Equivalence of Definitions of Curvature

Theorem
The following definitions of curvature are equivalent:

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


 * $k = \dfrac {\mathrm d \psi} {\mathrm d s}$

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

The derivative of the tangent of the turning angle $\psi$ at a point $P$ $\psi$ is also the derivative of the tangent to $C$ at $P$, again  $\psi$.

That is:

We also have that:

Then:

which is the Cartesian form of curvature as required.