Page 107 - Vector Analysis

§4.1 The Line Integrals 103

and ε
1)(b
››γ 1(s) ´ γ 1(t)››Rn ă 8(M + ´ a) whenever s, t P [a, b], |s ´ t| ă δ .

żb ( )
Moreover, since f ˝ γ and γ 1 are both continuous on [a, b], the integral f γ(t) }γ 1(t)}Rn dt

a

exists; thus there exists a partition P = ta = t0 ă t1 ă ¨ ¨ ¨ ă tk = bu of [a, b] such that

k( sup () ( )) ε (4.2)
ÿ f (γ(s))}γ 1(s)}Rn inf f (γ(s))}γ 1(s)}Rn |ti 2

´ sP[ti´1,ti] ´ ti´1| ă .

i=1 sP[ti´1,ti]

By choosing of a refinement of P if necessary, we can assume that }P} ă δ. Let si, ri P

[ti´1, ti] be such that

sup ( (γ(t))}γ 1(t)}Rn ) = f (γ(si))››γ 1(si)››Rn and sup f (ξ) = f (γ(ri)) .
f

tP[ti´1,ti] ξPγ([ti´1,ti])

Moreover, by Theorem 4.10 and the mean value theorem for integrals, there exists qi P
[ti´1, ti] such that

ż ti

ℓ(γ([ti´1, ti])) = ››γ 1(s)››Rn ds = ››γ 1(qi)››Rn|ti ´ ti´1| ;

ti´1

thus ε
1)(b
ˇ ˇ
ti´1|ˇˇ
ˇℓ(γ ([ti´1, ti ])) ´ ››γ 1(si)››Rn|ti ´ ď 8(M + ´ a) |ti ´ ti´1| .
ˇ

Therefore, by the fact that si, ri, qi P [ti´1, ti] and |ti ´ ti´1| ă δ,

ˇ( ) ˇ

ˇ sup f (γ(s))}γ 1(s)}Rn |ti ´ ti´1| ´ sup f (ξ)ℓ(γ([ti´1, ti]))ˇˇ
ˇ

sP[ti´1,ti] ξPγ([ti´1,ti])

ˇˇ

= ˇf (γ(si))››γ 1(si)››Rn ´ f (γ(ri))}γ 1 (qi )››Rn ˇˇ|ti ´ ti´1|
ˇ

ď ˇˇf (γ(si)) ´ f (γ(ri)ˇˇ››γ 1(si)››Rn|ti ´ ti´1| + ˇˇf (γ(ri))ˇˇ}γ 1(si) ´ γ 1(qi)››Rn|ti ´ ti´1|
ε

ă 4(b ´ a) |ti ´ ti´1| ,

and summing the inequality above over i we obtain that

k () k ˇ ε
f (γ(s))}γ 1(s)}Rn |ti 4
ˇÿ sup ÿ sup
ˇ
´ ti´1| ´ f (ξ)ℓ(γ ([ti´1, ti ]))ˇ ă .
ˇ ˇ

i=1 sP[ti´1,ti] i=1 ξPγ([ti´1,ti])

Similarly,

ˇ k ( ) k ˇ ε
ˇ f |ti ]))ˇ 4
ˇ ÿ inf (γ (s))}γ 1 (s)}Rn ´ ti´1| ´ ÿ inf f (ξ )ℓ(γ ([ti´1 , ti ă ;
ˇ
i=1 sP[ti´1,ti ] i=1 ξPγ([ti´1 ,ti])