→ ℝ^{p}be a function. Thenγis of bounded variation if

{ }
n∑
sup |γ (ti)− γ(ti− 1)| : a = t0 < ⋅⋅⋅ < tn = b ≡ V (γ, [a,b]) < ∞
i=1

where the sums are taken over all possible lists,

{a = t0 < ⋅⋅⋅ < tn = b}

. The set of pointsγ

([a,b])

willalso be denoted byγ^{∗}. Whenγis one to one on [a,b) and continuous on

[a,b]

we callγ^{∗}asimplecurve.

The idea is that it makes sense to talk of the length of the curve γ

([a,b])

, defined as V

(γ,[a,b])

. For this
reason, in the case that γ is continuous, such an image of a bounded variation function is called a
rectifiable curve.

Definition 4.1.2Letγ :

[a,b]

→ ℝ^{p}be of bounded variation and let f : γ^{∗}→ ℝ^{p}. Letting P≡

{t0,⋅⋅⋅,tn}

where a = t_{0}< t_{1}<

⋅⋅⋅

< t_{n} = b, define

∥P ∥ ≡ max {|tj − tj−1| : j = 1,⋅⋅⋅,n}

and the Riemann Steiltjes sum by

∑n
S (P ) ≡ f (γ (τj))⋅(γ(tj)− γ(tj− 1))
j=1

where τ_{j}∈

[tj−1,tj]

. (Note this notation is a little sloppy because it does not identify the specific point,τ_{j}used. It is understood that this point is arbitrary.) Define∫_{γ}f ⋅ dγas the unique numberwhich satisfies the following condition. For all ε > 0 there exists a δ > 0 such that if

∥P∥

≤ δ,then

|∫ |
|| f ⋅dγ − S(P)||< ε.
|γ |

Sometimes this is written as

∫
f ⋅dγ ≡ lim S(P ).
γ ∥P∥→0

The set of points in the curve,γ

([a,b])

will be denoted sometimes byγ^{∗}. Also, when convenient, I willwrite∑_{P}to denote a Riemann sum.

Then γ^{∗} is a set of points in ℝ^{p} and as t moves from a to b,γ

(t)

moves from γ

(a)

to γ

(b)

. Thus γ^{∗}
has a first point and a last point.

Note that from the above definition, it is obvious that the line integral is linear. Simply let P_{n} refer to a
uniform parition of

[a,b]

and let τ_{j}^{n} be the midpoint of

[ ]
tnj−1,tnj

. Then for a,b scalars and f,g vector
valued functions,

∫ ∫ ( ∑ ( ( n)) ( (n) ( n )) )
a f ⋅dγ + b g ⋅dγ = limn→ ∞ a ∑Pn f γ( τ(j )⋅) γ( tj( )− γ t(j−1 ))
γ γ + limn→ ∞ b Pn g γ τnj ⋅ γ tnj − γ tnj−1
= lim ∑ (af (γ(τn))+ bg(γ (τn)))⋅(γ(tn)− γ (tn ))
n→ ∞ P j j j j−1
∫ n
≡ (af + bg)⋅dγ
γ

Another issue is whether the integral depends on the parametrization associated with γ^{∗} or only on γ^{∗} and
the direction of motion over γ^{∗}. If ϕ :

[c,d]

→

[a,b]

is a continuous nondecreasing function, then
γ∘ϕ :

[c,d]

→ ℝ^{p} is also of bounded variation and yields the same set of points in ℝ^{p} with the same first
and last points. The next theorem explains that one can use either γ or γ∘ ϕ and get the same
integral.