Page 185 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 3
P 
i
r  x e , ...(3)

 
i
i 1

Exercise 1.1: Using (1), (2) and (3) derive the following formula relating the coordinates of the
point P in the two coordinate systems in Fig. 16.1 :
3
i
i
i
x  a   S x .  j ...(4)
j
j 1

Exercise 2.1: Derive the following inverse formula for (4):
3
i
i
i

x   a   T x . j ...(5)
j
j 1

i
Prove that ai in (4) and a  in (5) are related to each other as follows:
3 3
i
i
i
i
a    T x . j a    S a .  j ...(6)

j
j
j 1 j 1


Explain the minus signs in these formulas. Formula (4) can be written in the following expanded
form:
1
1
1
3
1
1
1
2
 x  S x  S x  S x  a ,



2
3
1
 2 2 2 2 2 3 2




 x  S x1 S x  S x  a , ...(7)
1
3
2
 x  S x  S x  S x  a .
2
3
3
3
3
3
3
1
 1  2  3 
This is the required specialization. In a similar way we can expand (5) :
2
1
1
1
1
3
1
1
 x  T x  T x  T x   a ,
3
2
1
 2 2 1 2 2 2 3 2

 x  T x  T x  T x   a , ...(8)
3
1
2
 x  T x  T x  3 3 a .
1
3
2
3
3
3
  1 2 T x  
3
This is the required specialization. Formulas (7) and (8) are used to accompany the main
transformation formulas.
16.3 Differentiation of Tensor Fields
In this section we consider two different types of derivatives that are usually applied to tensor
fields: differentiation with respect to spacial variables x , x , x and differentiation with respect
1
3
2
to external parameters other than x , x , x , if they are present. The second type of derivatives are
1
3
2
simpler to understand. Let’s consider them to start. Suppose we have tensor field X of type (r, s)
and depending on the additional parameter t (for instance, this could be a time variable). Then,
upon choosing some Cartesian coordinate system, we can write
1
3
2
2
3
1


 X i1...ir  lim X 1 i ...i r s (t h,x ,x ,x ) X 1 i ...i r s (t,x ,x ,x ) . ...(1)
1 j ...j
1 j ...j
j1...js
t  h 0 h
The left hand side of (1) is a tensor since the fraction in right hand side is constructed by means
of tensorial operations. Passing to the limit h  0 does not destroy the tensorial nature of this
fraction since the transition matrices S and T are all time-independent.
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