxld

AArAkAjAiAA

r

rzyx

),,( zr

zdzdrrdrdl

zAArAA

zzryrx

zr

ˆˆˆ

ˆˆˆ

sincos

Figure 1.36 Figure 1.42

1.4.1 General Coordinates

Cartesian coordinate 1,1,1,, ii hzyxq

iii

zyx

zyx

edqh

edqhedqhedqh

edzedyedxld

ˆ

ˆˆˆ

ˆ1ˆ1ˆ1

332211

kjjiiijk

z

edqhdqhad

edqhdqhedydxad

ˆ

ˆˆ 32211

321321

332211

dqdqdqhhh

dqhdqhdqhdzdydxd

ze

ze

ye

xe

1ad

dxdy

dz

1.4.1 (2)

Cylindrical coordinate 1,,1,, rhzrq ii

)( rs

ˆˆ ˆˆ

ˆ

s z

i i i

dl ds dl sd dl dz

dl dr r r d dz z h dq e

da rd dz r

321321 dqdqdqhhh

dzhdhdrh

dzrddr

dldldld

zr

zr

1.4.1 (3)

Spherical coordinate sin,,1,, rrhrq ii sin

ˆ ˆˆ sin

ˆ ˆsin

ˆ ˆ

ˆ ˆ

sin

r

r

r

r

dl dr dl rd dl r d

dl drr rd r d

da dl dl r r d d r

da dl dl r dr d

da dl dl r dr d

d dl dl dl r dr d d

21

2

3

2

(C) Figure 1.38

(a)

(b)

Figure 1.39

1.4.1 (4)

i

iii edqhld ˆ

321321

ˆ

dqdqdqhhhd

edqhdqhad kjjiiijk

In summary:

Gradient

iii

ii

iii

iiiii

q

T

he

dqh

dqqTeT

edqhTdqq

T

ldTdT

1ˆ

)(ˆ

)ˆ()(

)(

1.4.2 Gradient

So for

sinˆˆˆ),,,(

ˆˆˆ),,,(

r

Te

r

Te

r

TeTr

z

Te

r

Te

r

TeTzr

r

zr

ˆ ˆ ˆ( , , ),T T T

x y z T x y zx y z

for

for

( )V s

V d V da

1.4.3 Divergence

332211

221131133233221

dqhdqhdqhdqhdqhVdqhdqhVdqhdqhV

V

( ) i iV h dq V da jjiik

kijk qdhqdhV

221131133233221

332211332211 )()(

dqhdqhVdqhdqhVdqhdqhV

AVAVAVdqhdqhdqhV

])()()([

3

3

2

2

1

1321 2133123211

A

q

A

q

A

qhhh hhVhhVhhV

( )d dd d s

V d V da

1.4.3 (2)

( )k kkl l

l

V V dah q

( )k i i j j ijkkl l

l

V h q h qh q

( )V h q h q V h q h q V h q h qh h h q q q

1 2 2 3 3 2 3 3 1 1 3 1 1 2 21 2 3 1 2 3

V h h V h h V h hh h h q q q

1 2 3 2 3 1 3 1 2

1 2 3 1 2 3

1

zV

y

V

xV zyxV

zyxqi ,,

,,rqi

zrqi ,,

)](

)sin()sin([ 2

sin1

2

rV

rVrVV rrr

)()1()(1 rVVrVV zzrrr

( ) ˆijkV h q h q e V h qi i j j j j jk j

kqqijkhhh eV khjVjhjqjikjiˆ)]([1

kekhi

jj

kji q

hV

ijkhhh ˆ)(1

)()()( ldVadVldVadVs da

dadd

c

1.4.4 Curl

zyx ,,

zr ,,

,,r

ˆ])([

ˆ)]([

ˆ])(sin[

1

sin11

sin1

r

r

vrr

rv

r

vr

rv

rv

rvv

zrv

rv

r

zrz

vrr

rv

zv

z

vvr

ˆ])([

ˆ)(ˆ)(

1

1

z

yxv

yv

x

v

xv

zv

z

v

yv

xy

zxyz

ˆ)(

ˆ)(ˆ)(

1.4.4 (2)

1.5 The Direc Delta Function

1.5.2 Some Properties of the Delta Function

1.5.3 The Three-Dimensional Delta Function

1.5.4 The Divergence of 2/ˆ rr

1.5.1 The Definition of the Delta Function

The definition of Delta function ：0)( x 0x

)(x 0x

1)( dxx

1

1

)(xf1

212

1

1

)(xg

1.5.1 The Definition of the Delta Function

)()(lim0

xxf

)()(lim

0xxg

Figure 1.46

dyayayday )(1)()(

0)()(

ayxayx

ay

)()( ayxayx

ay

1.5.1 (2)

• Definition with shifted variable.

1.5.2 Some Properties of the Delta Function

• )()( 1 ykyk

dykykkydkydxx )()()()(1

1)()(1

dykykdyy

• ( ) ( ) (0) ( ) (0) ( ) (0)f x x dx f x dx f x dx f

• ( ) '( ) ( ) ( ) '( ) ( ) '(0)f x x dx f x x f x x dx f

0

•

•

)()()()( axafaxxf

)()()( afdxaxxf

• 1| |( ) | | ( ) ( ) ( )ky k k y k y y

1.5.3 The Three-Dimensional Delta Function

)()()()(3 zyxr

zyx ezeyexr ˆˆˆ

3( )

( ) ( ) ( )

1

all spacer d

x y z dxdydz

3( ) ( ) ( )all space

f r r a d f a

1.5.4 The Divergence of 2/ˆ rr

1 ( )rrrV Vr

22

2ˆV rr

)0( r

21 (1) 0rr

V s

adVdV

2

21( ) ( sin ) 4

s RR d d

Fig. 1.44

1.5.4 (2)

14 1

VV d

314 ( ) ( ) ( ) ( )V r x y z

231 ˆ

4 ( ) ( )rr

r

23ˆ( ) 4 ( )r

rr

23ˆ( ) 4 ( )r

rr

0rrr

22 3ˆ1 1( ) ( ) 4 ( )r

rr r r

1.6 Theory of Vector Fields

1.6.1 The Helmholtz Throrem

1.6.2 Potentials

CF

DF

AVF

requirement：

0 C 2

0limr

Cr

02

0limr

Dr

0, ,

( ')'

14 'D r

V r rV d

( ')'

14 'C r

V r rA d

2 ( ')14 '

( ) 'VD rr r

F V A d

3 ( ) ( )' ' 'V

D Dr r r d

0

1.6.1 The Helmholtz Theorem

Helmholtz theorem:

Proof : Assume

)()( AVF

CAA

)(20

2 214 '

( ')| '|V

A d CC rr r

'''

'1)(

rriirr

ceiirr

rC ce ii

' 'ˆ ˆi i i i r r r r

c e e c

1 1

V rrVrr

rC drCdA '1'')(''

'

)(4

0)( '1'''1'' V rrrrsdrCadC

)( ''1''1

rrrrd

1.6.1 (2)

'312 4' rrrr

)()()( 00 xfdxxfxx0,0, ' crcrs

s r

c

srdr 2lim rc 1

ss rrrdrc ln11

2

031 r

c

nr

r

c

rc

1

2 0

When n >20

r

1.6.1 (3)

Curl-less fields :

( ) ( )s

l

F da

F dl

3 0

0

( ) ( ) ( )b

aF dl b a 4

1.6.2 Potentials

( ) F 1 0

0 .F F V V is a scalar potential

( )F V 2

Divergence-less fields :

(1) 0F

AF

)2(

( )sF da 3 0

is independent of surface

1.6.2 (2)

.0 potentialvectoraisAAFF

(4)sF da