(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)