CH2. COORDINATE SYSTMES AND
TRANSFORMATION
1
Coordinate Systems
➢
A point or vector can be represented in any curvilinear
coordinate system
Orthogonal coordinate system: mutually perpendicular
– Nonorthogonal coordinate system : not mutually perpendicular
–
➢
Orthogonal systems are almost exclusively used in reality
–
–
➢
Three-dimensional orthogonal coordinate system is specified by three
orthogonal unit vectors 𝒂1 , 𝒂2 , 𝑎𝑛𝑑 𝒂3 .
A vector 𝑨 can be represented by linear combination of three orthogonal
vectors. 𝑨 = 𝐴1 𝒂1 + 𝐴2 𝒂2 + 𝐴3 𝒂3 → 𝐴𝑖 = 𝑨 ⋅ 𝒂𝑖 only valid in orthogonal
coordinate system. 𝐴𝑖 → ith component of the vector 𝑨
Widely used orthogonal coordinates in physics:
Cartesian coordinate (=rectangular coordinate)
– Circular cylindrical coordinate (or cylindrical coordinate in short)
– Spherical coordinate
–
➢
Choosing an appropriate coordinate system for a given problem
will save a considerable amount of work and time!
2
Cartesian Coordinate (I)
➢
Unit vectors 𝒂𝑥 , 𝒂𝑦 , 𝒂𝑧 are constant → position independent
➢
➢
A point P can be represented as (𝑥, 𝑦, 𝑧)
A vector A can be represented as 𝑨 = 𝐴𝑥 𝒂𝑥 + 𝐴𝑦 𝒂𝑦 + 𝐴𝑧 𝒂𝑧
➢
Especially, the position vector is 𝑷 = 𝑥 𝒂𝑥 + 𝑦 𝒂𝑦 + 𝑧𝒂𝑧
Constant surfaces
−  x  
−  y  
−  z  
3
Cartesian Coordinate (II)
➢
Differential displacement:
dl = dxa x + dya y + dza z
➢
Differential normal surface:
dS x = dy dz
dS y = dx dz
dS z = dx dy
➢
Differential volume:
dv = dx dy dz
4
Metric Coefficient
➢
➢
The metric coefficient ℎ𝑖 is a proportional constant which
converts the differential change of a coordinate 𝑑𝑢𝑖 to the
differential length 𝑑ℓ𝑖 .→ Differential length 𝑑ℓ𝑖 = ℎ𝑖 𝑑𝑢𝑖
Cartesian coordinate
–
–
Coordinate change → Length change
𝑢1 , 𝑢2 , 𝑢3 = 𝑥, 𝑦, 𝑧 → ℎ1 , ℎ2 , ℎ3 = (1,1,1)
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Cylindrical Coordinate (I)
➢
A point P can be represented as (𝜌, 𝜑,z)
0    ,
0    2 , −   z  
➢
Unit vectors are 𝒂𝜌 , 𝒂𝜑 , 𝒂𝑧 → 𝒂𝜌 , 𝒂𝜑 are position dependent
➢
A vector 𝐀 = 𝐴𝜌 𝒂𝜌 + 𝐴𝜑 𝒂𝜑 + 𝐴𝑧 𝒂𝑧 or simply (𝐴𝜌 , 𝐴𝜑 , 𝐴𝑧 )
➢
The position vector is 𝑷 = 𝜌𝒂𝜌 + 𝑧𝒂𝑧 → no 𝒂𝜑 component exists.
➢
Properties
A =
A2 + A2 + Az2
a   a  = a  a = a z  a z = 1
a   a = a  a z = a z  a  = 0
a   a = a z
a  a z = a 
a z  a  = a
Constant surfaces
6
Cylindrical Coordinate (II)
• Metric coefficient
(h1 , h2 , h3 ) = (1,  ,1)
• Differential displacement:
dl = d  a  +  d a + dz a z
• Differential normal surface:
dS  =  d dz
dS = d  dz
dS z =  d  d
• Differential volume:
dv =  d  d dz
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Cylindrical Coordinate (III)
➢
Transformation to Cartesian coordinate: point
y
x
 = x2 + y 2 ,
 = tan −1 , z = z
x =  cos  ,
y =  sin , z = z
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Cylindrical Coordinate (IV)
➢
Transformation to Cartesian coordinate: vector
–
–
Unit vectors
a x = cos  a  − sin  a
a  = cos  a x + sin  a y
a y = sin  a  + cos  a
a = − sin  a x + cos  a y
az = az
az = az
Vector components
 Ax  cos 
  
 Ay  = sin 
 A   0
 z
− sin  0  A 
 
cos  0  A 
0
1   Az 
 A  cos 
  
 A  =  − sin 
 A   0
 z
sin 
cos 
0
0   Ax 
 
0   Ay 
1   Az 
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Spherical Coordinate (I)
A point P can be represented as (𝑟, 𝜃, 𝜑)
➢ A vector A can be represented as 𝐴𝑟 𝒂𝑟 + 𝐴𝜃 𝒂𝜃 + 𝐴𝜑 𝒂𝜑 or
simply (𝐴𝑟 , 𝐴𝜃 , 𝐴𝜑 )
➢
➢
The position vector is 𝑷 = 𝑟𝒂𝑟 → no 𝒂𝜃 , 𝒂𝜑 component exist.
➢
Properties
A =
Ar2 + A2 + A2
a r  a r = a  a = a  a = 1
0r
0  
0    2
a r  a = a  a = a  a r = 0
a r  a = a
a  a = a r
a  a r = a
Constant surfaces
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Spherical Coordinate (II)
(h1 , h2 , h3 ) = (1, r , r sin  )
➢
Metric coefficient
➢
Differential displacement:
➢
Differential normal surface:
dl = dr ar + r d a + r sin  d a
dS r = r 2 sin  d d , dS = r sin  dr d , dS = r dr d
➢
Differential volume:
dv = r 2 sin  dr d d
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Spherical Coordinate (III)
➢
Transformation to Cartesian coordinate: point
r= x +y +z ,
2
2
2
 = tan
−1
x2 + y 2
y
,  = tan −1
z
x
x = r sin  cos  , y = r sin  sin  , z = r cos 
12
Spherical Coordinate (IV)
➢
Transformation to Cartesian coordinate: vector
–
Unit vectors
a x = sin  cos  a r + cos  cos  a − sin  a
a r = sin  cos  a x + sin  sin  a y + cos  a z
a y = sin  sin  a r + cos  sin  a + cos  a
a = cos  cos  a x + cos  sin  a y − sin  a z
a z = cos  a r − sin  a
a = − sin  a x + cos  a y
–
Vector components
 Ax  sin  cos 
  
 Ay  = sin  sin 
 A  cos 
 z
cos  cos 
cos  sin 
− sin 
 Ar  sin  cos  sin  sin 
  
 A  = cos  cos  cos  sin 
 A   − sin 
cos 
 
− sin    Ar 
 
cos    A 
0   A 
Ax = ( a r a x ) Ar + ( a a x ) A + ( a a x ) A
cos    Ax 
 
− sin    Ay 
0   Az 
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Summary of Differential Dimension
Cartesian
Coordinate
h1
Cylindrical
Coordinate
Spherical
Coordinate
1
1

r
h3
1
1
1
1
r sin 
dS1
dy dz
Differential area
dS2
r 2 sin  d d
r sin  dr d
dS3
dx dz
dx dy
 d dz
d  dz
 d  d
Differential volume
dv
dx dy dz
 d  d dz
r 2 sin  dr d d
Metric coefficient
h2
r dr d
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Example 2.1
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Example 2.2
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Example 2.4
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