Kenneth Kuttler

120 CHAPTER 5. LINE INTEGRALS AND CURVES

Proof: Let Sn be the closure of all Riemann sums for f corresponding to a partitionP which has ∥P∥ ≤ 1/n. Let P={t0, ..., tm} be a partition and let ∑

mi=1 f (τ i)(ti− ti−1)

and ∑mi=1 f (τ̂ i)(ti− ti−1) be two Riemann sums for such a partition P. Then, since f is

decreasing,∣∣∣∣∣ m

∑i=1

f (τ i)(ti− ti−1)−m

∑i=1

f (τ̂ i)(ti− ti−1)

∣∣∣∣∣ ≤ m

∑i=1

( f (ti−1)− f (ti))(ti− ti−1)

≤ ∥P∥( f (a)− f (b))

It follows that the diameter of Sn is no more than 1n ( f (a)− f (b)). Therefore, there is a

unique point in ∩∞n=1Sn and from the definition, lim∥P∥→0 S (P) exists and is the integral. ■

The same proof shows that increasing functions are Riemann integrable, and then thisgeneralizes to any function which is either increasing or decreasing on each of finitely manynon-overlapping intervals whose union is [a,b] will also be Riemann integrable. Thus allreasonable real valued functions are Riemann integrable.

5.2 Estimates and ApproximationsThe following theorem follows easily from the above definitions and theorem.

Theorem 5.2.1 Let f ∈ C (γ∗) and let γ : [a,b]→ Rp be of bounded variation andcontinuous. Let M be at least as large as the maximum of |f| on γ∗. That is,

M ≥max{|f◦ γ (t)| : t ∈ [a,b]} . (5.8)

Then ∣∣∣∣∫γ

f ·dγ

∣∣∣∣≤MV (γ, [a,b]) . (5.9)

Also if {fn} is a sequence of functions continuous on γ∗ which is converging uniformly tothe function f on γ∗, then

limn→∞

∫γ

fn ·dγ =∫

f ·dγ. (5.10)

Proof: Let 5.8 hold. From the proof of Theorem 5.1.5 on existence, when ∥P∥ <δ m,

∣∣∣∫γf ·dγ−S (P)

∣∣∣≤ 2mV (γ, [a,b]) and so

∣∣∣∫γf ·dγ

∣∣∣≤ |S (P)|+ 2mV (γ, [a,b]) Then by the

triangle inequality and Cauchy Schwarz inequality,

≤n

∑j=1

M∣∣γ (t j)− γ

(t j−1

)∣∣+ 2m

V (γ, [a,b])

≤ MV (γ, [a,b])+2m

V (γ, [a,b]) .

This proves 5.9 since m is arbitrary. To verify 5.10 use the above inequality to write∣∣∣∣∫γ

f ·dγ−∫

fn ·dγ

∣∣∣∣= ∣∣∣∣∫γ

(f− fn) ·dγ (t)∣∣∣∣

≤max{|f◦ γ (t)− fn ◦ γ (t)| : t ∈ [a,b]}V (γ, [a,b]) .

Since the convergence is assumed to be uniform, this proves 5.10. ■As an easy example of a curve of bounded variation, here is an easy lemma.

120 CHAPTER 5. LINE INTEGRALS AND CURVESProof: Let S, be the closure of all Riemann sums for f corresponding to a partitionP which has ||P|| < 1/n. Let P={to,...,tm} be a partition and let Y”, f (ti) (tj -—ti-1)and )°"_, f (7;) (t; —ti-1) be two Riemann sums for such a partition P. Then, since f isdecreasing,f(t) (i-t-1)} < YGGi)-f())G-t-1)I i=]I|PIl (F(a) — f (4)It follows that the diameter of S,, is no more than 4 (f(a) — f(b)). Therefore, there is aunique point in ;_,S;, and from the definition, limypj_.9S (P) exists and is the integral. HiThe same proof shows that increasing functions are Riemann integrable, and then thisgeneralizes to any function which is either increasing or decreasing on each of finitely manynon-overlapping intervals whose union is [a,b] will also be Riemann integrable. Thus allreasonable real valued functions are Riemann integrable.Ms:Y F(a (t;-ti-1) —lIA5.2 Estimates and ApproximationsThe following theorem follows easily from the above definitions and theorem.Theorem 5.2.1 Let f € C(y*) and let y: [a,b] + R? be of bounded variation andcontinuous. Let M be at least as large as the maximum of |f| on y*. That is,M > max {|foy(t)|:t € [a,b] }. (5.8)[earYAlso if {f,} is a sequence of functions continuous on ¥° which is converging uniformly tothe function f on y*, thenThen< MV (y, (a,b). (5.9)lim [t-ay= [tar (5.10)cy cyn—yooProof: Let 5.8 hold. From the proof of Theorem 5.1.5 on existence, when ||P|| <Sms [yf-dy—S(P)| < 2V(y,[a,b]) and so |J,f-dy| < |S(P)|+2V (7, [a,b]) Then by thetriangle inequality and Cauchy Schwarz inequality,lAYM inte) —r (1+ jV (releb)< MV(y,la.b)) +=V (1,[a,b).This proves 5.9 since m is arbitrary. To verify 5.10 use the above inequality to write[tar [a7 = [tm -are)< max {|fo y(t) —f, 0 y(t)| :t € [a,b]} VV (y, [a,5]).Since the convergence is assumed to be uniform, this proves 5.10. MlAs an easy example of a curve of bounded variation, here is an easy lemma.