Differentiable manifold: Orthogonal curvilinear coordinates, cylindrical and spherical
A differentiable manifold has a differential operator (del operator)
  xˆ



 yˆ
 zˆ
x
y
z

eˆ 
eˆ1 
eˆ 
 2
 3
h1 u1 h2 u 2 h3 u 3
In which h1, h2, h3 are scale factors for the given (orthogonal) curvilinear coordinate system as
defined by a volume in space dV  h1 h2 h3 du 1 du 2 du 3
The scaling relationship is defined by change of position



 r
r
r
dr 
du1 
du 2 
du 3
u1
u 2
u 3


 
r
r
For which magnitude ds  dr  dr  

du i du j
u i u j
i, j
2
 
 dr  dr   g ij du i du j
i, j


r
r
where gij 
is defined as the metric of the (u1, u2, u3) coordinates

u i u j
If the coordinates are orthogonal then
gij 


r
r
=0

u i u j
for i  j
ds 2  g 11du12  g 22 du 22  g 33 du 32
and length (squared) is
and then
ds1  g11 du1  h1 du1
ds 2  g 22 du 2  h2 du 2
ds3  g 33 du 3  h3 du 3
And this is how the scale factors h1, h2, h3 for the coordinate system can be realized, since

position is also defined by r  xxˆ  yyˆ  zzˆ
= fixed reference basis coordinate system,
which allows the ui scaling metric to be determined relative to the fixed reference (x,y,z)
coordinates by
i.e.
 x k
g    i
k  u
2
ii



2
from which we can get the (h1, h2, h3) scale factors
Example: Cylindrical coordinates (r,,z) may be related to the fixed (x,y,z) frame by
x  r cos 
y  r sin 
 y 
 x 
 z 
g      
 r 
 r 
 r 
2
2
2
 cos 2   sin 2   0
2
11
2
2
g
2
22
 x 
 y 
 z 
        
  
  
  
g
2
133
 y 
 x 
 z 
     
 z 
 z 
 z 
2
2
z=z
=1
2
 r 2 sin 2   r 2 cos 2   0
= r2
2
=1
For which the scale factors are then (h1, h2, h3) = (1, r, 1)
Other coordinate system relationships are
Spherical:
x = R cossin
For which the scale factors are
Hyperbolic:
z = R cos
(h1, h2, h3) = (1, R, Rsin )
x = b cossinh
For which the scale factors are
y = R cossin
y = b sin sinh
(h1, h2, h3) =
z = c cosh
a2 + b2 = c2
The vector space derivatives in orthogonal fixed and curvilinear coordinates (cylindrical and
spherical) are then
1. Divergence

F 
i
 F
Fy Fz
F  x 

x
y
z
1
  h1 h2 h3 i 

F 
h1 h2 h3 u i  hi

1  R 2  FR
1 sin   F 
1 F
 2


R
R sin 

R sin  
R

1 rFr  1 F Fz



r r
r 
z

2. Curl
eˆi

 ijk hi
hk F k
j
h
h
h

u
i , j ,k 1 2 3


 F  
rˆ
1 

r r
Fr
rˆ


rF
zˆ

z
Fz
xˆ


F 
x
Fx

Rˆ
1

 2
R sin  R
Fr
Rˆ


RF
yˆ

y
Fy
zˆ

z
Fz
R sin  ˆ


R sin   F
3. Laplacian
2 g  
i
1
  h1 h2 h3 g 


h1 h2 h3 u i  hi2 u i 
2 g 
2g 2g 2g


x 2 y 2 z 2

1   g  1  2 g  2 g

r  
r r  r  r 2  2 z 2

g 
2g
1   2 g 
1
 
1
R

sin






R  R 2 sin   
  R 2 sin 2   2
R 2 R 