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Lesson 5Vector Analysis
楊尚達 Shang-Da YangInstitute of Photonics TechnologiesDepartment of Electrical EngineeringNational Tsing Hua University, Taiwan
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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
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Outline
Vector algebraOrthogonal coordinate systems1. Cartesian2. Cylindrical3. SphericalVector calculus1. Gradient2. Divergence3. Curl4. Laplacian
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Sec. 5-1Vector Algebra
1. Addition2. Products of two vectors3. Products of three vectors
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Vector addition/subtraction
BACvvv
+=
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Dot (inner) product
ABBABA θcosvvvv
⋅=⋅],0[ π=
• Commutative:
• Distributive:
ABBAvvvv
⋅=⋅
CABACBAvvvvvvv
⋅+⋅=+⋅ )(
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Cross (outer) product
],0[ π=
• Not commutative:
• Not associative:
• Distributive:
ABn BAa
BA
θsin⋅⋅
=×vvv
vv
ABBAvvvv
×≠×
( ) ( ) CBACBAvvvvvv
××≠××
CABACBAvvvvvvv
×+×=+× )(
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Scalar triple product
( ) ( ) ( )BACACBCBAvvvvvvvvv
×⋅=×⋅=×⋅
Volume of the parallelepiped
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Vector triple product
( ) ( ) ( )BACCABCBAvvvvvvvvv
⋅⋅−⋅=××scalar scalar
vector vector
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Sec. 5-2Orthogonal Coordinate Systems
1. Cartesian2. Cylindrical3. Spherical
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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
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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.
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Application-2
,321 321AaAaAaA uuu
vvvv++= 321 321
BaBaBaB uuuvvvv
++=
vs.
321
321
321
BBBAAAaaa
BAuuuvvv
vv=×
ABn BAaBA θsin⋅⋅=×vvvvv
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Cartesian
( ) ( )zyxuuu ,,,, 321 ={ }3 2, ,1 , =ia
iuv
All base vectors
are independent of the observation point
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Cylindrical-1
( ) ( )zruuu ,,,, 321 φ=
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Cylindrical-2
2 base vectors vary with of obs. point{ }φaarvv , φ
φφ sincos yxr aaa vvv +=
P
φφφ cossin yx aaa vvv +−=
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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,
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Cylindrical-4
( ) ( )zdrzr ,,,, φφφ +→
Observation point moves along byφav φφ drdl ⋅=
Metric coefficient of φ
( ) ( )dzzddrrzr +++→ ,,,, φφφ
dzrdrddv φ=…Differential volume
φav
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Transformation of position representation-1
( ) ( )zyxPzrP ,,,, →φ
⎪⎩
⎪⎨
⎧
===
zzryrx
φφ
sincos
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Transformation of position representation-2
( )⎪⎪⎩
⎪⎪⎨
⎧
==
+=−
zzxy
yxr1
22
tanφ
( ) ( )zrPzyxP ,,,, φ→
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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 −=+°
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Spherical-1
( ) ( )φθ ,,,, 321 Ruuu =
azimuthal polar
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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 +−=
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Spherical-3
( ) ( )φθθφθ ,,,, dRR +→
observation point moves alongby
θav
θθ dRdl ⋅=
metric coefficient of θ
x y
z
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φθφ dRdl ⋅= sin
Spherical-4
( ) ( )φφθφθ dRR +→ ,,,,
observation point moves alongby
φav
metric coefficient of φ
x y
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Spherical-5
x y
( ) ( )φφθθφθ dddRRR +++→ ,,,,
…differential volumeφθθ ddRdRdv ⋅= sin2
z
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Transformation of position representation-1
( ) ( )zyxPRP ,,,, →φθ
⎪⎩
⎪⎨
⎧
===
θφθφθ
cossinsincossin
RzRyRx
x y
x y
z
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Transformation of position representation-2
( ) ( )φθ ,,,, RPzyxP →
( )( )⎪
⎪⎩
⎪⎪⎨
⎧
=
+=
++=
−
−
xy
zyx
zyxR
1
221
222
tan
tan
φ
θ
x y
z
θ
φ
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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 ++=
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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
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⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
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
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Sec. 5-3Vector Calculus
1. Gradient2. Divergence3. Curl4. Laplacian
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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.
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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
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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.
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Gradient - Proof of formula
( ) ( ) ( ) ldVdVaVaaVdndV
dldVdndlPP
lln
vvvv ⋅∇=⇒⋅∇=⋅∇=
=⇒==
,
cos ,cos31 α
α
dzzVdy
yVdx
xV
∂∂
+∂∂
+∂∂
dzadyadxa zyxvvv ++
zVa
yVa
xVaV zyx ∂
∂+
∂∂
+∂∂
=∇ vvv
By comparison, ⇒
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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
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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
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( ) ( )
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
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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
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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.
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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
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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
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( ) ( )
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
⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
+≈
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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⋅−≈
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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
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Curl - Proof of formula (4)
Find y-, z- components by evaluating the total circulation over contour , :
zyx
zyx
AAAzyx
aaaA ∂∂∂∂∂∂=×∇
vvvv
yC zC
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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
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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
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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:
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Vector Laplacian
( ) AAAvvv
×∇×∇−⋅∇∇≡∇2
( ) ( ) ( )zzyyxx AaAaAaA 2222 ∇+∇+∇=∇ vvvv
In Cartesian coordinate system:
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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)
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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
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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×∇+−∇=+=