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

328 Chap. 9 • EUCLIDEAN MANIFOLDS
U is the coordinate neighborhood for a chart ( )
for all x∈U ∩U 1
, where U , x
1
. The scalar fields
A are the covariant components of A and under a change of coordinates obey the
i1 ... iq
transformation rule
1 ˆ
A
i1
∂x
∂x
= ⋅⋅⋅
∂y
∂y
iq
A
k1... kq
k1
kq
i1...
iq
(45.22)
Equation (45.22) is a relationship among the component fields and holds at all points
x∈U1∩U2∩U where the charts involved are ( U , xˆ
1 ) and ( U ˆ ) 2, y . We encountered an example of
(45.22) earlier with (44.48). Equation (44.48) shows that the g
ij
are the covariant components of a
tensor field I whose value is the identity or metric tensor, namely
I = g g ⊗ g = g ⊗ g = g ⊗ g = g g ⊗g (45.23)
i j j j ij
ij j j i j
for points ∈
U , x
1
. Equations (45.23) show that the
components of a constant tensor field are not necessarily constant scalar fields. It is only in
Cartesian coordinates that constant tensor fields have constant components.
x U , where the chart in question is ( )
1 ˆ
Another important tensor field is the one constructed from the positive unit volume tensor
i , which has positive orientation, E is given by (41.6),
E . With respect to an orthonormal basis { j }
i.e.,
E = i ∧⋅⋅⋅∧ i = i ⊗⋅⋅⋅⊗i
(45.24)
1 N
εi 1 ⋅⋅⋅ iN i1
iN
Given this tensor, we define as usual a constant tensor field E : E → Tˆ
( V ) by
N
E( x)=
E (45.25)
for all
x ∈E . With respect to a chart ( U , x ) , it follows from the general formula (42.27) that
1 ˆ
E = E ⊗⋅⋅⋅⊗ = ⊗⋅⋅⋅⊗
(45.26)
i1
iN
i1
iN
i1 i
E ⋅⋅⋅
⋅⋅⋅
g g g
N
i
g
1
iN