11.8 DIVERGENCE THEOREM AND RELATED THEOREMS
vector fieldφ∇ψwe obtain
∮
S
φ∇ψ·dS=
∫
V
∇·(φ∇ψ)dV
=
∫
V
[
φ∇^2 ψ+(∇φ)·(∇ψ)
]
dV. (11.19)
Reversing the roles ofφandψin (11.19) and subtracting the two equations gives
∮
S
(φ∇ψ−ψ∇φ)·dS=
∫
V
(φ∇^2 ψ−ψ∇^2 φ)dV. (11.20)
Equation (11.19) is usually known as Green’s first theorem and (11.20) as his
second. Green’s second theorem is useful in the development of the Green’s
functions used in the solution of partial differential equations (see chapter 21).
11.8.2 Other related integral theorems
There exist two other integral theorems which are closely related to the divergence
theorem and which are of some use in physical applications. Ifφis a scalar field
andb is a vector field and bothφandbsatisfy our usual differentiability
conditions in some volumeVbounded by a closed surfaceSthen
∫
V
∇φdV=
∮
S
φdS, (11.21)
∫
V
∇×bdV=
∮
S
dS×b. (11.22)
Use the divergence theorem to prove (11.21).
In the divergence theorem (11.18) leta=φc,wherecis a constant vector. We then have
∫
V
∇·(φc)dV=
∮
S
φc·dS.
Expanding out the integrand on the LHS we have
∇·(φc)=φ∇·c+c·∇φ=c·∇φ,
sincecis constant. Also,φc·dS=c·φdS,soweobtain
∫
V
c·(∇φ)dV=
∮
S
c·φdS.
Sincecis constant we may take it out of both integrals to give
c·
∫
V
∇φdV=c·
∮
S
φdS,
and sincecis arbitrary we obtain the stated result (11.21).
Equation (11.22) may be proved in a similar way by lettinga=b×cin the
divergence theorem, wherecis again a constant vector.