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

306 Chap. 9 • EUCLIDEAN MANIFOLDS
Section 44 Coordinate Systems
r
Given a Euclidean point space E of dimension N , we define a C -chart at x ∈E to be a
N
r
U , xˆ
, where U is an open set in E containing x and xˆ : U →R is a C diffeomorphism.
i
U , xˆ
, there are N scalar fields xˆ : U →R such that
pair ( )
Given any chart ( )
1
N
( )
xˆ( x) = xˆ ( x),..., xˆ
( x )
(44.1)
for all x ∈U . We call these fields the coordinate functions of the chart, and the mapping ˆx is also
called a coordinate map or a coordinate system on U . The set U is called the coordinate
neighborhood.
N
N
Two charts xˆ : U1
→R and yˆ : U2
→R , where U1∩U 2
≠∅, yield the coordinate
−1
−1
transformation yˆ xˆ : xˆ( U ˆ
1∩U2) → y( U1∩U2)
and its inverse xˆ yˆ : yˆ( U ˆ
1∩U2) → x( U1∩U2)
.
Since
1
N
( )
yˆ( x) = yˆ ( x),..., yˆ
( x )
(44.2)
The coordinate transformation can be written as the equations
( ˆ 1 ˆ N
) ˆ ˆ −
( ),..., ( ) 1 ( ˆ 1 ( ),..., ˆ N
y x y x = y x x x x ( x)
)
(44.3)
and the inverse can be written
( ˆ 1 ˆ N
) ˆ ˆ −
( ),..., ( ) 1 ( ˆ 1 ( ),..., ˆ N
x x x x = x y y x y ( x)
)
(44.4)
The component forms of (44.3) and (44.4) can be written in the simplified notation
j j 1 N j k
y = y ( x ,..., x ) ≡ y ( x )
(44.5)