Page 190 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 16: Tensor Fields Differentiation of Tensors

(r + s + 1)-dimensional array with one extra dimension due to index q. We write it as a lower Notes

 X 1 i ...i r
index in Y qj 1 i ...i r s due to the following theorem concerning Y qj 1 i ...i r s   x 1 j ...j s .
1 ...j
q
1 ...j
Divergency is the second differential operation of vector analysis. Usually it is applied to a
vector field and is given by formula:
3
div X    i X . i
i 1

16.7 Self Assessment


1. .................... in space and hence can represent P by its radius-vector r  OP and by its
p
coordinates x , x , x :
1
2
3
X 1 i ...i r s  X 1 i ...i r s (x ,x ,x ).
2
3
1
1 j ...j
1 j ...j
2. A .................... X 1 i ...i r  X 1 i ...i r (x ,x ,x ) is a coordinate representation of a tensor field
1
2
3
1 j ...j
1 j ...j
X = X(P). s s
3
2
3
1
2
1
 X 1 i ...i r X 1 i ...i r (x  h,x ,x ) X 1 i ...i r (x ,x ,x )

3. .................... 1 j ...j s  lim 1 j ...j s 1 j ...j s , taken as a whole, form an
t  h 0 h
(r + s + 1)-dimensional array with one extra dimension due to index q. We write it as a
1 j ...j
lower index in Y qj 1 i ...i r s due to the following theorem concerning Y qj 1 i ...i r s   X x 1 i ...i r s .
1 ...j
q
1 ...j

16.8 Review Questions
  3 3 3
1. Using r  OO r  , a  OO   a e i , r  x e , and e  S e derive the following
i
j

P 

i 
i


 

i
P
P
j
i
i 1 i 1 j 1



formula relating the coordinates of the point P in the two coordinate systems.
3
i
i
i
x  a   S x .  j
j
j 1

3 3
j
i
i
j
i
j
i
Compare x  a   S x  with x   S x .  Explain the differences in these two formulas.
j
i
j 1 i 1


 X i1...ir
j1...js
2. Give a more detailed explanation of why the time derivative =
t 
X 1 i ...i r (t h,x ,x ,x ) X 1 i ...i r (t,x ,x ,x )
3
2
1
3
1
2


lim 1 j ...j s 1 j ...j s represents a tensor of type (r, s)
h 0 h
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