Page 74 - Vector Analysis

70 CHAPTER 3. Multiple Integrals

A collection of rectangles P is called a partition of A if there exists partitions P(k) of
[ak, bk], k = 1, ¨ ¨ ¨ , n, P(k) = ␣ak = x0(k) ă x(1k) ă ¨ ¨ ¨ ă xN(kk) = bk(, such that

! ˇ [x(i11), xi(11+) 1] [x(i22), xi(22+) 1] ˆ [x(inn), x(n+1) ]
∆i1i2¨¨¨in in+1 ,
P = ˇ ∆i1 i2¨¨¨in = ˆ ¨¨¨ ˆ
ˇ

)
ik = 0, 1, ¨ ¨ ¨ , Nk ´ 1, k = 1, ¨ ¨ ¨ , n .

The mesh size of the partition P, denoted by }P}, is defined by

g
fn
! ˇ )
}P } = max ÿ (x(ikk+) 1 x(ikk))2 ˇ ik = 0, 1, ¨ ¨ ¨ , Nk 1, k = 1, ¨ ¨ ¨ ,n .
f ˇ
´ ´
e

k=1

d
n

The number ř (x(ikk+) 1 ´ x(ikk))2 is often denoted by diam(∆i1i2¨¨¨in), and is called the di-
k=1

ameter of the rectangle ∆i1i2¨¨¨in.

Definition 3.4. Let A Ď Rn be a bounded set, and f : A Ñ R be a bounded function. For
any partition

! ˇ [x(i11), xi(11+) 1] [x(i22), x(i22+) 1] ˆ [xi(nn), x(n+1) ]
∆i1i2¨¨¨in in+1 ,
P = ˇ ∆i1 i2¨¨¨in = ˆ ¨¨¨ ˆ
ˇ

)
ik = 0, 1, ¨ ¨ ¨ , Nk ´ 1, k = 1, ¨ ¨ ¨ , n ,

the upper sum and the lower sum of f with respect to the partition P, denoted by
U (f, P) and L(f, P) respectively, are numbers defined by

U (f, P) = ÿ sup f A(x)νn(∆) ,

∆PP xP∆

L(f, P) = ÿ inf f A(x)νn(∆) ,
xP∆
∆PP

where νn(∆) is the n-dimensional volume of the rectangle ∆ given by

νn(∆) = (x(i11+) 1 ´ x(i11))(x(i22+) 1 ´ xi(22)) ¨ ¨ ¨ (xi(nn+) 1 ´ xi(nn))

if ∆ = [x(i11) ´ x(i11+) 1] ˆ [x(i22) ´ x(i22+) 1] ˆ ¨ ¨ ¨ ˆ [xi(nn) ´ xi(nn+) 1 ] and fA is the extension of f by
,

zero outside A given by

f A(x) = " f (x) x P A , (3.1)
0 x R A.