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

336 Chap. 9 • EUCLIDEAN MANIFOLDS
aa 1 2...
aq
12
a1
... aq
( aa )
1 1 aqaq
−
( ) ( )
q q
A = g ⋅⋅⋅g A
12 12
a2
... aq
aa 1 1 aa 2 2 a a a1
= g g ⋅⋅⋅g A
⋅
⋅
⋅
−
( ) ( )
12 12
aq
aa 1 1 aq−1aq−1 aqaq a1...
aq−1
= g ⋅⋅⋅g g A
(46.19)
In mathematical physics, tensor fields often arise naturally in component forms relative to
e are fields of bases,
product bases associated with several bases. For example, if { } a
ˆ b
e and { }
possibly anholonomic, then it might be convenient to express a second-order tensor field A as a
field of linear transformations such that
bˆ
a aeb
A e = A ˆ , a = 1,..., N
(46.20)
In this case A has naturally the component form
bˆ
a b
a
A = A eˆ
⊗ e
(46.21)
a
Relative to the product basis { eˆ
⊗ b
e } formed by { e ˆ b } and { e
a
}
basis of { e a } as usual. For definiteness, we call { ˆ ⊗ a
b }
with the bases { e ˆ b } and { e
a
}. Then the scalar fields
called the composite components of A , and they are given by
, the latter being the reciprocal
e e a composite product basis associated
ˆb
A
a
defined by (46.20) or (46.21), may be
bˆ
a
b
A = A ( eˆ
⊗e
)
(46.22)
a
Similarly we may define other types of composite components, e.g.,
ba ˆ
b a
Aˆ = A( eˆ , e ), A = A ( eˆ
, e )
(46.23)
ba
b
a
etc., and these components are related to one another by