lesson 5 vector analysis - 清華大學電機系-nthueesdyang/courses/em/lesson05_slides.pdf ·...
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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×∇+−∇=+=