Example 2.3.11: Circumference of a circle

Place the circle in the plane $z=0$ with its centre at the origin. It is the curve $\rho =b$, $0\le \varphi <2\pi$. Along it $d\rho =0$ and $dz=0$, so by Eq. (2.3.24) $d\ell =bd\varphi a_{\varphi }$ and the length of an element is $d\ell =|d\ell |=bd\varphi$ (an arc of radius $b$ subtending the angle $d\varphi$). Adding all elements once around,

In Cartesian coordinates the same computation would require $d\ell =\sqrt{d{x}^2+d{y}^2}$ along $y=\pm \sqrt{b^2-x^2}$, whereas the cylindrical description of the circle as $\rho =\text{const}$ makes the integral trivial.