Chapter 7 Vector Analysis
7-1 Vector Functions

One-variable vector function: R(t )  x(t )iˆ  y(t ) ˆj  z(t )kˆ

Multi-variable vector function: F ( x, y, z)  F1 ( x, y, z)iˆ  F2 ( x, y, z) ˆj  F3 ( x, y, z)kˆ


dR(t ) dR(t ) dt
dR(t ) df (t )
The derivatives of vector functions:



/
df (t )
dt df (t )
dt
dt




F ( x, y, z ) F ( x, y, z )
x
F ( x, y, z )
y
F ( x, y, z )
z






g ( x, y, z )
x
g ( x, y, z )
y
g ( x, y, z )
z
g ( x, y, z )



Eg. For R(t )  2tiˆ  cos(3t ) ˆj  t 3 kˆ , 0  t  1, find R' (t )  dR(t ) / dt . Let s(t)=

1
2
4
,
then
find
d
R
(t ) / ds(t ) .
4

9
sin
(
3
t
)

9
t

dt
0

(Sol) dR(t ) / dt  2iˆ  3 sin( 3t ) ˆj  3t 2 kˆ


dR(t ) dR(t ) dt
2iˆ  3 sin( 3t ) ˆj  3t 2 kˆ




ds (t )
dt ds (t )
4  9 sin 2 (3t )  9t 4
Some theorems of derivatives of vector functions:
 
   
1. ( F  G)'  F 'G  F  G'
 
   
2. ( F  G)'  F 'G  F  G'
 
 
3. ( F  F ' )'  F  F ' '
 
   
 
(Proof) ( F  F ' )'  F 'F ' F  F "  F  F "


4. For R(t )  x(t )iˆ  y(t ) ˆj  z(t )kˆ , if R(t ) does not change direction, then


R(t )  R' (t )  0 , and vice versa.

5. Let R(t ) denote the position of a particle at time t. If the particle moves so


that equal areas are swept out in equal times, then we have R(t )  R" (t )  0 , and
vice versa. (Kepler's law)
(Proof)


1
Area  R 2 and | R(t  t )  R(t ) | R
2



∴ 2 area  R 2  R(t )  [ R(t  t )  R(t )]





If area  0  R(t )  [ R(t  t )  R(t )]  0  R(t )  R' (t )  t  0


 R(t )  R' (t )  0
Equal area in equal time






 R(t )  R' (t )  constant  [ R(t )  R' (t )]'  0  R(t )  R" (t )  0
53
～～
7-2 Differential Geometry

Position vector: R (t )  x (t ) iˆ  y (t ) ˆj  z (t ) kˆ

dx(t ) ˆ dy (t ) ˆ dz (t ) ˆ

Velocity: v (t )  R (t ) 
i
j
k
dt
dt
dt

t2 
ds

Arc length: s (t )   | R (t ) | dt ,
| R (t ) || v (t ) |
t1
dt
2
2
2
d x (t ) ˆ d y (t ) ˆ d z (t ) ˆ


Acceleration: a (t )  v (t ) 
i
j
k
dt 2
dt 2
dt 2



R(t )
v (t )
dR(t )
dTˆ
ˆ
Curvature:  
, where T  
 

ds
| R(t ) | | v (t ) | ds(t )

Eg. C: R(t )  t iˆ  (t  2) ˆj  (3t  1)kˆ is a straight line.

v
iˆ  ˆj  3kˆ
dTˆ
dTˆ dt
ˆ
,  


 0.
T  
ds
dt ds
|v |
11

Eg. C : R(t )  2 cos(t ) iˆ  2 sin( t ) ˆj  4kˆ is a circle of radius 2 at z=4.
 2 sin( t ) iˆ  2 cos (t ) ˆj
dTˆ dt
1 1
Tˆ 
,  

  .
2
dt ds
2 r


 d | v | ˆ | v |2 ˆ
Theorem a 
T
N  tangential acceleration+centripetal acceleration
dt



d | v (t ) | ˆ 
dTˆ
dv (t ) d 

ˆ
 T  | v (t ) | 
 [| v (t ) | T ] 
(Proof) a (t ) 
dt
dt
dt
dt


ˆ
ˆ
 ds dTˆ  d | v (t )
d | v (t ) | ˆ 
ˆ  | v (t ) | 2 dT . Define Nˆ   dT


T

 T  | v (t ) |   

dt
ds
ds
dt
 dt ds 

 2
d | v (t ) | ˆ | v (t ) | ˆ
d (Tˆ  Tˆ )

0
 a (t ) 
T
N ( Nˆ  Tˆ  Tˆ  Tˆ  1) ,
ds
dt


Eg. For R(t )  [cos(t )  t sin( t )]iˆ  [sin( t )  t cos(t )] ˆj  t 2 kˆ , t>0, we have

v (t )  t cos(t )iˆ  t sin( t ) ˆj  2tkˆ

a (t )  [cos(t )  t sin( t )]iˆ  [sin( t )  t cos(t )] ˆj  2kˆ

v (t )
1
1
2 ˆ
ˆ
T

cos(t )iˆ 
sin( t ) ˆj 
k
| v(t ) |
5
5
5

1


1
dTˆ dt

dt ds

1
dTˆ
1
 
dt | v (t ) |
 5t
dTˆ
dt dTˆ
 dTˆ
Nˆ  
 
 

  sin (t ) iˆ  cos (t ) ˆj
ds
ds dt | v (t ) | dt

∴ a (t )  5 Tˆ  t Nˆ , at= 5 , an=t
54
～～
Binormal vector: Bˆ  Tˆ  Nˆ
 dTˆ
Nˆ
 Nˆ 
(1)

ds


 dNˆ
Fernet formulae: 
 Tˆ  Bˆ
( 2)
ds

 dBˆ
 Nˆ
(3)

 ds
Torsion of a curve: τ
dBˆ
d ˆ ˆ
dTˆ ˆ ˆ dNˆ
 Tˆ  ( Tˆ )  Tˆ  (Bˆ )  Nˆ
(Proof of (3))

(T  N ) 
 N T 
ds ds
ds
ds
Note: Tˆ , Nˆ , and Bˆ are unit vectors.
Basic theorems of curvature and torsion:
 
  
  
| R   R  |
1   
 3 , τ= [Tˆ , Nˆ , Nˆ ' ]  2 [ R , R , R ] , where [ A, B, C ]  A  ( B  C )
κ=

| R |

Eg. For C : R (t )  3 cos(t ) iˆ  3 sin (t ) ˆj  4t kˆ , find Tˆ , Nˆ , Bˆ , κ, τ, and ρ.
 

  R  | 3
|
R
1 25
v
3
3
4
(Sol.) Tˆ     sin( t ) iˆ  cos(t ) ˆj  kˆ ,  

, 
3
25

3
|v |
5
5
5
| R |
dTˆ

dTˆ
4
4
3
Nˆ  
 

  cos (t ) iˆ  sin (t ) ˆj , Bˆ  Tˆ  Nˆ  sin (t ) iˆ  cos (t ) ˆj  kˆ
5
5
5
ds | v (t ) | dt
dBˆ
1
dBˆ
4
4
 

cos (t ) iˆ  sin (t ) ˆj   Nˆ   ( cos ( t ) î  sin ( t ) ĵ )
ds | v (t ) | dt 25
25

4
25
Eg. Consider the curve: rˆ  a cos (t ) iˆ  a sin (t ) ˆj  bt kˆ , 0  t  2π. What is the
equation of tangential vector at t=π/2. [1991 中山機研]


 dr

 a sin (t ) iˆ  a cos (t ) ˆj  bkˆ . At t   v  a iˆ  bkˆ
(Sol.) v 
dt
2

v
 aiˆ  bkˆ
Tˆ   
|v |
a2  b2
諾貝爾物理獎得主楊振寧作詩「贊陳氏級」稱讚數學大師/美國加州柏克萊大學
教授陳省身對微分幾何學的貢獻：「天衣豈無縫，匠心剪接成，渾然歸一體，廣
邃妙絕倫。造化愛幾何，四力纖維能，千古寸心事，歐高黎卡陳。」
55
～～
7-3 Gradient, Divergence, and Curl in the Rectangular Coordinate System
 ( x, y, z ) ˆ  ( x, y, z ) ˆ  ( x, y, z ) ˆ
i
j
k is a
x
y
z
vector function if  ( x, y, z ) is a scalar function.



Eg. For  ( x, y, z )  e xyz ,  ( x, y, z )  yze xyz i  xze xyz j  xye xyz k .
Gradient  ( x, y, z ) :  ( x, y, z ) 


 F1 ( x, y, z )  F2 ( x, y, z )  F3 ( x, y, z )
Divergence   F ( x, y, z ) :   F ( x, y, z ) 


x
y
z

is a scalar function if F ( x, y, z)  F1 ( x, y, z) iˆ  F2 ( x, y, z) ˆj  F3 ( x, y, z)kˆ is a vector
function.

Curl   F ( x, y, z ) :
kˆ
 F F 
 F F   F F 

  3  2  iˆ   1  3  ˆj   2  1  kˆ is a vector
z y z 
z x   x y
F3

function if F ( x, y, z)  F1 ( x, y, z, ) iˆ  F2 ( x, y, z) ˆj  F3 ( x, y, z) kˆ is a vector function.



Eg. F  x iˆ  y ˆj  z kˆ , compute   F and   F .
iˆ


F 
x
F1
ˆj

y
F2
ˆj
iˆ
kˆ





 0iˆ  0 ˆj  0kˆ  0
(Sol.)   F  1  1  1  3 ,   F 
x y z
x
y
z



Eg. F ( x, y, z )  x 2 iˆ  2 x 2 yˆj  2 yz 4 kˆ , find   F and   F at (1,-1,1). [1991 中
山電研]


(Sol.)   F  2 x  2 x 2  8 yz 3 , at (1,1,1)    F  8


  F  2 z 4 iˆ  4 xykˆ, at (1,1,1)    F  2iˆ  4kˆ
Laplacian operator:  2      (
2
2
2


)
 x2  y2  z2
Eg. For  ( x, y, z )  e xyz ,  2 ( x, y, z )  y 2 z 2 e xyz  x 2 z 2 e xyz  x 2 y 2 e xyz .



Theorems (a)   (  F )  (  F )   2 F


(b)   (  F )  0,  F  C 2
(c)   ( )  0,   C 2


 
 


 
(d)   ( F  G)  (G  ) F  ( F  )G  (  G) F  (  F )G




 

 

(e) ( F  G)  ( F  ) G  (G  ) F  F  (  G)  G  (  F )


 


(f)   ( F  G)  G  (  F )  F  (  G)



(g)   ( F )  ( )  F   (  F )
56
～～
Theorem  ( x, y, z )  the surface of  (x,y,z)=constant.
(Proof) ∵  (x,y,z)=constant
∴ d  ( x, y , z )  0 
  ( x, y , z )
  ( x, y , z )
  ( x, y , z )
dx 
dy 
dz
x
y
z

  ˆ  ˆ  ˆ
 
i
j
k   (dxiˆ  dyˆj  dzkˆ)    dR
y
z 
x


∴  ( x, y, z )  dR , and dR is the tangential increment on the surface  (x,y,z)
Eg. Find the tangential plane and normal line to z=x2+y2 at (2,-2,8).
(Sol.) Let  (x,y,z)=z-x2-y2,   2 xiˆ  2 yˆj  kˆ , and (2,-2,8) is on the surface.
For z-x2-y2=0, the normal vector at (2,-2,8) is  4iˆ  4 ˆj  kˆ .
The tangential plane at (2,-2,8) is  4( x  2)  4( y  2)  ( z  8)  0  4 x  4 y  z  8
x 2 y  2 z 8


.
4
4
1
Eg. Find a unit normal vector of z2=4(x2+y2) at (1,0,2). [1993 中山電研、2007 師大
光電所]

Theorem   F ( x, y, z ) is the outward flux per unit volume of the flow at point
(x,y,z) and time t.

(Proof) F at P  F iˆ  F ˆj  F kˆ
The normal line is
x
y
z



 Fx
1  Fx
1  Fx 
The x-direction flux   Fx 
 x y z   Fx 
x y z 
x y z
2 x
2 x
x




The y-direction flux 
 Fy
y
x y z , and the z- direction flux 
Fy Fz
 F
∴ Total flux   x 

 x y z
Fz
x y z
z


  x y z / x y z    F

  
 1


Theorem If T  F  R and R  xiˆ  yˆj  zkˆ , then F    T .
2
ˆj kˆ
iˆ

 
(Proof)   T    ( F  R)    F1 F2 F3
x
y
z
   [( F2 z  F3 y) iˆ  ( F3 x  F1 z) ˆj  ( F1 y  F2 x) kˆ]
ˆj
iˆ
kˆ

 1





 2 ( F1iˆ  F2 ˆj  F3 kˆ)  2 F  F    T
x
y
z
2
F2 z  F3 y F3 x  F1 z F1 y  F2 x
57
～～
7-4 Line Integrals & Surface Integrals

Line integral:  Curve C: R(t )  x(t )iˆ  y(t ) ˆj  z (t )kˆ, a  t  b

 Vector F ( x, y, z)  F1 ( x, y, z)iˆ  F2 ( x, y, z) ˆj  F3 ( x, y, z)kˆ
 

b 
(a)  F  dl   F [ x(t ), y (t ), z (t )]  R (t )dt ,
a
C
(b)


 F  dl  
b
a


F [ x(t ), y (t ), z (t )]  R (t )dt ,
C
(c)
 f ( x, y, z)dl  
b
a

f [x(t ), y (t ), z (t )] | R (t ) | dt
C

Eg. For C : R(t )  x(t ) iˆ  y(t ) ˆj  z(t )kˆ  cos(t ) iˆ  sin( t ) ˆj  kˆ , 0  t  2π, and

 


F ( x, y, z )  x iˆ  yˆj  z kˆ , determine  F  d l and  F  d l .
C
C


(Sol.) C : R(t )  cos(t ) iˆ  sin( t ) ˆj  kˆ , R (t )   sin( t ) iˆ  cos(t ) ˆj

F ( x(t ), y (t ), z (t ))  x(t ) iˆ  y (t ) ˆj  z (t ) kˆ = cos(t ) iˆ  sin( t ) ˆj  kˆ
  

F  dl  F ( x(t ), y(t ), z(t ))  R' (t )dt  (cos(t ) iˆ  sin( t ) ˆj  kˆ)  ( sin( t ) iˆ  cos(t ) ˆj )dt
 ( cos(t ) sin( t )  sin( t ) cos(t )) dt  0
iˆ
  

F  dl  F ( x(t ), y (t ), z (t ))  R' (t )dt  cos(t )
ˆj
kˆ
sin( t ) 1 dt  [ cos(t )iˆ  sin( t ) ˆj  kˆ ] dt
 sin( t ) cos(t ) 0
∴
 
F
  dl  0 ,


2
F

d
l


 [ cos(t )iˆ  sin( t ) ˆj  kˆ]dt 2kˆ
C
C
0
 
 ˆ  ˆ
Surface integral:  Surface S :  ( x, y, z )  z   ( x, y )  0 . N 
iˆ 
j
k
x
y
z


 ˆ  ˆ ˆ
=
i
j  k is a normal vector on S, and Nˆ  N / | N | .
x
y

 Vector F ( x, y, z)  F1 ( x, y, z)iˆ  F2 ( x, y, z) ˆj  F3 ( x, y, z)kˆ
2
2
 

     
  
 dxdy ,
(a)  F  dA   F ( x, y, z )  Nˆ 1  

x

y




S
D
2
2



     
  
 dxdy ,
(b)  F  dA   F ( x, y, z )  Nˆ 1  
x  y
S
D
(c)
 f ( x, y, z)dA  
S
D
2
     
 dxdy ,
f ( x, y , z ) 1  
  
 x   y 
2


Note: dl  Tˆdl is parallel to the tangential direction of the curve, but dA  Nˆ dA is
normal to the surface.
58
～～

Eg. For S : x 2  y 2  z 2  1, z  0 , and F ( x, y, z )  x iˆ  yˆj  z kˆ , determine

 

and
.
F

d
A
F

d
A


S
S
z
x
x


x
z
1 x2  y2
(Sol.) S : x 2  y 2  z 2  1  z  0  z  1  x 2  y 2 ,

z
z ˆ ˆ x ˆ y ˆ ˆ
z
y
y
jk i  jk,

   N   iˆ 
x
y
z
z
y
z
1 x2  y2


N
Nˆ    x iˆ  yˆj  z kˆ , F ( x, y, z )  x iˆ  yˆj  z kˆ , and x 2  y 2  z 2  1
|N|
2
2
  
x2  y2  z 2
dxdy
 z   z 
 F  dA  F  Nˆ 1       dxdy 
dxdy 
z
 x   y 
1 x2  y2
iˆ
2
2
 




z

z
 
 F  dA  F  Nˆ 1       dxdy  x
 x   y 
x
∴
 
F
  dA 
S

dxdy
x 2  y 2 1
1 x2  y2

2
0
1
Eg. f  , r  x 2  y 2  z 2 , find
r
1
rdrd 
0
1 r2

ˆj kˆ
y
z
y
z
dxdy
0
z
 2 ,


F

d
A
0

S
  f  nˆdA
for S: x2+y2+z2=a2. [1990 中山
S
電研]
(Sol.) S : x 2  y 2  z 2  a 2  z   a 2  x 2  y 2 ,

z
z ˆ ˆ x ˆ y ˆ ˆ
For z  a 2  x 2  y 2  N   iˆ 
jk i  jk,
x
y
z
z

2
2
a
N
1 ˆ ˆ
 z   z 
ˆ
ˆ
N    ( x i  yj  z k ) , 1       
z
|N| a
 x   y 
1
1
1
1
f   ( )   (
)
( xiˆ  yˆj  zkˆ)  3 ( xiˆ  yˆj  zkˆ)
2
2
2
2
2
2
3
r
a
x y z
( x y z )
2
2
2
2
 dxdy
 ( 1 )  nˆdA  ( 1 )  Nˆ 1   z    z  dxdy   x  y3  z dxdy 
r
r
a z
 x   y 
a a2  x2  y2
2
  f  nˆdA =
z a2  x2  y2

x 2  y 2 a 2
 dxdy
a a  x  y
2
2
2

=-2π. Similarly, for z   a 2  x 2  y 2 ,
 1 2 a rdrd
 2



0
0
2
2
a
a
a r
 
z  a  x  y
2
∴
  f  nˆdA =(-2π)+(-2π)=-4π
S
59
～～
2
f  nˆ dA =-2π
2

a
0
rdr
a2  r 2
Eg.
Compute
the
surface
 ( x  y  z)dA
integral
,
where
S
S : z  x  y , 0  y  x , 0  x  1 . [1990 台大農工研]
2
 z   z 
1        3 ,
 x   y 
2
(Sol.)
1 x
 ( x  y  z)dA = 

0 0
S
2( x  y)  3dydx = 3
Green’s theorem Let C be a regular, closed, positively-oriented curve enclosing a

region D, F ( x, y )  F1 ( x, y ) iˆ  F2 ( x, y ) ˆj .
 F2 ( x, y ) F1 ( x, y ) 

dxdy
x
y 
 F ( x, y)dx  F ( x, y)dy   
1
2
C
D
 yiˆ  xˆj
Eg. Fˆ  2
, find
x  y2
 
F
  dr , where C is any closed curve.
C
F2 F1

 0, ( x, y)  0 ,
x
y
 
Else,  F  dr  2
 
F
  dr  0 if C does not enclose 0.
(Sol.)
C
Stokes’s theorem Let S be a regular surface with coherently oriented boundary C,


 
(


F
)

d
A

F

  d .
S
C
Divergence theorem Let S be a regular, positive-oriented closed surface,
 

enclosing a region V,  F  dA   (  F )dxdydz .
S
Eg.
V
 
Compute  F  dA ,
2

2
2
2
2 3ˆ
F  ( y  z ) i  sin( x 2  z ) ˆj  e x  y kˆ ,
where
S
x2 y2 z2


 1 . [1990 成大工程科學所]
a2 b2 c2
 


(Sol.)   F  0   F  dA   (  F )dxdydz  0
S:
S
V
1
Eg. f  , r  x 2  y 2  z 2 , find
r
  f  nˆdA
for S: x2+y2+z2=a2. [1990 中山
S
電研]

(Sol.) F  f  (
 3
F  3 ,
a
1
x2  y2  z2
)
1
1
( xiˆ  yˆj  zkˆ)  3 ( xiˆ  yˆj  zkˆ) ,
a
( x 2  y 2  z 2 )3
 

 3 4 a 3
ˆ

f

n
dA

F

d
A

(


F
)
dxdydz


  4
S
S

3
3
a
V
60
～～
Green’s identities

(a) [ 2     ]dxdydz  ( )  dA
S
 
V
 ,  are scalars, 
.
2
2
(
b
)
[







]
dxdydz

[







]

d
A
 
S

V
Hyperbolic Paraboloid
𝑥2
𝑎2
𝑦2
− 𝑏2 = 𝑐𝑧
Egg Equation
𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑧1.5
Mobius Belt/Strip
7-5 Potential Theory



Potential:  is called a potential for the vector field F if F   or F  


Test for a potential: If F and   F are continuous in a simply-connected domain


Ω, then F has a potential function.    F  0


(Proof) 〝  〞: F      F  0 .

 


〝  〞:   F  0 ,  F  d    (  F )  dA  0
C
S

Let C be the boundary of  (x,y,z)=constant, then we choose F   , then
 

F

d


0

is
always
valid.
has a potential function.
F

C

Eg. Check (a) F  2 xyiˆ  z 2 ˆj  ( x  y  z )kˆ ,

and (b) F  ( yze xyz  4 x) iˆ  ( xze xyz  z ) ˆj  ( xye xyz  y )kˆ , which does have a
potential? [1991 成大機研]

(Sol.) (a)   F  (2 z  1) iˆ  ˆj  2 xkˆ  0 , ∴ no potential.

(b)   F  0 , ∴ there exists a potential.

 ˆ  ˆ  ˆ
F   
i
j
k  ( yze xyz  4 x)iˆ  ( xze xyz  z ) ˆj  ( xye xyz  y )kˆ
x
y
z
  ( x, y, z )  e xyz  2 x 2  I ( y, z )  e xyz  yz  J ( x, z )  e xyz  yz  K ( x, y)
  ( x, y, z )  e xyz  2 x 2  zy  Constant
61
～～

Eg. F  ( x 2  y 2  z 2 ) n ( xiˆ  yˆj  zkˆ) , find a scalar potential  (x,y,z) so that

 ( x 2  y 2  z 2 ) n 1
F   . [1990 台大材研] (Ans.)  ( x, y, z ) 
C
2(n  1)


Theorem If F has a potential, then the line integral of F is independent of
 
 
path in Ω. That is,  F  d    F  d  , whenever C and K are regular curves in Ω
C
K
with the same initial point and the same terminal point.
 
 
 
(Proof)  F  d    F  d    F  d 
C
C
K

If F has a potential, then






 F  d  0 , ∴  F  d   F  d .
C
C
K

Theorem Let F be a 2-D vector field of a simply-connected domain Ω. Then


F
F
F has a potential on Ω.  2  1 . (In this case, F  F1 ( x, y )iˆ  F2 ( x, y ) ˆj )
x
y

 ( x, y )
 ( x, y ) F1  2
 2 F2
(Proof) If F   , then F1 
,
, F2 



x
y
y yx xy x
By
Green’s
theorem,
 F2 F1 
C F1dx  F2 dy  D  x  y dxdy  0 ,
∴

F
is
conservative on Ω.
Eg. Evaluate
 

2
2
v
  dR , where v  2 x iˆ  2 yzˆj  ( y  3)kˆ

and Γ is some
complicated path from (0,0,0) to (0,0,4). [2016 成大電研]



(Sol.)   v  0 , ∴ v is conservative. , ∴  ( x, y, z ) such that v  
  ( x, y , z ) 
 
2x 3
 y 2 z  3z   F  dr   ( x, y, z ) (( 00,,00,,04))  12
3

y
xy
x
Eg. (a) Is F ( x, y, z )   iˆ  ˆj  2 kˆ conservative in the region z>0? (b)
z
z
z
 

y ˆ x ˆ xy ˆ
Evaluate  F  dr , where F   i  j  2 k , C is a piecewisely smooth curve
C
z
z
z
from (1,1,1) to (2,-1,3) and not crossing the xy-plane. [1990 成大化工所]


(Sol.) (a)   F  0 in the region z>0, ∴ F is conservative.


(b)   F  0 , ∴  ( x, y, z ) such that F  
 
xy
5
  ( x, y , z )  
  F  dr   ( x, y , z ) ((12,,1,11),3) 
z
3
62
～～
7-6 Curvilinear Coordinates
 x  x (q1 , q 2 , q3 )
q1  q1 ( x, y, z )


Coordinate transformation:  y  y (q1 , q 2 , q3 )  q 2  q 2 ( x, y, z )
 z  z (q , q , q )
q  q ( x, y, z )
3
1
2
3
 3

Scalar factors: If (q1,q2,q3) are orthogonal system, then
0 , i 

hij   x

 qi
j
2
  y
  
  qi
2
  z
  
  qi



2



12
,i j
Eg.
Rectangular
coordinate
Spherical

coordinate

2 12


r  x 2  y 2  z 2
hr  h11   x    y    z    1
x

r
sin

cos




 r   r   r  



1
2
2
2
2
2
12
x y z
 y  r sin  sin     sin x  y

 x 2  y 2  z  2 
 h  h22            r
 z  r cos 

1
2
2


         
  cos x x  y

12



2
2

 x 2  y  2  z 2 

h

h

        

33
         

 r sin 
Differential length vector in the spherical coordinate:

dl  aˆ r dr  aˆ rd  aˆ r sin d
Eg. Rectangular coordinate  Cylindrical
coordinate
r  x 2  y 2
 x  r cos 


1
y

r
sin



  tan  y x 
z  z
z  z


63
～～
12

 x  2  y  2  z  2 
hr  h11            1

 r   r   r  

12

 x  2  y  2  z  2 
 h  h22            r

         

12
 x  2  y  2  z  2 

hz  h33            1

 z   z   z  

Differential length vector in the cylindrical coordinate: dl  aˆr dr  aˆ rd  aˆ z dz
Differential arc length: ds  [( h1 dq1 ) 2  (h2 dq 2 ) 2  (h3 dq3 ) 2 ]1 2
Differential elements of area on the qiqj-plane: dAij  dsi ds j  hi h j dqi dq j
Differential element of volume: dV  ds1ds 2 ds3  h1h2 h3 dq 1dq2 dq3
Eg. Spherical coordinate:


 ds  dr 2  r 2 d 2  r 2 sin 2 d 2 1 2

2
dA ,  r sin dd , dAr ,  rdrd , etc.

2
 dV  r sin drdd


 ds  dr 2  r 2 d 2  dz 2 1 2

Eg. Cylindrical coordinate: dA , z  rddz , dAr ,  rdrd , dAr , z  drdz .
 dV  rdrddz

Jacobian determinants: (p1,p2,p3)  (q1,q2,q3)
 pi

qi
dpi dp j  det 
pi

 q j
p j

qi 
  dqi dq j
p j

q j 
64
～～
 p1
q1

p1
and dp1 dp 2 dp3  det  q
2
p
 1
q3

p 2
p 2
p 2
q1
q 2
q3
p3

q1 

p3
 dq1 dq 2 dq3
q 2 

p3

q 2 
Eg. Let I=  f ( x, y, z )dxdydz . Transform the integral from (x,y,z) into (r,θ,φ).
V
 sin  cos 
| J | det  r cos  cos 
 r sin  sin 
sin  sin 
r cos  sin 
r sin  cos 
cos  
 r sin    r 2 sin  cos 2  cos 2 
0 
 r 2 sin 3  sin 2   r 2 sin  cos 2  sin 2   r 2 sin 3  cos 2  | r 2 sin  cos 2   r 2 sin 3  | r 2 sin   h1 h2 h3
∴ I   f (r sin  cos  , r sin  sin  , r cos  )  r 2 sin drdd
V
Del operators in (q1,q2,q3) coordinate system:
1 
1 
1 
 q1 , q 2 , q3  
uˆ1 
uˆ 2 
uˆ 3
h1 q1
h2 q 2
h3 q3

  F q1 , q2 , q3  

F 

1  

F1h2 h3    F2 h1h3    F3 h1h2 
h1h2 h3  q1
q2
q3

h1uˆ1
1

h1h2 h3 q1
F1h1
 2 q1 , q 2 , q3  
h2 uˆ 2

q2
F2 h2
1
h1 h2 h3
h3uˆ3

q3
F3h3
   h2 h3  


 

 q1  h1 q1  q 2

where F  F1uˆ1  F2 uˆ 2  F3uˆ 3
 h1 h3 

 h2 q 2
 
 
 q3
 h1 h2  

 ,
h

q
3 
 3
Eg. Spherical coordinate system: hr=1, hθ=r, hφ=rsinθ

F  Fr uˆ r  F uˆ  F uˆ
 1 
1

sin   F   1   F 
  F  2  r 2 Fr  

r r
r sin  
r sin  
uˆ r ruˆ
r sin uˆ

1




1 
1 
 F  2
,  
uˆ r 
uˆ 
uˆ

r sin  r 
r
r 
r sin  
Fr rF r sin   F
 2 
1   2  
1
 
 
1
 2

r


sin







r  r 2 sin   
  r 2 sin 2   2
r 2 r 
65
～～
Eg. Cylindrical coordinate system: hr=1, hθ=r, hz=1

F  Fr uˆ r  F uˆ  Fz uˆ z
 1 
rFr   1 F  Fz
F 
r r
r 
z
  1 Fz F 
F
F 
 F
1 
rF   1  uˆ z
 F  

uˆ r     z uˆ  
z 
r 
r  
 r 
 z
 r r
 
1   

1 

uˆ r 
uˆ 
uˆ z ,  2 
r
r
r 
z
r r  r
2
2
 1  



2
2
z 2
 r 
Unit vector conversions between distinct coordinates:
Eg. Rectangular coordinate  Spherical coordinate
(Proof) xˆ  rˆ  sin  cos  , yˆ  rˆ  sin  sin  ,
zˆ  rˆ  cos , ∴ rˆ  xˆ sin  cos   yˆ sin  sin   zˆ cos 
Similarly, xˆ  ˆ  cos  cos  , yˆ  ˆ  cos  sin  ,
zˆ  ˆ   sin  , ∴ ˆ  xˆ cos  cos   yˆ cos  sin   zˆ sin  ,
And xˆ  ˆ   sin  , yˆ  ˆ  cos  , zˆ  ˆ  0 ,
∴ ˆ   xˆ sin   yˆ cos 
 rˆ   sin  cos  sin  sin 
ˆ  cos  cos  cos  sin 
  
ˆ   sin 
cos 
cos    xˆ 
 sin     yˆ  ,
0   zˆ 
 xˆ  sin  cos  cos  cos 
or  yˆ    sin  sin  cos  sin 
 zˆ   cos 
 sin 
 sin    rˆ 
cos    ˆ
0  ˆ
Eg. Rectangular coordinate  Cylindrical coordinate
 xˆ  cos 
 yˆ    sin 
  
 zˆ   0
 sin 
 rˆ   cos 
ˆ   sin 
  
 zˆ   0
cos 
0
sin 
cos 
0
66
～～
0  rˆ 
0  ˆ or
1  zˆ 
0  xˆ 
0   yˆ 
1  zˆ 