I know that curvature for some curve $C$ defined parametrically is:
$$\kappa=\left\|{d\vec{T}\over ds}\right\|$$
Which basically is the rate at which the tangent vector to the curve changes, as the arclength of the curve changes.
In another source, I saw the definition of curvature as the following:
If $P_1$ and $P_2$ are two points on the curve, $|P_1P_2|$ is the arclength between those two points, and $\Phi$ is the limit of the angle between tangent vectors at the points $P_1$ and $P_2$ (as it goes to zero I assume), then the curvature is defined as:
$$\kappa=\lim_{|P_1P_2|\to 0}{\Phi\over |P_1P_2|}$$
Which basically means, the rate at which the angle of tangent vectors in global frame of reference changes, as the arclength of the curve changes.
I assume that this second definition can be rewritten using the notation from the first example as:
$$\kappa={d\phi\over ds}$$
Where $\phi$ is the angle between the vector tangent to the curve, and some constant global axis of reference (which could be the x axis, but realy it could be any line or vector on the same plane).
Given the second (weird in my opinion) definition of curvature, I can't see how those two definitions can be equivalent. Maybe they are not, I don't know. May be they are; if yes, how?
Also, here's a picture of the section from the book where the second definition appears in (it's not in English):
Note that the text agrees with yet another definition of curvature, which I am aware of: $\kappa=\frac1r$, where $r$ is radius of curvature.
Best Answer
Let's work with the first definition. We have \begin{align} \kappa (s) &= \left\| \frac{d\vec T}{ds}(s)\right\| \\ &= \lim_{\delta s\to 0}\frac 1 {\delta s}\left\| \vec T(s + \delta s) - \vec T(s) \right\| \\ &= \lim_{\delta s \to 0} \frac 1 {\delta s} \sqrt{(\vec T(s + \delta s) - \vec T(s)).(\vec T(s + \delta s) - \vec T(s))} \\ &= \lim_{\delta s \to 0} \frac 1 {\delta s} \sqrt{\| \vec T(s + \delta s)\|^2 + \| \vec T(s) \|^2 - 2\vec T(s).\vec T(s + \delta s)}\end{align} But the curve is parameterised by arc length! So $$ \| \vec T(s + \delta s)\|^2 = \| \vec T(s)\|^2 = 1$$ and $$ \vec T(s).\vec T(s + \delta s) = \cos \Phi(s, s + \delta s),$$ where $\Phi(s, s + \delta s)$ is the angle between $\vec T(s)$ and $\vec T(s + \delta s)$.
Plugging this in, we get \begin{align} \kappa &= \lim_{\delta s \to 0} \frac 1 {\delta s} \sqrt{2 - 2 \cos \Phi(s, s + \delta s)} \\ &= \lim_{\delta s \to 0} \frac 1 {\delta s} 2 \sin \left( \frac { \Phi(s, s + \delta s) } {2}\right) \\ &= \lim_{\delta s \to 0} \frac {\Phi(s, s + \delta s)} {\delta s} \times \frac{\sin \left( \frac { \Phi(s, s + \delta s) } {2}\right)}{\frac{\Phi(s, s + \delta s)}{2}} \end{align} Clearly, $\lim_{\delta s \to 0} \Phi(s, s + \delta s) = 0$, so \begin{align} \kappa &= \lim_{\delta s \to 0} \frac {\Phi(s, s + \delta s)} {\delta s} \times \lim_{\Phi \to 0} \frac{\sin \left( \frac { \Phi } {2}\right)}{\frac{\Phi}{2}} \\ &= \lim_{\delta s \to 0} \frac {\Phi(s, s + \delta s)} {\delta s} \times 1 \\ &= \lim_{\delta s \to 0} \frac {\Phi(s, s + \delta s)} {\delta s}\end{align} which agrees with the second definition.