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

310 Chap. 9 • EUCLIDEAN MANIFOLDS
j
where the operator r⋅
i between vector fields is defined pointwise in a similar fashion as in
1 2 N
i , i ,..., i be another basis for V related to the original basis
(44.17). As an illustration, let { }
by
j k
= Q k
j
i i (44.19)
and let 0 E
be another fixed point of E . Then by (44.13)
z
= ( x−0 ) ⋅ i = ( x−0 ) ⋅ i + ( 0 −0 ) ⋅ i
j j j j
E E E E
= Q ( x−0 ) ⋅ i + ( 0 −0 ) ⋅ i = Q z + c
j k j j k j
k E E E
k
(44.20)
where (44.19) has been used. Also in (44.20) the constant scalars
k
c , k 1,...,
= N , are defined by
k
k
c = ( 0E −0E ) ⋅ i
(44.21)
1 2
If the bases { , ,..., N
1 2 N
i i i } and { , ,..., }
i i i are both orthonormal, then the matrix
orthogonal. Note that the coordinate neighborhood is the entire space E .
⎡
⎣
j
Q k
⎤
⎦ is
curve
The j th coordinate curve which passes through 0 E
of the Cartesian coordinate system is the
λ() t = ti j
+ 0 E
(44.22)
1 2
N
Equation (44.22) follows from (44.10), (44.15), and the fact that for x = 0 E
, z = z =⋅⋅⋅= z = 0 .
As (44.22) indicates, the coordinate curves are straight lines passing through 0 E
. Similarly, the j th
coordinate surface is the plane defined by