(i) Scalar source functions; {h (t, r, 0,) h, (t, r, 0, ) h (t, r, 0,)},
(ii) Vector source functions; {hso (t, r, 0, ) h, (t, r, 0, ), hS (t, r,, ) ,
h^ (t: r! d) },
(iii) Tensor source functions; {hSo (t, r, 0) hs, (t, r, 0, ) h (t, r, 0, Q)}.
The regularization parameters for the three scalars {h,, ht, h,} can be calculated
directly from the scalar source functions (i) above. However, the cases of vectors
{ht, h6, h r, h00} and tensors {hS, hS, h4} are more complicated. Their
regularization parameters cannot be determined directly from the source functions
(ii) and (iii) above. They must reflect the distinct properties under a rotation on
the two-dimensional submanifold E as shown in the previous Subsection. Then
the proper functional forms for vectors and tensors, from which our regularization
parameters are calculated, must be determined by considering their geometrical
properties. Detweiler and Whiting [11] present clear analyses on this, which
will provide us with the framework for the calculations of vector and tensor
regularization parameters. The brief summary follows:
(A) Vectors
An arbitrary vector field Aa (E {h hs hs, hs}), defined on the two-
dimensional submanifold Z, may be represented by two scalar fields Aev and
Aod as
A = 'V7bAev + CabV Aod, (5-52)
where Aev and Aod are called the even and odd .'n, :/;/ potentials of Aa, and are
each unique up to the addition of a constant. The components Aev and Aod
are determined by
VaVaAev V a (5-53)
and
VaVaAod -abVaAA.
(5-54)