6.2 Arc length  (Page 2/3)

Next we develop a definition of the length of a parameterized curve from a purely mathematical or geometric point of view.Happily, it will turn out to coincide with the physically intuitive definition discussed above.

Let $C$ be a piecewise smooth curve joining the points $z_1$ and $z_2,$ and let $\phi :[a,b]\to C$ be a parameterization of $C.$ Let $P=\{a=t_0<t_1<...<t_n=b\}$ be a partition of the interval $[a,b].$ For each $0\le j\le n$ write $z_j=\phi \left(t_j\right),$ and think about the polygonal trajectory joining these points $\left\{z_j\right\}$ in order. The length $L_P^{\phi }$ of this polygonal trajectory is given by the formula

and this length is evidently an approximation to the length of the curve $C.$ Indeed, since the straight line joining two points is the shortest curve joining those points, these polygonal trajectories all should have a length smaller than or equal to the length of the curve.These remarks motivate the following definition.

Let $\phi :[a,b]\to C$ be a parameterization of a piecewise smooth curve $C\subset C.$ By the length $L^{\phi }$ of $C$ , relative to the parameterization $\phi ,$ we mean the number $L^{\phi }={sup}_P{L}_P^{\phi },$ where the supremum is taken over all partitions $P$ of $[a,b].$

Of course, the supremum in the definition above could well equal infinity in some cases.Though it is possible for a curve to have an infinite length, the ones we will study here will have finite lengths.This is another subtlety of this subject. After all, every smooth curve is a compact subset of $R^2,$ since it is the continuous image of a closed and bounded interval, and we think of compact sets as being “finite” in various ways.However, this finiteness does not necessarily extend to the length of a curve.

Of course, we again face the annoying possibility that the definition of length of a curve will depend on the parameterization we are using.However, the next theorem, taken together with [link] , will show that this is not the case.

If $C$ is a piecewise smooth curve parameterized by $\phi :[a,b]\to C,$ then

specifically meaning that one of these quantities is infinite if and only if the other one is infinite.

We prove this theorem for the case when $C$ is a smooth curve, leaving the general argument for a piecewise smooth curve to the exercises.We also only treat here the case when $L^{\phi }$ is finite, also leaving the argument for the infinite case to the exercises.Hence, assume that $\phi =u+iv$ is a smooth function on $[a,b]$ and that $L^{\phi }<\infty .$

Let $ϵ>0$ be given. Choose a partition $P=\{t_0<t_1<...<t_n\}$ of $[a,b]$ for which

Because $\phi$ is continuous, we may assume by making a finer partition if necessary that the $t_j$ 's are such that $|\phi \left(t_1\right)-\phi \left(t_0\right)|<ϵ$ and $|\phi \left(t_n\right)-\phi \left(t_{n-1}\right)|<ϵ.$ This means that

The point of this step (trick) is that we know that ${\phi }^{\text{'}}$ is continuous on the open interval $(a,b),$ but we will use that it is uniformly continuous on the compact set $[t_1,t_{n-1}].$ Of course that means that $|{\phi }^{\text{'}}|$ is integrable on that closed interval, and in fact one of the things we need to prove is that $|{\phi }^{\text{'}}|$ is improperly-integrable on the open interval $(a,b).$