Potential Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 407

ter) is convergent and

( , ) < ε .

F M P d S

S
δ

The generalized integral in Eq. (7.147) is called uniformly convergent at point M 0
on S .

Theorem 1. If the integral in Eq. (7.147) is uniformly convergent at the point M 0 on
the surface S ,then v is continuous at M 0 .

Proof . By the definition of uniform convergence, we can choose a sufficiently small
neighboring region V of M 0 such that its intersecting part with S is on S δ .The
integral in Eq. (7.147) thus converges at all points M in V that are sufficiently close
to M 0 . By the definition of the uniform convergence, we have, for M 0 and M

( M )

∈ V ,

< 3 , < 3 .

F ( M 0 , P ) d S F ( M , P ) d S

S S

δ δ

Since F ( M , P ) is a uniformly continuous function of M ∈ V for P ∈ S \ S δ ,wehave,

for M sufficiently close to M 0 ,

< 3 .

| ( , ) − ( , ) |

F M P F M 0 P d S

S \ S

δ

Thus

| v ( M ) − v ( M 0 ) | = F ( M , P ) d S − F ( M 0 , P ) d S

S S

≤ F ( M , P ) d S + F ( M 0 , P ) d S

S S

δ δ

+ [ F ( M , P ) − F ( M 0 , P )] d S < ε ,

S \ S

δ

( )

so v is continuous at M 0 .

By a similar approach, we can show that this result is also valid for triple integrals
and line integrals.

M

Theorem 2 .Let S be a bounded smooth surface. Suppose that F ( M , P ) is continuous

when M = P . If, for a point M 0 on S , there exists a neighboring region V of M 0 and