Mathematical Methods for Physics and Engineering : A Comprehensive Guide

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.