Vector Calculus: An Introduction

The length of the line segment connecting $\mathit{\gamma }(t_{i-1})$ to $\mathit{\gamma }(t_i)$ is $\Vert \mathit{\gamma }(t_i)-\mathit{\gamma }(t_{i-1})\Vert \text{,}$ so the total length of the polygon is

The length of the polygon will always be smaller than the length of the curve as we always take a `shortcut’ from one point to the next on the polygon. If we take a finer partition, that is, make $max(t_i-t_{i-1})$ smaller, we expect to get a closer approximation of the length of $C\text{.}$ This motivates the following definition.

The above definition is not very handy to compute the length of $C\text{.}$ Using that the curve is smooth we want to derive a formula to compute its length. Setting $\mathrm{\Delta }{t}_i:=t_i-t_{i-1}$ we can rewrite (7.2) as

As $\mathrm{\Delta }{t}_i\to 0$ the $i$-th fraction in the above expression tends to ${\mathit{\gamma }}^{\prime }(t_{i-1})\text{,}$ so the above sum is very close to

if $max\mathrm{\Delta }{t}_i$ is small. The last sum is a Riemann sum for the function $t\mapsto \Vert {\mathit{\gamma }}^{\prime }(t)\Vert \text{,}$ and converges to

as $max\mathrm{\Delta }{t}_i$ tends to zero. With some effort involved to estimate the error terms in the above procedure one can show that in fact the following is true.

Let us finally look at two special cases of the above formula. First assume that $f$ is a continuously differentiable function on the interval $[a,b]\text{.}$ Then the function

defines a smooth curve in ${\mathbb{R}}^2\text{,}$ namely the graph of $f\text{.}$ It follows that ${\mathit{\gamma }}^{\prime }(t):=(1,f^{\prime }(t))\text{,}$ and applying Proposition 7.8 we see that the

Next we suppose that the curve is given in polar coordinates, that is, the radius as a function of the angle: $r=f(\phi )\text{,}$ $\phi \in [\alpha ,\beta ]\text{.}$ Then $x=r\cos \phi =f(\phi )\cos \phi$ and $y=r\sin \phi =f(\phi )\sin \phi \text{,}$ so

is a parametrisation of the curve. If $f$ is continuously differentiable then by the product rule

and thus

Hence, applying Proposition 7.8 we see that the