6.3 Integration with respect to arc length

We introduce next what would appear to be the best parameterization of a piecewise smooth curve, i.e., a parameterization by arc length.We will then use this parameterization to define the integral of a function whose domain is the curve.

Let $C$ be a piecewise smooth curve of finite length $L$ joining two distinct points $z_1$ to $z_2.$ Then there exists a parameterization $\gamma :[0,L]\to C$ for which the arc length of the curve joining $\gamma \left(t\right)$ to $\gamma \left(u\right)$ is equal to $|u-t|$ for all $t<u\in [0,L].$

Let $\phi :[a,b]\to C$ be a parameterization of $C.$ Define a function $F:[a,b]\to [0,L]$ by

In other words, $F\left(t\right)$ is the length of the portion of $C$ that joins the points $z_1=\phi \left(a\right)$ and $\phi \left(t\right).$ By the Fundamental Theorem of Calculus, we know that the function $F$ is continuous on the entire interval $[a,b]$ and is continuously differentiable on every subinterval $(t_{i-1},t_i)$ of the partition $P$ determined by the piecewise smooth parameterization $\phi .$ Moreover, $F^{\text{'}}\left(t\right)=\left|{\phi }^{\text{'}}\left(t\right)\right|>0$ for all $t\in (t_{i-1},t_i),$ implying that $F$ is strictly increasing on these subintervals. Therefore, if we write $s_i=F\left(t_i\right),$ then the $s_i$ 's form a partition of the interval $[0,L],$ and the function $F:(t_{i-1},t_i)\to (s_{i-1},s_i)$ is invertible, and its inverse $F^{-1}$ is continuously differentiable. It follows then that $\gamma =\phi \circ F^{-1}:[0,L]\to C$ is a parameterization of $C.$ The arc length between the points $\gamma \left(t\right)$ and $\gamma \left(u\right)$ is the arc length between $\phi \left(F^{-1}\left(t\right)\right)$ and $\phi \left(F^{-1}\left(u\right)\right),$ and this is given by the formula

which completes the proof.

If $\gamma$ is the parameterization by arc length of the preceding theorem, then, for all $t\in (s_{i-1},s_i),$ we have $|{\gamma }^{\text{'}}\left(s\right)|=1.$

We just compute

as desired.

We are now ready to make the first of our three definitions of integral over a curve. This first one is pretty easy.

Suppose $C$ is a piecewise smooth curve joining $z_1$ to $z_2$ of finite length $L,$ parameterized by arc length. Recall that this means that there is a 1-1 function $\gamma$ from the interval $[0,L]$ onto $C$ that satisfies the condidition that the arc length betweenthe two points $\gamma \left(t\right)$ and $\gamma \left(s\right)$ is exactly the distance between the points $t$ and $s.$ We can just identify the curve $C$ with the interval $[0,L],$ and relative distances will correspond perfectly. A partition of the curve $C$ will correspond naturally to a partition of the interval $[0,L].$ A step function on the dcurve will correspond in an obvious way to a step function on the interval $[0,L],$ and the formula for the integral of a step function on the curve is analogous to what it is on the interval.Here are the formal definitions:

Let $C$ be a piecewise smooth curve of finite length $L$ joining distinct points, and let $\gamma :[0,L]\to C$ be a parameterization of $C$ by arc length. By a partition of $C$ we mean a set $\{z_0,z_1,...,z_n\}$ of points on $C$ such that $z_j=\gamma \left(t_j\right)$ for all $j,$ where the points $\{t_0<t_1<...<t_n\}$ form a partition of the interval $[0,L].$ The portions of the curve between the points $z_{j-1}$ and $z_j,$ i.e., the set $\gamma (t_{j-1},t_j),$ are called the elements of the partition.

A step fucntion on $C$ is a real-valued function $h$ on $C$ for which there exists a partition $\{z_0,z_1,...,z_n\}$ of $C$ such that $h\left(z\right)$ is a constant $a_j$ on the portion of the curve between $z_{j-1}$ and $z_j.$