518 CHAPTER 25. LINE INTEGRALS

and so∣∣R′ (s)

∣∣= 1 as claimed. R(l) = r(φ−1 (l)

)= r

(φ−1(∫ b

a |r′ (τ)|dτ

))= r (b) and

R(0) = r(φ−1 (0)

)= r (a) and R delivers the same set of points in the same order as r

because dsdt > 0.

The arc length parameter is just like any other parameter, in so far as considerationsof line integrals are concerned, because it was shown above that line integrals are inde-pendent of parametrization. However, when things are defined in terms of the arc lengthparametrization, it is clear they depend only on geometric properties of the curve itself andfor this reason, the arc length parametrization is important in differential geometry.

Definition 25.1.7 Recall piecewise smooth curves are just smooth curves joinedtogether at a succession of points p1,p2, · · · ,pm. If C is such a curve which goes from p1then to p2 then to p3 etc. one defines∫

CF ·d R≡

∫Cp1p2

F ·d R+∫

Cp2p3

F ·d R+ · · ·+∫

Cp(n−1)P n

F ·dR

25.2 Conservative Fields and NotationConservative vector fields are the gradient of some scalar function.

Proposition 25.2.1 Suppose C is a piecewise smooth curve which goes from p to q.Also suppose that F (x) = ∇φ (x) . Then

∫C F ·dR= φ (q)−φ (p) .

Proof: Say r (t) , t ∈ [ai,bi] is a parametrization for C going from xi−1 to xi and r isa parameterization for the smooth curve from xi−1 to xi with x0 = p and xm = q. Then,from the chain rule,∫

CF ·dR =

m

∑i=1

∫ bi

ai

∇φ (ri (r)) ·r′i (t)dt =m

∑i=1

∫ bi

ai

ddt

(φ (ri (t)))dt

=m

∑i=1

φ (xi)−φ (xi−1) = φ (q)−φ (p)

Note how this says that the integral is path independent, depending only on the valuesof the function φ , called a potential function, at the end points.

Definition 25.2.2 Let F (x,y,z) = (P(x,y,z) ,Q(x,y,z) ,R(x,y,z)) and let C be anoriented curve. Then another way to write

∫C F ·dR is

∫C Pdx+Qdy+Rdz

This last is referred to as the integral of a differential form, Pdx+Qdy+Rdz. Thestudy of differential forms is important. Formally, d R= (dx,dy,dz) and so the integrandin the above is formally F ·dR. Other occurrences of this notation are handled similarly in2 or higher dimensions.

25.3 Exercises1. Let r (t) =

(ln(t) , t2

2 ,√

2t)

for t ∈ [1,2]. Find the length of this curve.

2. Let r (t) =( 2

3 t3/2, t, t)

for t ∈ [0,1]. Find the length of this curve.