3.2  Some Properties of the Kernel Functions

3.2.1  Derivatives of G0 with respect to r

In two dimensions we have

G0
r
 = - 1
2 p
 1
r
   ,
(18)

2 G0
r2
 = 1
2 p
 1
r2
   .
(19)
In three dimensions we have

G0
r
 = - 1
4 p
 1
r2
   ,
(20)

2 G0
r2
 = 1
2 p
 1
r3
   .
(21)

3.2.2  Expressions for the normal derivative of r

The derivatives of r with respect to vp and nq may be written as follows:

r
nq
 = - r.nq
r
   ,
(22)

r
vp
 = r.vp
r
   ,
(23)

2 r
vp nq
 = - 1
r
 ( vp.nq + r
vp
 r
nq
 )  .
(24)

3.2.3  Expressions for [(2 G0)/( vp nq)]

2 G0
vp nq
 = 1
2 pr2
 ( vp.nq + 2 r
vp
 r
nq
 )  in two dimensions,
(25)

G0
vp nq
 = 1
4 pr3
 ( vp.nq + 3 r
vp
 r
nq
 )  in three dimensions.
(26)

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