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

318 Chap. 9 • EUCLIDEAN MANIFOLDS
Based upon (44.44), the curvilinear coordinate system is orthogonal if g
ij
= 0 when i ≠ j. The
symbol g denotes a scalar field on U defined by
g( x) = det ⎡
⎣g ij
( x ) ⎤
⎦
(44.45)
for all
x ∈U .
At the point
x ∈U , the differential element of arc ds is defined by
2
ds = d ⋅ d
x x (44.46)
and, by (44.8), (44.33), and (44.39),
∂x
∂x
ˆ
∂x
∂x
i j
= g ( x)
dx dx
2
i j
ds = ( x( x)) ⋅ ( x( x))
dx dx
i
j
ij
ˆ
(44.47)
If ( U , x ) and ( , y )
1 ˆ
U are charts where U1∩U 2
≠∅, then at x ∈U1∩
U
2
2
ˆ
h ( x) = h ( x) ⋅h ( x)
ij i j
∂x
∂x
= ∂ y
∂ y
k
l
1 N
1 N
( y ,..., y ) ( y ,..., y ) gkl
( x)
i
j
(44.48)
Equation (44.48) is helpful for actual calculations of the quantities g ( x ) . For example, for
the transformation (44.23), (44.48) can be arranged to yield
ij
k
k
∂z
1 N ∂z
1 N
gij( x ) = ( x ,..., x ) ( x ,..., x )
(44.49)
∂
i j
x ∂ x
since k l kl
⋅ =δ i i . For the cylindrical coordinate system defined by (44.25) a simple calculation
based upon (44.49) yields