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

316 Chap. 9 • EUCLIDEAN MANIFOLDS
The values of the vector fields { g ,..., 1
g
N } form a linearly independent set of vectors
{ g x g x } at each x ∈U . To see this assertion, assume x ∈U and
( ),..., ( )
1 N
N
λ g ( x) + λ g ( x) +⋅⋅⋅+ λ g ( x)
= 0
1 2
1 2
N
for
1 N
j
λ ,..., λ ∈R . Taking the inner product of this equation with g ( x ) and using equation (44.35)
j
, we see that λ = 0 , j = 1,..., N which proves the assertion.
Because V has dimension N, { g ( ),..., ( )
1
x g
N
x } forms a basis for V at each ∈
Equation (44.35) shows that { g 1 ( x),..., g N ( x )}
is the basis reciprocal to { ( ),..., ( )
1 N }
of the special geometric interpretation of the vectors { }
N
x U .
g x g x . Because
g ( ),..., ( )
1
x g
N
x and { g 1 ( x ),..., g ( x ) }
mentioned above, these bases are called the natural bases of ˆx at x . Any other basis field which
cannot be determined by either (44.31) or (44.32) relative to any coordinate system is called an
anholonomic or nonintegrable basis. The constant vector fields { i ,..., 1
i
N } and { i 1 ,..., i N
} yield the
natural bases for the Cartesian coordinate systems.
U are two charts such that U1∩U 2
≠∅, we can determine the
transformation rules for the changes of natural bases at x ∈U1∩
U
2
in the following way: We shall
j
let the vector fields h , j = 1,..., N , be defined by
If ( U , x ) and ( , y )
1 ˆ
2
ˆ
j
j
h = grad yˆ
(44.36)
Then, from (44.5) and (44.31),
∂y
h x = x = x
∂ x
j
∂y
1 N i
= ( x ,..., x ) g ( x)
∂
i
x
j
j j 1 N i
( ) grad yˆ
( ) ( x ,..., x )grad xˆ
( )
i
(44.37)
for all x ∈U1∩U 2. A similar calculation shows that