Page 79 - Vector Analysis

§3.1 The Double Integrals 75

∆k = ∆(k1) ˆ ∆k(2) ˆ ¨ ¨ ¨ ˆ ∆k(n) as well as ∆1k = ∆1k(1) ˆ ∆k1(2) ˆ ¨ ¨ ¨ ˆ ∆k1(n). By the
definition of the upper sum,

N

U (f, P) = ÿ sup f A(x)ν(∆k)

k=1 xP∆k

= ÿ sup f A(x)ν(∆k) + ÿ sup f A(x)ν(∆k)

1ďkďN with xP∆k 1ďkďN with xP∆k

yj(i)R∆k(i)for all i, j yj(i)P∆(ki)for some i, j

and similarly,

U (f, P1) = ÿ sup f A(x)ν(∆k1 ) + ÿ sup f A(x)ν(∆k1 ) .
1ďkďN 1 with xP∆1k xP∆k1
1ďkďN 1 with

yj(i)R∆1k(i)for all i, j yj(i)P∆k1(i)for some i, j

By the fact that ∆k P P1 if yj(i) R ∆1k(i) for all i, j, we must have

ÿ ÿ ν(∆k1 ) .
ν(∆k) =

1ďkďN with 1ďkďN 1 with
yj(i)P∆k(i)for some i, j yj(i)P∆k1(i)for some i, j

The equality above further implies that

U (f, P)´U (f, P1) = ÿ sup f A(x)ν(∆k)´ ÿ sup f A(x)ν(∆k1 )
1ďkďN with xP∆k 1ďkďN 1 with xP∆k1

yj(i)P∆k(i)for some i, j yj(i)P∆1k(i)for some i, j

ď ( sup f A(R) ´ inf f A(R)) ÿ
ν(∆k) .

1ďkďN with
yj(i)P∆(ki)for some i, j

Moreover, for each fixed i, j,

ď [ r, r ]i´1 [yj(i) yj(i) ] [ r r ]n´i
´ δ 2
∆k Ď 2 2 ˆ ´ δ, + ˆ ´ , 2 ;

1ďkďN

yj(i) P∆k(i)

thus

ÿ ν(∆k) ď 2δrn´1 @ 1 ď i ď n, 1 ď j ď mi .

1ďkďN with
yj(i) P∆k(i)