Begin2.DVI

Figure 8-14. Multiply-connected region.

Such a region can be converted to a simply connected region by introducing cuts

Γi, i = 1 ,... , n. Observe that one can integrate along C 0 until one comes to a cut,

say for example the cut Γ 1 in figure 8-14. Since it is not possible to cross a cut, one

must integrate along Γ 1 to the curve C 1 , then move about C 1 clockwise and then

integrate along Γ 1 back to the curve C 0 .Continue this process for each of the cuts

one encounter as one moves around C 0 .Note that the line integrals along the cuts

add to zero in pairs (i.e. from C 0 to Ciand from Cito C 0 for each i= 1, 2 ,.. .n ), then

one is left with only the line integrals around the curves C 0 , C 1 ,... , C n in the sense

illustrated in figure 8-14(b).

Green’s First and Second Identities

Two special cases of the divergence theorem, known as Green’s first and second

identities, are generated as follows.

In the divergence theorem, make the substitution F
=ψ∇φto obtain

∫∫∫

V

∇F dV
 =

∫∫∫

V

∇(ψ∇φ)dV =

∫∫

S

ψ∇φ·dS
=

∫∫

S

ψ

∂φ

∂n dS (8 .60)

where

∂φ

∂n =∇φ·

ˆen is known as a normal derivative at the boundary. Using the

relation

∇(ψ∇φ) = ψ∇^2 φ+∇ψ·∇ φ