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

332 Chap. 9 • EUCLIDEAN MANIFOLDS
Section 46. Anholonomic and Physical Components of Tensors
In many applications, the components of interest are not always the components with
g
j . For definiteness let us call the components of a
respect to the natural basis fields { } i
g and { }
tensor field A ∈T ∞
q
( U ) is defined by (45.20) the holonomic components of A . In this section, we
shall consider briefly the concept of the anholonomic components of A ; i.e., the components of
A taken with respect to an anholonomic basis of vector fields. The concept of the physical
components of a tensor field is a special case and will also be discussed.
Let
1
e a
denote a set of N vectors fields on U
1
, which are
linearly independent, i.e., at each x ∈U 1
, { e a } is a basis of V . If A is a tensor field in T ∞ q
( U )
where U ∩ ≠∅
1
U , then by the same type of argument as used in Section 45, we can write
U be an open set in E and let { }
a
a
1
q
A aa 1 2 ... a
e e
q
A = ⊗⋅⋅⋅⊗
(46.1)
or, for example,
b1
... b q
eb
e
1
bq
A = A ⊗⋅⋅⋅⊗
(46.2)
where { a
}
e is the reciprocal basis field to { }
e defined by
a
a
a
e ( x) ⋅ e ( x ) = δ
(46.3)
b
b
for all x ∈U 1
. Equations (46.1) and (46.2) hold on U ∩U 1
, and the component fields as defined by
A
...
= A ( e ,..., e )
(46.4)
aa 1 2 aq
a1
aq
and
b1 ... bq
b b
1
q
A = A ( e ,..., e )
(46.5)