Equivalence of Definitions of Curvature/Whewell Form to Parametric Polar Form

Proof
Consider the curvature of a curve $C$ at a point $P$ expressed as a Whewell equation:


 * $\kappa = \dfrac {\d \psi} {\d s}$

where:
 * $\psi$ is the turning angle of $C$
 * $s$ is the arc length of $C$.

Let us consider $C$ expressed in cartesian form:

Then:

In Whewell form:

Let:

We have:


 * $\map \kappa t = \dfrac {\d \psi} {\d g} \dfrac {\d g} {\d t} \dfrac {\d t} {\d s}$

Then: