lesson 5 vector analysis - 清華大學電機系-nthueesdyang/courses/em/lesson05_slides.pdf ·...

Lesson 5Vector Analysis

楊尚達 Shang-Da YangInstitute of Photonics TechnologiesDepartment of Electrical EngineeringNational Tsing Hua University, Taiwan

Introduction

Physical quantities in EM could be:1. Scalar: charge, current, energy 2. Vector: EM fieldsEM laws are independent of coordinate system of useVector analysis is not necessary for EM, but can lead to elegant formulations

Outline

Vector algebraOrthogonal coordinate systems1. Cartesian2. Cylindrical3. SphericalVector calculus1. Gradient2. Divergence3. Curl4. Laplacian

Sec. 5-1Vector Algebra

1. Addition2. Products of two vectors3. Products of three vectors

Vector addition/subtraction

BACvvv

+=

Dot (inner) product

ABBABA θcosvvvv

⋅=⋅],0[ π=

• Commutative:

• Distributive:

ABBAvvvv

⋅=⋅

CABACBAvvvvvvv

⋅+⋅=+⋅ )(

Cross (outer) product

],0[ π=

• Not commutative:

• Not associative:

• Distributive:

ABn BAa

BA

θsin⋅⋅

=×vvv

vv

ABBAvvvv

×≠×

( ) ( ) CBACBAvvvvvv

××≠××

CABACBAvvvvvvv

×+×=+× )(

Scalar triple product

( ) ( ) ( )BACACBCBAvvvvvvvvv

×⋅=×⋅=×⋅

Volume of the parallelepiped

Vector triple product

( ) ( ) ( )BACCABCBAvvvvvvvvv

⋅⋅−⋅=××scalar scalar

vector vector

Sec. 5-2Orthogonal Coordinate Systems

1. Cartesian2. Cylindrical3. Spherical

Definition

A point is located as the intersection of 3 mutually perpendicular surfaces

3 base vectors satisfy with:{ }iuav

consant=iu

,kji uuu aaa vvv =× )2,1,3(),1,3,2(),3,2,1(),,( =kji

⎩⎨⎧

=≠

==⋅jiji

aa ijuu ji if ,1 if ,0

δvv

Application-1

,321 321AaAaAaA uuu

vvvv++= 321 321

BaBaBaB uuuvvvv

++=

( ) ( )321321 321321BaBaBaAaAaAaBA uuuuuu

vvvvvvvv++⋅++=⋅

( ) ( ) ( )332211

332111 332111...

BABABA

BAaaBAaaBAaa uuuuuu

++=

⋅+⋅+⋅= vvvvvv 1 1

[ ] ∑=

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡×=⋅

3

13

2

1

321 i

ii BABBB

AAABAvv

ABBABA θcosvvvv

⋅=⋅

vs.

Application-2

,321 321AaAaAaA uuu

vvvv++= 321 321

BaBaBaB uuuvvvv

++=

vs.

321

321

321

BBBAAAaaa

BAuuuvvv

vv=×

ABn BAaBA θsin⋅⋅=×vvvvv

Cartesian

( ) ( )zyxuuu ,,,, 321 ={ }3 2, ,1 , =ia

iuv

All base vectors

are independent of the observation point

Cylindrical-1

( ) ( )zruuu ,,,, 321 φ=

Cylindrical-2

2 base vectors vary with of obs. point{ }φaarvv , φ

φφ sincos yxr aaa vvv +=

P

φφφ cossin yx aaa vvv +−=

Cylindrical-3

What does -dependent base vector mean?φ

φφφ sincos)( yxrr aaaa vvvv +==e.g.

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=∂∂

⇒ rrrrrr AaAazrAaφφ

φφ

vvv ),,(

φφφ aaa yxvvv =+− cossin

⎟⎠⎞

⎜⎝⎛∂∂

=∂∂

xxxx Ax

azyxAax

vv ),,(In contrast,

Cylindrical-4

( ) ( )zdrzr ,,,, φφφ +→

Observation point moves along byφav φφ drdl ⋅=

Metric coefficient of φ

( ) ( )dzzddrrzr +++→ ,,,, φφφ

dzrdrddv φ=…Differential volume

φav

Transformation of position representation-1

( ) ( )zyxPzrP ,,,, →φ

⎪⎩

⎪⎨

⎧

===

zzryrx

φφ

sincos

Transformation of position representation-2

( )⎪⎪⎩

⎪⎪⎨

⎧

==

+=−

zzxy

yxr1

22

tanφ

( ) ( )zrPzyxP ,,,, φ→

Transformation of vector components

zzyyxxzzrr AaAaAaAaAaAaV vvvvvvv++=++= φφ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

z

r

z

y

x

AAA

AAA

φφφφφ

1000cossin0sincos

( )zxzxrxr

xzzrrxx

AaaAaaAaaaAaAaAaaVA)()()( vvvvvv

vvvvvv

⋅+⋅+⋅=

⋅++=⋅=

φφ

φφ

( ) φφ sin90cos −=+°

Spherical-1

( ) ( )φθ ,,,, 321 Ruuu =

azimuthal polar

Spherical-2

3 base vectors vary withof the observation point

{ }φθ ,{ }φθ aaaRvvv ,,

θφθφθ

cossinsincossin

zy

xR

aaaa

vv

vv

++=

θφθφθθ

sinsincoscoscos

zy

x

aaaa

vv

vv

−+=

φφφ cossin yx aaa vvv +−=

Spherical-3

( ) ( )φθθφθ ,,,, dRR +→

observation point moves alongby

θav

θθ dRdl ⋅=

metric coefficient of θ

x y

z

φθφ dRdl ⋅= sin

Spherical-4

( ) ( )φφθφθ dRR +→ ,,,,

observation point moves alongby

φav

metric coefficient of φ

x y

Spherical-5

x y

( ) ( )φφθθφθ dddRRR +++→ ,,,,

…differential volumeφθθ ddRdRdv ⋅= sin2

z

Transformation of position representation-1

( ) ( )zyxPRP ,,,, →φθ

⎪⎩

⎪⎨

⎧

===

θφθφθ

cossinsincossin

RzRyRx

x y

x y

z

Transformation of position representation-2

( ) ( )φθ ,,,, RPzyxP →

( )( )⎪

⎪⎩

⎪⎪⎨

⎧

=

+=

++=

−

−

xy

zyx

zyxR

1

221

222

tan

tan

φ

θ

x y

z

θ

φ

Transformation of vector components

zzyyxxRR AaAaAaAaAaAaV vvvvvvv++=++= φφθθ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

φ

θ

θθφφθφθφφθφθ

AAA

AAA R

z

y

x

0sincoscossincossinsinsincoscoscossin

( )φφθθ

φφθθ

AaaAaaAaaaAaAaAaaVA

xxRxR

xRRxx

)()()( vvvvvv

vvvvvv

⋅+⋅+⋅=

⋅++=⋅=

θφθφθ cossinsincossin zyxR aaaa vvvv ++=

Transformation of position vector (cylindrical)

( ) ( ) ( )zrAAAzaraPzrP zrzr ,0,,, , means ,, =⇒+= φφ vvv

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

zrr

z

r

AAA

z

y

x

φφ

φφφφ

sincos

01000cossin0sincos

⎪⎩

⎪⎨

⎧

===

zzryrx

φφ

sincos

consistent with

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

θφθφθ

θθφφθφθφφθφθ

cossinsincossin

00

0sincoscossincossinsinsincoscoscossin

RRR

R

AAA

z

y

x

Transformation of position vector (spherical)

( ) ( ) ( )0,0,,, , means ,, RAAARaPRP RR =⇒= φθφθ vv

⎪⎩

⎪⎨

⎧

===

θφθφθ

cossinsincossin

RzRyRx

consistent withx y

z

Sec. 5-3Vector Calculus

1. Gradient2. Divergence3. Curl4. Laplacian

Overview: Derivative of functions

1D only has 1 variable and 2 directions (±x), can be represented by the sign of function values (e.g. the force function exerted on an object).≥2D has multiple variables and infinitely many directions:

: slope, rate of change

Slope of scalar function depends on the choice of differential path.“Slope” of vector function is even complicated, for the change of function “value” is a vector.

Gradient - Definition & meaning

The gradient of a scalar field V is a vector field whose magnitude and direction describes the max. space rate of increase of V

dndVaV n

v≡∇

Max. change of Voccurs along the normal direction of the equi-potential surface (zero change)

Equi-potential surfaces, nearly parallel if dV → 0

Gradient - 2D example

A scalar function of two variables V(x,y) can be fully represented by a contour plot:

Equi-potential lines, nearly parallel if dV→0

(x0, y0)

At (x0,y0), gradient is a vector along the steepest ascent direction, and the magnitude is the slope of V(x,y) in that direction.

Gradient - Proof of formula

( ) ( ) ( ) ldVdVaVaaVdndV

dldVdndlPP

lln

vvvv ⋅∇=⇒⋅∇=⋅∇=

=⇒==

,

cos ,cos31 α

α

dzzVdy

yVdx

xV

∂∂

+∂∂

+∂∂

dzadyadxa zyxvvv ++

zVa

yVa

xVaV zyx ∂

∂+

∂∂

+∂∂

=∇ vvv

By comparison, ⇒

Divergence - Definition & meaning

The divergence of a vector field is a scalar field describing the net outward flux per unit volume

v

sdAA S

v Δ

⋅≡⋅∇ ∫

→Δ

0

limvv

v

Av

…flow source( ) 0>⋅∇ Av

…flow sink( ) 0<⋅∇ Av

Divergence - Proof of formula (1)

zzyyxx AaAaAazyxA

vvv

v

++=),,(

2: 01

xxxS Δ+=

∫∫ ⋅=1 1 ),,(

SsdzyxAF vv

zyasd x ΔΔ= vv

),,2( 000 zyxxA Δ+≈v

( ) ( )zyzyxxAF x ΔΔ⋅Δ+≈ 0001 ,,2

( )2

,, 000x

xAzyxA

P

xx

Δ⎥⎦

⎤⎢⎣

⎡∂∂

+≈

( ) ( )

2

,, 0001

zyxxA

zyzyxAF

P

x

x

ΔΔΔ⎥⎦

⎤⎢⎣

⎡∂∂

+

ΔΔ⋅≈

vΔ=

zyasd x ΔΔ= vv

( ) ( )

2

,, 0002

vxA

zyzyxAF

P

x

x

Δ⎥⎦

⎤⎢⎣

⎡∂∂

+

ΔΔ⋅−≈

Divergence - Proof of formula (2)

2: 02

xxxS Δ−=

∫∫ ⋅=2 2 ),,(

SsdzyxAF vv

zyasd x ΔΔ−= vv

),,2( 000 zyxxA Δ−≈v

( ) ( )zyzyxxAF x ΔΔ−⋅Δ−≈ 0002 ,,2

( )2

,, 000x

xAzyxA

P

xx

Δ⎥⎦

⎤⎢⎣

⎡∂∂

−≈

zyasd x ΔΔ−= vv

Divergence - Proof of formula (3)

( )v

xAFF

zyxP

x Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

=+⇒000 ,,

21

vzA

yA

xAFsdA

zyxP

zyx

nnS

Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+∂∂

+∂∂

==⋅ ∑∫= ),,(

6

1

000

vv

Total flux of the cuboid:

⇒Δ

⋅≡⋅∇ ∫

→Δ ,lim

0 v

sdAA S

v

vvr

zA

yA

xAA zyx

∂∂

+∂∂

+∂∂

=⋅∇v

Divergence theorem

impliesv

sdAA S

v Δ

⋅≡⋅∇ ∫

→Δ

0lim

vvr ( )∫∫ ⋅∇=⋅

VSdvAsdA

vvv

microscopic macroscopic

Contributions of flux from the internal surfaces will cancel with one another.

Curl - Definition & meaning (1)

),,( zyxAv

Circulation: the work done by the force in moving some object (energy obtained by the object) around a contour

at point P,

⇒ is a non-conservativeforce, forming a vortexsource at P that drives flow circularly.

Av

C

∫ ⋅≡C

ldA nCirculatio

vv

ldv 0

≠⋅∫C

ldAvv

Av

Curl - Definition & meaning (2)

The curl of a vector field is a vector field whose: (1) magnitude: the net circulation per unit area,(2) direction: normal direction of a contour that maximizes the circulation.

Av

s

ldAaA Cn

s Δ

⎟⎠⎞⎜

⎝⎛ ⋅

≡×∇∫

→Δ

max

0

lim

vvvv

( ) uu

C

saA

s

ldAu

u

vvvv

⋅×∇=Δ

⋅⇒

∫→Δ

0

lim

( ) ( )

2

,, 0001

zyyA

zzyxAW

P

z

z

ΔΔ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+

Δ⋅≈

Curl - Proof of formula (1)

zzyyxx AaAaAazyxA

vvv

v

++=),,(

xC

),2,( 000 zyyxA Δ+≈v

sΔ=

∫ ⋅=1 1 ),,( ldzyxAW

vvzald zΔ= vv

( ) ( )zzyyxAW z Δ⋅Δ+≈ 0001 ,2,

( )2

,, 000y

yAzyxA

P

zz

Δ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+≈

Curl - Proof of formula (2)

xC

),2,( 000 zyyxA Δ−≈v

∫ ⋅=3 3 ),,( ldzyxAW

vvzald zΔ−= vv

( )2

,, 000y

yAzyxA

P

zz

Δ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−≈

( ) ( )zzyyxAW z Δ⋅Δ−−≈ 0003 ,2,

( ) ( )

2

,, 0003

syA

zzyxAW

P

z

z

Δ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+

Δ⋅−≈

Curl - Proof of formula (3)

Total circulation of the contour :

( )s

yAWW

zyxP

z Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

=+⇒000 ,,

31

( )s

zA

yAldA

zyxP

yz Δ⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−∂∂

=⋅∫000 ,,

3412

vv

xC

( )( )⎥

⎥⎦

⎤

⎢⎢⎣

⎡

∂

∂−

∂∂

=Δ

⋅=⋅×∇⇒

∫→Δ

000 ,,

0

lim

zyxP

yzC

sx zA

yA

s

ldAaA x

vv

vv

Curl - Proof of formula (4)

Find y-, z- components by evaluating the total circulation over contour , :

zyx

zyx

AAAzyx

aaaA ∂∂∂∂∂∂=×∇

vvvv

yC zC

Stokes’ theorem

implies

microscopic macroscopic

Contributions of work from the internal boundaries will cancel with one another.

( )∫∫ ⋅×∇=⋅SC

sdAldA

vvvv

s

ldAaA Cn

s Δ

⎟⎠⎞⎜

⎝⎛ ⋅

≡×∇∫

→Δ

max

0

lim

vvvv

2nd-order derivative

ΔΔ−−Δ+

=→Δ

)2/()2/(lim0

xfxfdxdf

Consider a scalar function of single variable : )(xf

( )ffxfxfxf

xfxfxfxf

xfxfdx

fd

−∝⎥⎦⎤

⎢⎣⎡ −

Δ−+Δ+Δ

=

⎥⎦⎤

⎢⎣⎡

ΔΔ−−

−Δ−Δ+

Δ=

ΔΔ−′−Δ+′

=

→Δ

→Δ

→Δ

)(2

)()(2lim

)()()()(1lim

)2/()2/(lim

20

0

02

2

Scalar Laplacian

( )VV ∇⋅∇≡∇ 2

zVa

yVa

xVaV zyx ∂

∂+

∂∂

+∂∂

=∇ vvv

zA

yA

xAA zyx

∂∂

+∂∂

+∂∂

=⋅∇v

2

2

2

2

2

22

zV

yV

xVV

∂∂

+∂∂

+∂∂

=∇

In Cartesian coordinate system:

Vector Laplacian

( ) AAAvvv

×∇×∇−⋅∇∇≡∇2

( ) ( ) ( )zzyyxx AaAaAaA 2222 ∇+∇+∇=∇ vvvv

In Cartesian coordinate system:

Null identities

( ) 0=∇×∇ V A conservative (curl-free) vector field can be expressed as the gradient of a scalar field (electrostatic potential)

( ) 0=×∇⋅∇ Av A solenoidal (divergence-

free) vector field can be expressed as the curl of another vector field (magnetostatic potential)

Helmholtz’s theorem-1

A vector field is uniquely determined if both its divergence and curl are specified everywhere

⎩⎨⎧

=×∇

=⋅∇

)()(rGE

rgEvvv

vv

)(rE vv

In other words, a vector field is completely determined by its flow source and vortex source .

)(rg v

)(rG vv

Helmholtz’s theorem-2

scalar potential of

A vector field can be decomposed into:

( ) 0=∇×∇ V

( ) 0=×∇⋅∇ Av

Curl-free (irrotational) component , with:iFv

⎪⎩

⎪⎨⎧

=×∇

=⋅∇

0i

i

FgF

v

v

VFi −∇=v

Fv

vector potential of

Divergence-free (solenoidal) component , with:sFv

⎪⎩

⎪⎨⎧

=×∇

=⋅∇

GF

F

s

svv

v0

AFs

vv×∇=

Fv

Fv

AVFFF si

vvvv×∇+−∇=+=