ABSTRACT
Gauss' divergence theorem has several useful, easy formulas, as corollaries which are collectively known as Green's identities. Setting v = u grad f , we getZ
div(u grad f) d =
Z
u grad f n d = Z
u df
dn d: (G.1)
By the same token:Z
div(f grad u) d =
Z
f grad u n d = Z
f du
dn d: (G.2)
However
div(u grad f) = ur2f + grad u grad f (G.3) div(f grad u) = fr2u+ grad f grad u; (G.4)
so, by subtracting the last two dierential identities, we get
div(u grad f) div(f grad u) = ur2f fr2u; (G.5) so that Z
(ur2f fr2u)d = Z
u df
dn f du
dn
d: (G.6)
The name Green is also associated with the so-called Green's theorem, which can be summarized in the following formula (known as the Green's formula for the 2-D case)Z Z
P
x Q y
(x; y) =
Z P
(x; y)dx+Q(x; y)dy: (G.7)
As shown in Appendix D, this formula can be regarded as a particular case of the more general Stoke's formula, or as a particular case of Gauss' divergence theorem, when applied to a 3-D body of cylindrical shape with a body (of unit-height plane cross-section ) acting as a domain for the 2-D vector eld v = Qi P j.
