向量分析 (Vector Analysis)
Chapter 9
Vector Integration — Line Integrals
 
 V  dr

 d r
C
C
 
 V  dr
scalar integrals
vector integral
C

dr  x̂dx  ŷdy  ẑdz
 x̂dx  x̂  dx
C
C
only in Cartesian system


d
 r  x̂  ( x, y, z )dx  ŷ  ( x, y, z )dy  ẑ  ( x, y, z )dz
C
C
C
C
The integral with respect to x cannot be evaluated unless y and z are known in
terms of x and similarly for the integrals with respect to y and z.
The path of integration C must be specified, i.e., the integral depends on the
particular choice of contour C.
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Vector Integration — Line Integrals
 
W   F  d r   Fx ( x, y, z )dx   Fy ( x, y, z )dy   Fz ( x, y, z )dz

Example : the force exerted on a body is F  x̂y  ŷx , Calculate
the work done going from the origin to the point (1,1).
  1, 1
1
1
W  0, 0 F  d r  0, 0 (  ydx  xdy )   0 ydx  0xdy
1, 1
The integrals cannot be evaluated until we specify the values of y as x and x as y !
 
1
1
W  0, 0 F  d r   0 ydx  0xdy  0  1  1
1,1
(1,1)
For this force the work done depends on the choice
of path ! (this force is a nonconservative force)
(1,0)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Vector Integration — Line Integrals
 
 V  dr
C

V  Vx x̂  Vy ŷ  Vz ẑ
If
then




dr
V

d
r

(
V

)ds
C
C
ds
dz
dy
dx
dz
dy
dx
 x̂  ( Vy  Vz )ds  ŷ  ( Vz
 Vx )ds  ẑ  ( Vx
 Vy )ds
ds
ds
ds
ds
ds
ds
此運算的物理功能並不顯著
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Vector Integration — Surface Integrals and Volume Integrals
z
The most commonly encountered form
 
 V  d


d


n


V

d


y
A flow or flux through the given surface (divergence).
Area element
x
Right-hand rule for the
positive normal
 
d  ndA
Two conventions for choosing the positive directions :
1. For closed surface, the outward normal is positive.
2. For open surface, obey the right-hand rule.
For the volume element dτ is a scalar quantity, volume integrals are somewhat simpler !

 Vd  x̂  Vxd  ŷ  Vyd  ẑ  Vzd
V
V
V
V
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Integral Definitions of Gradient, Divergence, and Curl

 d
  lim 
 d  0  d
 

V  d

  V  lim
 d0  d
 

d  V

  V  lim
 d0  d
 d
is the volume of a small region of space
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
The proof of the integral definition of gradient :

d

d  x̂   d

For surface ABDC d  x̂   d


d

  lim 
 d  0  d
For surface EFHG
is outward
z
G
C
E
A

 dx
 dx

d



x̂
(


)
dydz

x̂
(


)dydz

EFHG

ABDC
x 2
x 2
 dy
 dy
H
 ŷ AEGC ( 
)dxdz  ŷ BFHD ( 
)dxdz
y 2
y 2
D
 dz
 dz
y
 ẑ ABFE ( 
)dxdy  ẑ CDHG ( 
)dxdy
z 2
z 2
F
B
x
Differential rectangular parallelepiped
(origin at center)
d  dxdydz
Using the first two terms of a Maclaurin expansion





d


(
x̂

ŷ

ẑ
)dxdydz

 x
y
z



 ( x̂
 ŷ
 ẑ )  d
x
y
z
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
The proof of the integral definition of divergence :

d
For surface EFHG
is outward
z
G
C
E
A

d  x̂   d

For surface ABDC d  x̂   d
 

V  d
  V  lim 
 d0  d
 
Vx dx
Vx dx
V

d



(
V

)
dydz

(
V


EFHG x x 2
ABDC x x 2 )dydz
V dy
V dy
H
 AEGC ( Vy  y )dxdz  BFHD ( Vy  y )dxdz
y 2
y 2
D
V dz
V dz
y
 ABFE ( VZ  Z )dxdy  CDHG ( VZ  Z )dxdy
z 2
z 2
F
B
x
Differential rectangular parallelepiped
(origin at center)
 
Vx Vy Vz
 V  d   ( x  y  z )dxdydz
(
Vx Vy Vz


)  d
x y z
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
The proof of the integral definition of curl :
 

d  V
  V  lim 
 d0  d
Vy dx
Vy dx
 
Vz dx
Vz dx
d


V


ẑ
(
V

)
dydz

ŷ
(
V

)
dydz

ẑ
(
V

)
dydz

ŷ
(
V

)dydz

EFHG y
EFHG z
ABDC y
ABDC z
x 2
x 2
x 2
x 2
 x̂ AEGC ( Vz 
 ŷ ABFE ( Vx 
Vz dy
V dy
V dy
V dy
)dxdz  ẑ AEGC ( Vx  x )dxdz  x̂ BFHD ( Vz  z )dxdz  ẑ BFHD ( Vx  x )dxdz
y 2
y 2
y 2
y 2
V dz
V dz
Vx dz
V dz
)dxdy  x̂ ABFE ( Vy  y )dxdy  ŷ CDHG ( Vx  x )dxdy  x̂ CDHG ( Vy  y )dxdy
z 2
z 2
z 2
z 2
z
G
H
C
Vy Vx
 
Vz Vy
Vx Vz
d


V

[
x̂
(

)

ŷ
(

)

ẑ
(

)]dxdydz

 y z
z x
x y
D
y
E
F
A
 [x̂(
V V
Vz Vy
V V

)  ŷ( x  z )  ẑ( y  x )] d
y z
z x
x y
B
x
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Gauss’s Theorem
Closed surface
Gauss’s theorem states the relation between a surface integral of a function and
the volume integral of the divergence of that function.
 

 V  d     Vd
S
For example :
V
 

 E  d     Ed
S
V
 

 V  d    V d
For each parallelepiped
six surfaces
Net rate of flow out =

(  (v))dxdydz
 
V  d terms cancel (pairwise) for all interior faces
Only the contributions of the exterior surface survive.
 

 V  d  volumes
   Vd
exterior surfaces
Number   , dimensions
 

 V  d     Vd
S
V
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Green’s Theorem
If u and v are two scalar function, we have the identities :
  (uv)  u  v  (u)  (v)
  ( vu)  v  u  (v)  (u)
  (uv)    ( vu)  u  v  v  u
   ( uv )    ( vu )d   ( u  v  v  u )d
V
V

 
Gauss’s Theorem :    Vd   V  d
V
For developing
Green’s functions
S

(
u

v

v

u
)

d

  ( u  v  v  u )d

S
Green’s Theorem
V
  (uv)  u  v  (u)  (v)

u

v

d

  ( u  v )d   (u  v )d

S
V
V
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Alternate Forms of Gauss’s Theorem
 

 V  d     Vd
S
Volume integral involving divergence
V
gradient ?


Suppose V( x, y, z )  V( x, y, z )a
curl ?

a is a vector with constant magnitude
and constant but arbitrary direction.




a   Vd     aVd  a   Vd
S
V
V


a  [  Vd   Vd]  0
S
V
Volume integral involving gradient



  (fV)  (f )  V  f  V




  (Va )  (V)  a  V  a  (V)  a

Vd

  Vd

S
V
(86清華化工,70成大電機)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Alternate Forms of Gauss’s Theorem
Volume integral involving curl ?
 

S V  d  V   Vd
  
 is a vector with constant magnitude
Suppose
a
V aP
and constant but arbitrary direction.



 
S (a  P)  d  V   (a  P)d
 

     

 
 (a  P)  d   a  P  d  a   P  d  a   d  P
S
S
S
S



 



   (a  P)d   P  (  a )d   a  (  P)d  a     Pd
V
V
V
V


 
 
 
 
  (a  P)  a  (a  P)   p  (a  P)  P  (a  a )  a  ( p  P)

 
(90成大機械,88交大機械,84清華動機)
 d  P     Pd
S
V
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Stokes’s Theorem
Stokes’s theorem states the relation between a line integral of the function and the
 
 
(open) surface integral of a curl of that function.
 V  d     V  d
S
y
x0, y0+dy
3
x0+dx, y0+dy
 
 
circulatio n1234   V  d    V  d
four sides
4
2
dλ
x0, y0
1
x0+dx, y0
x
Circulation around a differential loop
 
 
 V  d     V  d
exterior line
segments
rec tan gles
Exact cancellation on interior paths; no
cancellation on the exterior path.
 
 
 V  d  S  V  d
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Alternate Forms of Stokes’s Theorem


 d  Sd  
Suppose
 
V  a

a is a vector with constant magnitude
and constant but arbitrary direction.

 
 



 a  d  S  a  d
 V  d  S  V  d
 


 a  d  a   d


 
 
 
S  (a)  d  S  a  d  S  a  d  S  a  d






  Sa    d   Sa    d  a  S  d
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Alternate Forms of Stokes’s Theorem
 


S(d  )  P   d  P
Suppose
  
V aP

a is a vector with constant magnitude
 
 
 V  d  S  V  d
and constant but arbitrary direction.

 
 

 (a  P)  d  S  (a  P)  d

    
 a  P  d  a   P  d

 


 
 
 
S  (a  P)  d  Sd    (a  P)  Sd  [P  (  a )  a  (  P)]
 


 



  Sd  a  (  P)  a  S(  P)  d  a  S(d  )  P
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Potential Theory – Scalar Potential
If a force over a given simply connected region of space S can be expressed as the
negative gradient of a scalar function φ

F  
then we call φ a scalar potential that describes the force by one function instead
of three. The force F appearing as the negative gradient of a single-valued scalar
potential is labeled a conservative force. (gravitational and electrical force)

 F  0
 
 F  dr  0
Stokes’s theorem
 
 
0   F  d r     F  d  0
for every closed path in our simply connected region S.

  F      0
 

 F  d r      d r    d  0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Potential Theory – Scalar Potential
D
A
  conservative force
ACBDA F  d r  0
 
 
 
ACB F  d r   BDA F  d r  ADB F  d r
B
C
Physically, this means that the work done in
going from A to B is independent of the path
and the work done in going around a closed
path is zero.
Energy is conserved !
possible paths for doing work
 
work done by force  A F  dr  ( A)  ( B)
B
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Example
Gravitational Potential
The gravitational force on a unit mass m1 :

Gm1m 2
k
FG  
r̂


r̂
2
2
r
r
  
r 
G (r )  G ()   FG  dr  r FG  dr

r 


G (r )  G ()   Fapplied  dr
FG   Fapplied
 
W   F  dr
The potential is the work done in bringing the unit mass in from infinity
 G ()  0
kdr
k
Gm1m 2
 G ( r )   r 2    
r
r
r

The final negative sign is a consequence of the attractive force of gravity.
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Example
Centrifugal Potential
The centrifugal force per unit mass :
φ
 
C ( r )  C (0)   0 FC  dr

FC  2 rr̂
r
φSHO
φG
φC
 C ( 0)  0

2 r 2
r 
 C ( r )   0 FC  d r  
2
r


The simple harmonic oscillation : FSHO   kr

r 
SHO (r )  SHO (0)   0 FSHO  dr
 SHO (0)  0
 kr 2
r 
 SHO ( r )   0 FSHO  d r 
2
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Thermodynamics – Exact Differentials
In thermodynamics
if
df  P(x, y)dx  Q(x, y)dy
 ( P( x, y)dx  Q( x, y)dy )
depends only on the end points
df is indeed an exact differential
The necessary and sufficient condition is that
df 
f
f
dx  dy
x
y
or
P( x , y ) 
f
f
, Q( x , y ) 
x
y
P( x , y ) Q( x , y )

y
x

F is irrotational
Fx Fy

y x

 F  0
with
Fx 
f
f
, Fy 
x
y
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Potential Theory – Vector Potential



In electromagnetic theory
A is a vector potential
B  A




B is solenoidal
 V  0  B   A  0


Suppose B  x̂b1  ŷb2  ẑb3
A  x̂a1  ŷa 2  ẑa 3


a 2 a 1
a 3 a 2
a 1 a 3

 b3

 b1

 b2
B  A
x y
y z
z x

Assuming the coordinates have been chosen so that A is parallel to the yz-plane,
that is a 1  0
a
a 2
 3  b2
 b3
x
x
x
a 2   b3dx  f 2 ( y, z )
x0
x
a 3    b2dx  f 3 ( y, z )
x0
Where f2 and f3 are arbitrary functions of y and z but are not functions of x.
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Vector Potential
Using the Leibniz formula for the derivative of an integral
h()
d h()
f ( x, )
dh ()
dg ()
f ( x, )dx  
dx  f [h(), ]
 f [g(), ]

g()
d g (  )

d
d
x
a 3
 x
f 3 ( y, z )
b
f ( y, z )
   b 2dx 
   2 dx  3
x y
y
y x
y
y
0
0
a 2  x
f 2 ( y, z ) x b3
f ( y, z )
  b3dx 

dx  2
x z
z z x
z
z
0
0
x
a 3 a 2
b b
f ( y, z ) f 2 ( y, z )

   ( 3  2 )dx  3

x
y z
z y
y
z
x
 2a 2  2a 3
f 3 f 2 x b1
f 3 f 2
 (

)dx 


dx 

x
zx yx
y z x x
y z
0
0
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Vector Potential
a 3 a 2 x b1
f f
f f


dx  3  2  b1 ( x, y, z )  b1 ( x 0 , y, z )  3  2
y z x x
y z
y z
0
Remembering that f2 and f3 are arbitrary functions of y and z, we choose
f2  0
y
f 3   b1 ( x 0 , y, z )dy
y0
a 3 a 2

 b1 ( x, y, z )  b1 ( x 0 , y, z )  b1 ( x 0 , y, z )  b1 ( x, y, z )
y z
y
x
x

A  ŷ  b3 ( x, y, z )dx  ẑ[  b1 ( x 0 , y, z )dy   b2 ( x, y, z )dx ]
x0
y0
x0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Example
A Magnetic Vector Potential For a Constant Magnetic Field

B  ẑBz
a 3 a 2

0
y z
Bz is a constant
a 1 a 3

0
z x
a 2 a 1

 Bz
x y

Assuming the coordinates have been chosen so that A is parallel to the yz-plane,
that is a 1  0
y
x
x
x

A  ŷ  b3 ( x, y, z )dx  ẑ[  b1 ( x 0 , y, z )dy   b2 ( x, y, z )dx ]  ŷ  Bz dx  ŷxBz
x0
y0
x0

Assuming the coordinates have been chosen so that A is parallel to the xy-plane,
that is a 3  0
a 2
0
z
a 1
0
z
a 2 a 1

 Bz
x y
a 1  a 1 ( x, y)
a 2  a 2 ( x, y)
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
x
a 2  p  Bz dx  pxB z
y
a1  ( p  1)  Bz dy  ( p  1) yB z

A  x̂(p  1)yBz  ŷpxBz
with p any constant

A  ŷxBz

A is not unique

1
1
A   ( x̂y  ŷx ) Bz  ( p  )( x̂y  ŷx ) Bz
2
2
1
1
  xy
  ( x̂y  ŷx ) Bz  ( p  ) Bz 
2
2
1  
 (B  r )
2

' 
A  A  A  
a gauge transformation
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Gauge covariant derivative
The Schrödinger equation without magnetic induction field
{
1
( i) 2  V  E} 0  0
2m
The Schrödinger equation with magnetic induction field
 2
1
{ ( i  eA)  V  E}  0
2m

ie r   '  '

 ( r )  exp[  A( r )  d r ] 0 ( r )




ie r
'
ie 
( i  eA)  exp[  A  d r ]{( i  eA) 0  i 0 A}



ie r

 exp[  A  d r ' ]( i 0 )

Gauge covariant derivative
e  describes the coupling of a charged particle with the magnetic field.
  i( ) A

Minimal substitution
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Gauss’s Law
q

E
qr̂
40 r 2
Gauss’s theorem
s
 

 V  d     Vd
S
 
E
  d  0
V

q r̂  d
q
r̂



(
)d  0


40 S r 2
40 V
r2
S
Gauss’s law
q
s
  q
E
S  d 
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Gauss’s Law


r̂  d
r̂  d '
S r 2    2  0
S

d'  r̂2d
S’  spherical

r̂  d '
r̂  r̂ 2d
  2     2  4 
S
S
z
r̂
'
s’
s
q
'
 
q
q
E

d


4


S
40
0
y
x
q   d
Exclusion of the origin
'
V
 


E

d




E
d


S
V
V d
0
 
E 
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Poisson’s Equation
 
E 
0

E  
    

0
Poisson’s equation
0
    0
Laplace’s equation
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Dirac Delta Function
Example 1.6.1
 

Gauss’s theorem  V  d     Vd
df
f ( r )  r̂
dr
S
 4
1
r̂
V   ( r )d   V   ( r 2 )d  { 0
V
Include the origin
Not include the origin
1

 2 ( )  4( r )  4( x )( y)( z )
r
Dirac delta function :
( x )  0, x  0

f (0)   f ( x )( x )dx

 (x)dx  1

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
δ-sequence function

 (x)dx  1

0
y   n ( x )  {n
0
x
1
2n
1
1

x
2n
2n
1
x
2n
x
Convenient to differentiate
Hermite polynomials
y  n (x) 
x
n
exp( n 2 x 2 )

y  n (x) 
n
1

 1  n2x 2
Lorentzian
x
y  n (x) 
sin nx 1 n ixt

e dt

n
x
2
Useful in Fourier analysis
and quantum mechanics
1
sin[(
n

)x ]
1
2
n (x) 
2 sin( 1 x )
2
x
Dirichlet kernel
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
δ function



 ( x )f ( x )dx  lim
  n ( x )f ( x )dx  f (0)
n   
 (x)dx  1

From a mathematical point of view, lim  n ( x ) do not exist.
n 
The Dirac delta function must be even in x, ( x )  ( x )
1
(ax )  ( x ), a  0
a
( x  a )
'
g
(a )
g ( a ) 0
( g( x ))  
a,
g ' ( a ) 0
1  y
1
f
(
x
)

(
ax
)
dx

f
(
)

(
y
)
dy

f ( 0)


a  a
a

g( x )  g(a )  ( x  a )g' (a )  ( x  a )g' (a )

a 
 f ( x )(g( x ))dx    f ( x )(( x  a )g (a ))dx

a

a
'
a 
1 a 
f ( x )( x  a )dx

'
a 
g (a )
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
δ function

'
'
f
(
x
)

(
x

x
)
dx

f
(
x
)



'
'
f
(
x
)

(
x

x
)
dx

f
(
x
)

(
x

x
)dx  0



'
'

'
'
'
'
f
(
x
)

(
x

x
)
dx


f
(
x
)

(
x

x
)
dx


f
(
x
)


'
'
The definition of the derivative
A linear operator
 ( x  x )dx
0
' ( x )
f (x)
f (x 0 )

£(x0)f ( x )   ( x  x 0 )f ( x )dx  f ( x 0 )
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Uniqueness Theorem
A vector is uniquely specified by giving its divergence and its curl within a simply
connected region and its normal component over the boundary.

  V1  s
 
A source (charge) density   V1  c

 
Assuming
  V2  s
  V2  c


  
W  0
Let W  V1  V2
W  0
Laplace equation
using the Green’s theorem

W  
    0

u

v

d

  ( u  v )d   (u  v )d

S
Wn = V1n-V2n = 0
on the boundary
A circulation (current) density
V
V

S   d  V (   )d  V (   )d
  
 
W  V1  V2  0
0  0   (  )d   W  Wd
V
V
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Helmholtz’s Theorem


 
A vector V satisfying   V  s ,   V  c , with both source and circulation
densities vanishing at infinity may be written as the sum of the two parts, one of
which is irrotational, the other solenoidal.


V      A
irrotational

V is a known vector


  V  s( r )
solenoidal
  
  V  c( r )
We construct a scalar potential and a vector potential

 
 

1 s( r2 )
1 c( r2 )
( r1 ) 
d2
A( r1 ) 
d 2


4 r12
4 r12



If s = 0, then V is solenoidal
0
V  A
y
x
x

A  ŷ  b3 ( x, y, z )dx  ẑ[  b1 ( x 0 , y, z )dy   b2 ( x, y, z )dx ]
y

 x
 x
A0
If c = 0, then V is irrotational
V  
0
0
0
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Helmholtz’s Theorem
z
Field point
(x1,y1,z1)

r1

r12
Source point
(x2,y2,z2)

r2
If we can show that

  V    


  V    (  A)


   ( r1 )  s( r1 )
 
 
  (  A( r1 ))  c( r1 )
  
  V  c( r )


  V  s( r )


1
s( r2 )
1

1
  V         
d 2    s( r2 )12 ( )d2
4
r12
4
r12
y
x


V      A
Include the origin
 4
1
r̂
V   ( r )d   V   ( r 2 )d  { 0
Not include the origin
 
 
( r1  r2 )  0, r1  r2
1
 
2 ( )  4( r1  r2 )
  

f
(
r
)

(
r

r
)
d


f
(
r2 )
r12
 1 1 2 2
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Helmholtz’s Theorem
r12  [( x1  x 2 )2  ( y1  y2 )2  (z1  z 2 )2 ]1/ 2
1
1
 
 
12 ( )  22 ( )  4( r1  r2 )  4( r2  r1 )
r12
r12

1

1
1

1
1

 
  V    s( r2 )12 ( )d2    s( r2 )22 ( )d2    s( r2 )( 4)(r2  r1 )d 2
4
r12
4
r12
4

 s( r1 )


V      A





V

s
(
r)

1 s( r2 )
( r1 ) 
d2

4 r12
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
Chapter 9
向量分析 (Vector Analysis)
Helmholtz’s Theorem
 






1
c
( r2 )
A( r1 ) 
d 2
  V    (  A)    A  2 A

4 r12
 
 
1
4  A( r1 )   c( r2 )  11 ( )d 2
r12

 
 1
   1
   1
4  A x   c( r2 )  2
( )d2   2  [c( r2 )
( )]d2   [2  c( r2 )]
( )]d2  0
x 2 r12
x 2 r12
x 2 r12
=0
=0



1   2 1
1

1
2
  V   A    c( r2 )1 ( )d2
  V    s( r2 )12 ( )d 2
4
r12
4
r12
  
  V  c( r1 )


V      A
  
 
 
1 c( r2 )
  V  c( r )
A( r1 ) 
d


2
4 r12
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung
向量分析 (Vector Analysis)
Chapter 9
Helmholtz’s Theorem


 
A vector V satisfying   V  s ,   V  c , with both source and circulation
densities vanishing at infinity may be written as the sum of the two parts, one of
which is irrotational, the other solenoidal.


V      A
irrotational
solenoidal
Applied to the electromagnetic field
Irrotational electric field

E

Solenoidal magnetic induction field B

Source density s( r )
 
circulation density c( r )

E  


B  A
electric charge density divided by electric permittivity 
electric current density times magnetic permeability 
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung