Mathematics of Physics and Engineering

Laplace's and Poisson's Equations 159

Theorem 3.2.5 Let G be a domain in R^3. Assume that the boundary dG
of G consists of finitely many closed piece-wise smooth orientable surfaces.
As usual, we choose the orientation of dG so that every unit normal vector
to dG points outside of G. Let f, g be two scalar fields defined in an open
set G\ containing G.
(a) If f is continuously differentiable in G\ and g is twice continuously
differentiable, then

HI (V

2

<?) fdV = - HI Vf-VgdV + Hf^ da. (3.2.19)

G G dG

(b) If both f and g are twice continuously differentiable in G\, then

///(/V^- 9 W)^ = //(/|- 9 i) *. (3.2,0)

G dG

EXERCISE 3.2.11.c (a) Prove (3.2.19). Hint: apply Gauss's Theorem to the

vector field fVg. (b) Prove (3.2.20). Hint: in (3.2.19), switch f and g, then

take the difference of the two resulting identities.

Equalities (3.2.19) and (3.2.20) are known as Green's first and

second formulas (or identities).

EXERCISE 3.2.12? State and prove the two-dimensional versions of (3.2.18),

(3.2.19), and (3.2.20). Hint: you just have to replace triple integrals with

double, and surface integrals with line. For example, (3.2.19) becomes

II W^2 gdA = - II Vf -VgdA + j f ^da. (3.2.21)

G G dG

Laplace's and Poisson's equations are examples of partial differential

equations. While there exist many different methods for finding explicit so-

lutions of ordinary differential equations, most partial differential equations

are not explicitly solvable. As a result, theorems that ensure existence and

uniqueness of solutions of such equations are very useful. Below, we discuss

uniqueness of solution for the Poisson equation.

Theorem 3.2.6 Let S be a closed orientable piece-wise smooth surface.

Consider two functions fi, fi that are continuously differentiable in an open

set containing S and are harmonic everywhere in the domain Gs enclosed

byS.

(a) If fi = /2 on S, then f\ = fi everywhere in Gs-