Chapter 1 fundamental concepts :

Vectors

Scalar : magnitude only

Vector : magnitude(1), orientation(n-1)

• more than one component

Unit vector : vector with magnitude=1, only orientation, denoted by a hat

Unit coordinate vectors : (1,0,0), (0,1,0), (0,0,1), basis vectors

Scalar product, dot product –

projection, angle between

Vector product, cross product –

area,

zzyyxx

zyxzyx

BABABA

BBBAAAAB

zyxzyxBA ˆˆˆˆˆˆcos

zyx

zyx

BBB

AAA

zyx

BA

1.1 Vector Algebra

1.1.1 Vector operations

(i) Addition of two vectors

BABA

CBACBA

ABBA

: direction & magnitude

(iii) Dot product of two vectors (scalar)

. when ,0

cos

2

BABA

AA

CABACBA

ABBA

BA

A

AB

(ii) Multiplication by a scalar BABA aaa

productDot parallel)( cos A

product Cross

ular)(perpendic sin

A

(iv) Cross product of two vectors (vector)

0

,ˆ ˆsin

AA

BAAB

CABACBA

BAnnBA AB

1.1.2 Vector algebra: Component form

.ˆˆˆ

ˆˆˆ

zyxBA

zyxA

zzyyxx

zyx

BABABA

AAA

(i) Addition:

(ii) Multiplication: zyxA ˆˆˆzyx aAaAaAa

(iii) Dot product:

zzyyxx

zyxzyx

BABABA

BBBAAAAB

zyxzyxBA ˆˆˆˆˆˆcos

0 ,1 zyzxyxzzyyxx

. , 222222

zyxzyx AAAAAAA AA

BA

(iv) Cross product

zyx

zyxzyxBA

xyyxzxxzyzzy

zyxzyx

BABABABABABA

BBBAAA

. , ,

,

yzxxzxyzzyzxyyx

0zzyyxx

zyx

zyx

BBB

AAA

zyx

BA cf.

1.1.3 Triple product

(i) Scalar triple product:

CBACBA

BACACBCBA

:eInterchang-

:Rotation-

CBA

zyx

zyx

zyx

CCC

BBB

AAA

)( CBA

“Volume of the parallelepiped”

cosAheight

(ii) Vector triple product:

.;

.

DCBADCABDCBA

CBDADBCADCBA

CABCBABACCBA

BACCABCBA

1.1.4 Position, Displacement, and Separation Vectors

(i) Position vector

222

222

ˆˆˆˆ

ˆˆˆ

zyx

zyx

r

zyxr

zyx

zyxrr

r

zyxr

(ii) infinitesimal displacement vector zyxl ˆˆˆ dzdydxd

(iii) source point r’, field point r separation vector r-r’

zyxrr ˆ'ˆ'ˆ'' zzyyxx

source point

field point

1.1.5 How vectors transform

- When changing coordinates, vector components should be converted. Transformation

(i) Rotational transformation (about x-axis)

zyz

zyy

zy

AAAAAA

AAAAAA

AA

cossinsincoscossinsinsin

sincossinsincoscoscoscos

cos ,cos

z

y

z

y

A

A

A

A

cossin

sincos

For matrix notation,

For second rank tensor,

3

1

3

1k l

kljlikij TRRT

cf. n-th rank : 3n components, rank 0 : scalar, rank 1 : vector

More generally,

.or 3

1

j

jiji

z

y

x

zzzyzx

yzyyyx

xzxyxx

z

y

x

ARA

A

A

A

RRR

RRR

RRR

A

A

A

1.4 Curvilinear Coordinates

1.4.1 Spherical coordinates

1.4 Curvilinear Coordinates

1.4.1 Spherical coordinates

r : distance from the origin

: polar angle

: azimuthal angle

yxφ

zyxθ

zyxr

zyxAφθrA

ˆcosˆsinˆ

.sinˆsincosˆcoscosˆ

ˆcosˆsinsinˆcossinˆ

.ˆˆˆ cf. ˆˆˆ

cos ,sinsin ,cossin

zyxr AAAAAA

rzryrx

(i) coordinate

(ii) coordinate vector

ddrdrdldldld

drrddrd

drdlrddldrdl

r

r

sin

ˆsinˆˆ

sin , ,

2

φθrl

(iii) displacement

20:

0:

0:r

.ˆsin2

1 ra ddrdldld

(iv) surface

.2/for ˆ2 θa rdrddldld r

Example 1.13 Find the volume of a sphere of radius R.

R

rddrdrd

0 0

2

0

2 sin

(v) Gradient, divergence, curl, and Laplacian

.sin

1sin

sin

11

sin

ˆsinˆˆ

sin

1

sin

1sin

sin

11

ˆsin

1ˆ1ˆ

2

2

222

2

2

2

2

2

2

T

r

T

rr

Tr

rrT

Laplacian

vrrvvr

rr

r

Curl

v

rv

rvr

rr

Divergence

T

r

T

rr

TT

Gradient

r

r

φθr

v

v

φθr

1.4.2. Cylindrical Coordinate

1.4.2. Cylindrical Coordinate

s : distance from the origin

: azimuthal angle

z

zz

yxφ

yxs

ˆˆ

.ˆcosˆsinˆ

ˆsinˆcosˆ

,sin ,cos

zzsysx

(i) coordinate

(ii) coordinate vector

dzrdrddldldld

dzsddsd

dzdlrddldsdl

zs

zs

zφsl ˆˆˆ

, ,

(iii) displacement

:

20:

0:

z

s

(iv) Gradient, divergence, curl, and Laplacian

2

2

2

2

2

2 11

ˆˆˆ

1

11

ˆˆ1

ˆ

z

TT

ss

Ts

ssTLaplacian

vrvvzs

s

rCurl

z

vv

ssv

ssDivergence

z

TT

rs

TTGradient

zs

zs

zφs

v

v

zφs

The End