Page 199 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Formulas (4) are of no practical use because they express  through an external thing like
k
ij
transition matrices to and from the auxiliary Cartesian coordinate system. However, they will
help us below in understanding the differentiation of tensors.

17.7 Tensor Fields in Curvilinear Coordinates

As we remember, tensors are geometric objects related to bases and represented by arrays if
some basis is specified. Each curvilinear coordinate system provides us a numeric representation
for points, and in addition to this it provides the basis.
This is the moving frame. Therefore, we can refer tensorial objects to curvilinear coordinate
systems, where they are represented as arrays of functions:

1
X 1 i ...i r s  X 1 i ...i r s (y ,y ,y ). ...(1)
3
2
1 j ...j
1 j ...j
We also can have two curvilinear coordinate systems and can pass from one to another by means
of transition functions:
1
1
1
1
1
3
1
2
3
2
 y   y (y ,y ,y ),  y  y (y ,y ,y ),



 2 2 1 2 3  2 2 1 2 3



  y   y (y ,y ,y ),  y  y (y ,y ,y ), ...(2)
 3 y (y ,y ,y ),  3 y (y ,y ,y ).
1
3
3
3
3
1
2
2
  y    y    
1
3
2
If we call y ,y ,y  the new coordinates, and y , y , y the old coordinates, then transition
1
3
2


matrices S and T are given by the following formulas:
i
i
S   y i , T   y  i . ...(3)
j
j
 y  j  y j
They relate moving frames of two curvilinear coordinate systems:
3 3
j
i 

i 
j 
E  S E , E  T E . i ...(4)
j
i
j
j 1 i 1


Exercise 1.1: Derive (3) from (4) and (2) using some auxiliary Cartesian coordinates with basis e ,
1
e , e as intermediate coordinate system:
2
3
S
S 
(E ,E ,E )
(E ,E ,E )  (e ,e ,e )   1  2  3 ...(5)
3 
3 
1
2
1
2
T
T
Transformation formulas for tensor fields for two curvilinear coordinate systems are the same:
3 3
3
...
X 1 i ...i r (y ,y ,y )    T ...T S ...S X h k 1 ...h r (y ,y ,y ), ...(6)
s k
2
1
1 i
r i
3
2
k
1


1


1 j
h
r h
s j
1 ...k
1 j ...j
s
h 1 , ... r h 1 s
1 k , ... ks
3 3
2
1
3
r i
1 i
k
3  
X 1 i ...i r (y ,y ,y )  ... S ...S T ...T X h 1 ...h r (y ,y ,y ). ...(7)
s k 




1
1 j ...j s 1 2 ... h 1 r h 1 j s j k 1 ...k s
h 1 , r h
k 1 , ... ks
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