Mathematical Physics Vol 1

7.7 Green formulas

371

and then the relation (7.223) can be written in the form y V div ( u · grad v ) d V = x S

u · grad v · d S ⇒

y V

( grad u · grad v + u · divgrad v ) d V = x S

u · grad v · n · d S ⇒

y V

u · ∆ v d V = x S

u · grad v · n · d S − y V

grad u · grad v · d V

(7.226)

From here follows the first Green formula y V u · ∆ v d V = x S u ·

d S − y V

∂ v ∂ n

∇ u · ∇ v d V .

(7.227)

Assume now that a = v · grad u , and we obtain similarly y V v · ∆ u d V = x S v · ∂ u ∂ n d S − y V

∇ v · ∇ u d V .

(7.228)

Subtracting equation (7.228) from (7.227) we obtain y V ( u · ∆ v − v · ∆ u ) d V = x S u ∂ v ∂ n − v ∂ u ∂ n d S ,

(7.229)

the second Green formula . Note that the relation

∆ u = f ( x , y , z )

(7.230)

is called the Poisson equation , where u is the unknown function, while f is known. Equation of the form ∆ u = 0 (7.231) as already mentioned, is called the Laplace equation. Each continuous function u = u ( x , y , z ) , which satisfies the Laplace equation (7.231), is called a harmonic function . Note that it is assumed that its first and second derivatives (which appear in the expression (7.231)) are also continuous functions. Let us now state some theorems, which we will use later in our proofs:

Theorem25 If S is a closed surface and u and v are harmonic functions, then x S ( u · gradv ) d S = x S ( v · gradu ) d S ,

(7.232)

or, written differently

x S

dS = x S

∂ v ∂ n

∂ u ∂ n

u ·

v ·

dS .

(7.233)