262 CHAPTER 14. VECTOR VALUED FUNCTIONS OF ONE VARIABLE
14.4.1 Arc Length And OrientationsThe application of the integral considered here is the concept of the length of a curve.
Definition 14.4.1 C is a smooth curve in Rn if there exists an interval [a,b]⊆R and func-tions xi : [a,b]→ R such that the following conditions hold
1. xi is continuous on [a,b].
2. x′i exists and is continuous and bounded on [a,b], with x′i (a) defined as the derivativefrom the right,
limh→0+
xi (a+h)− xi (a)h
,
and x′i (b) defined similarly as the derivative from the left.
3. For p(t)≡ (x1 (t) , · · · ,xn (t)), t→ p(t) is one to one on (a,b).
4. |p′ (t)| ≡(
∑ni=1 |x′i (t)|
2)1/2ΜΈ= 0 for all t ∈ [a,b].
5. C = ∪{(x1 (t) , · · · ,xn (t)) : t ∈ [a,b]}.
The functions xi (t), defined above are giving the coordinates of a point in Rn and thelist of these functions is called a parametrization for the smooth curve. Note the naturaldirection of the interval also gives a direction for moving along the curve. Such a directionis called an orientation. The integral is used to define what is meant by the length of such asmooth curve. Consider such a smooth curve having parametrization (x1, · · · ,xn). Forminga partition of [a,b], a = t0 < · · · < tn = b and letting pi = ( x1 (ti), · · · , xn (ti) ), you couldconsider the polygon formed by lines from p0 to p1 and from p1 to p2 and from p3 to p4etc. to be an approximation to the curve C. The following picture illustrates what is meantby this.
p0
p1
p2
p3
Now consider what happens when the partition is refined by including more points.You can see from the following picture that the polygonal approximation would appear tobe even better and that as more points are added in the partition, the sum of the lengthsof the line segments seems to get close to something which deserves to be defined as thelength of the curve C.
•
p0
p1
p2
p3
Thus the length of the curve is approximated byn
∑k=1|p(tk)−p(tk−1)| .