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

334 Chap. 9 • EUCLIDEAN MANIFOLDS
a
( ) ˆ c a
T x T ( x ) = δ
(46.9)
c b b
for all x ∈U1∩U 2. It follows from (46.4) and (46.7) that
A = T ⋅⋅⋅ T Aˆ
(46.10)
ˆb
b
1 ˆ q
a1... aq a1 aq b1...
bq
where
A ˆ = A ( eˆ ,..., eˆ
)
(46.11)
b1...
bq
b1
bq
Equation (46.10) is the transformation rule for the anholonomic components of A . Of course,
(46.10) is a field equation which holds at every point of U1∩U2∩
U . Similar transformation rules
for the other components of A can easily be derived by the same type of argument used above.
We define the physical components of A , denoted by
A
a1 ,..., aq
, to be the anholonomic
components of A relative to the field of orthonomal basis { g i } whose basis vectors g are unit
i
vectors in the direction of the natural basis vectors g
i
of an orthogonal coordinate system. Let
( , x )
U by such a coordinate system with g = 0, i ≠ j. Then we define
1 ˆ
ij
g ( x) g ( x) g ( x ) (no sum)
(46.12)
i
=
i
i
at every x ∈U 1
. By (44.39), an equivalent version of (46.12) is
g x = g x g x (46.13)
12
( )
i( ) (
ii( )) (no sum)
i
Since { } i
g is orthogonal, it follows from (46.13) and (44.39) that { i }
g is orthonormal:
g ⋅ g = δ
a b ab
(46.14)