Page 162 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 14: Notation and Summation Convention

It follows from equations (5) and (7) that Notes

 x' x 
a  k  i , ...(8)
ki
 x i  x' k
and from equation (4) that

e i x  i  e ' j x  ' j  e , ...(9)
'
k
 x' k dx ' k
since x/x =  , and from equations (8) and (9) that
k
j
jk
'
e  a e , ...(10)
k
ki i
and
'
e  a e . ...(11)
i
ki k
Equations (10) and (11) are the transformation rules for base vectors. The nine elements of a are
ij
not all independent, and in general,
a  a .
ik
ki
A relation similar to equations (5) and (7),
'
u  a u , and u  a u ' k ...(12)
ki
k
i
ki
i
'
'
is obtained for a vector u since u e  u e , which is similar to equation (4) except that the ui are
k
k
i i
the components of a vector and the x are coordinates of a point.
i
The magnitude |u| = (u u ) of the vector u is independent of the orientation of the coordinate
1/2
i
i
system, that is, it is a scalar invariant; consequently,
'
'
u u = u  a a u . ...(13)
k
ki
k
ji
i
i
Eliminating u from equation (12) gives
i
'
'
u  a a u ,
k
ki
j
ji
'
'
and since u   kj u ,
j
k
a a =  . ...(14)
ki ji
kj
Similarly, eliminating u from equation (12) gives
k
a a =  . ...(15)
ik jk
ij
It follows from equation (14) or (15) that
{det[a ]} = 1, ...(16)
2
ij
where det [a ] denotes the determinant of a . The negative root of equation (16) is not considered
ij
ij
unless the transformation of axes involves a change of orientation since, for the identity
transformation x = x , a =  and det[ ] = 1. Consequently, det[a ] = 1, provided the
'
i
ik
ik
ik
i
ik
transformations involve only right-handed systems (or left-handed systems).
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