5.2. ESTIMATES AND APPROXIMATIONS 123

Let η ≡ γh. ■This is a very useful theorem because if γ is C1 ([a,b]) , it is easy to calculate

∫γ

f · dγ

and the above theorem allows a reduction to the case where γ is C1. The next theoremshows how easy it is to compute these integrals in the case where γ is C1.

Theorem 5.2.5 If f : γ∗→Rp is continuous and γ : [a,b]→Rp is in C1 ([a,b]) , then∫γ

f ·dγ =∫ b

af(γ (t)) · γ ′ (t)dt. (5.18)

Proof: Let P = {t0, · · · , tn} . Then

S (P) =n

∑k=1

f(γ (σ k)) · (γ (tk)− γ (tk−1)) =n

∑k=1

p

∑i=1

fi (γ (σ k))(γ i (tk)− γ i (tk−1))

By the mean value theorem, this is ∑nk=1 ∑

pi=1 fi (γ (σ k))γ ′i

(τ i

k

)(tk− tk−1) ,τ

ik ∈ (tk−1, tk) .

This isn

∑k=1

p

∑i=1

fi (γ (σ k))γ′i (σ k)(tk− tk−1)+ e(∥P∥)

where lim∥P∥→0 e(∥P∥) = 0. This follows from the uniform continuity of γ ′i. Then

∫γ

f ·dγ = lim∥P∥→0

S (P) = lim∥P∥→0

(n

∑k=1

p

∑i=1

fi (γ (σ k))γ′i (σ k)(tk− tk−1)+ e(∥P∥)

)

=∫ b

af(γ (t)) · γ ′ (t)dt ■

5.2.1 Finding the Length of a C1 Curve

It is very easy to find the length of a C1 curve.

Proposition 5.2.6 Let γ : [a,b]→ Rp be C1. Then V (γ) =∫ b

a |γ ′ (t)|dt

Proof: Let P̂ = {t0, · · · , tm} be such that

V (γ)− ε ≤m

∑j=1

∣∣γ (t j)− γ(t j−1

)∣∣≤V (γ)

Now using the same notation, let P = {t0, · · · , tn} be a partition containing P̂ so that theabove inequality will hold for all such P.

n

∑j=1

∣∣γ (t j)− γ(t j−1

)∣∣= n

∑j=1

∣∣∣∣∫ t j

t j−1

γ′ (s)ds

∣∣∣∣= n

∑j=1

(p

∑k=1

(∫ t j

t j−1

γ′k (s)ds

)2)1/2

=n

∑j=1

(p

∑k=1

(γ′k(sk j)(

t j− t j−1))2

)1/2

=n

∑j=1

(p

∑k=1

γ′k(sk j)2

)1/2 (t j− t j−1

)