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

308 Chap. 9 • EUCLIDEAN MANIFOLDS
1
( ,..., N
j
x x ) ( x )
(44.8)
x = x = x
where x is a diffeomorphism
x : xˆ
( U)
→U
.
r
r
A C -atlas on E is a family (not necessarily countable) of -
where I is an index set, such that
{ , x I
α α
α ∈ }
C charts ( ˆ )
U ,
E
=
∪ Uα
(44.9)
α∈I
Equation (44.9) states that E is covered by the family of open sets { U
α
α ∈ I}
. A
r
C -Euclidean
r
∞
manifold is a Euclidean point space equipped with a C -atlas . A C -atlas
and a C ∞ -Euclidean
manifold are defined similarly. For simplicity, we shall assume that E is C ∞ .
A C ∞ curve in E is a C ∞ mapping λ : ( ab , ) →E , where ( ab , ) is an open interval of R .
A C ∞ curve λ passes through x ∈ 0
E if there exists a c∈
( a , b ) such that λ( c ) = x
0
. Given a chart
( U , xˆ
) and a point x ∈U
, the 0 jth coordinate curve passing through x
0
is the curve λ
j
defined by
1 j− 1 j j+
1 N
λ
j( t) = x ( x0,..., x0 , x0 + t, x0 ,..., x0
)
(44.10)
1 j− 1 j j+
1 N
k
for all t such that ( x ˆ
0,..., x0 , x0 + t, x0 ,..., x0 ) ∈x( U ) , where ( x ˆ
0) = x( x
0)
. The subset of U
obtained by requiring
x
j
j
= xˆ ( x ) = const
(44.11)
is called the j th coordinate surface of the chart
Euclidean manifolds possess certain special coordinate systems of major interest. Let
i i be an arbitrary basis, not necessarily orthonormal, for V . We define N constant vector
{ 1 ,..., N
}
j
fields i : E →V , j = 1,..., N , by the formulas