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

326 Chap. 9 • EUCLIDEAN MANIFOLDS
With respect to the chart ( U , y)
2
ˆ
, we have also
k
v=υ
h
k
(45.12)
k
where υ : U ∩U2
→R. From (45.12), (45.11), and (45.8), the transformation rule for the
components of v relative to the two charts ( U , x ) and ( , y )
1 ˆ
U is
2
ˆ
υ
∂x
y
i
i
j
= υ
(45.13)
∂
j
for all x ∈U1∩U2∩U .
As in (44.18), we can define an inner product operation between vector fields. If
v1: U1
→V and v2: U2
→V are vector fields, then v1⋅
v
2
is a scalar field defined on U1∩U 2
by
v ⋅ v ( x) = v ( x) ⋅v ( x),
x∈U ∩U (45.14)
1 2 1 2 1 2
Then (45.10) can be written
i
υ =
i
v ⋅ g (45.15)
Now let us consider tensor fields in general. Let T ∞ q
( U ) denote the set of all tensor fields of
order q defined on an open set U in E . As with the set F ∞ ( U ) , the set T ∞ q
( U ) can be assigned
an algebraic structure. The sum of A : U → Tq( V ) and B : U → Tq( V ) is a C ∞ tensor field
A+ B : U →T ( V ) defined by
q
( A+ B)( x) = A( x) + B ( x)
(45.16)
for all
x∈U . If f ∈ F ∞ ( U ) and A ∈T ∞ ( U ), then we can define fA ∈T ∞ ( U ) by
q
q