Introduction to Vectors and Tensors Vol 2 (Bowen 246). - Index of

312 Chap. 9 • EUCLIDEAN MANIFOLDS
3
z
3
z
i
3
r
r
3
i
1
z
i
1
0 E
i
2
2
z
1
z
1
i
0 E
2
i
2
z
Figure 8
1 N
Given any rectangular coordinate system ( zˆ
,..., z ˆ ) , we can characterize a general or a
curvilinear coordinate system as follows: Let ( , xˆ
)
U be a chart. Then it can be specified by the
coordinate transformation from ẑ to ˆx as described earlier, since in this case the overlap of the
coordinate neighborhood is U = E ∩U . Thus we have
z z = zˆ
xˆ
x x
1 N
−1 1 N
( ,..., ) ( ,..., )
x x = xˆ
zˆ
z z
1 N
−1 1 N
( ,..., ) ( ,..., )
(44.23)
−1
where zˆ xˆ
: xˆ( U) → zˆ( U)
and
of (44.23) are
ˆ ˆ : ˆ( ) → ˆ( U)
are diffeomorphisms. Equivalent versions
−1
x z zU
x
z = z x x = z x
j j 1 N j k
( ,..., ) ( )
x = x z z = x z
j j 1 N j k
( ,..., ) ( )
(44.24)