2  Some Properties of the Kernel Functions

The results given in this section are extracted mainly from references [8], [9], and [14]. In the following r = p - q and r = |r|, G_k = G_k(p,q), G₀ = G₀(p,q).

2.1  Derivatives of G₀ with respect to r

In two dimensions we have

2.2  Derivatives of G_k (k ¹ 0) with respect to r

In two dimensions we have

In three dimensions we have

2.3  Expressions for the normal derivatives of G_k

The following expressions hold in both two and three dimensions and for all k:

2.4  Expressions for the normal derivative of r

The derivatives of r with respect to v_p and n_q may be written as follows:

2.5  Expressions for ¶² G₀/ ¶v_p ¶n_q

The following results can be derived from the substitution of (25) and (12),(13) or (14),(15) into (22) with k = 0:

2.6  Asymptotic Properties

In the following results, p, q Î G where G is a surface and G is smooth at p: