On the youtube lectures on differential equations by Claudi Arezzo the curvature is defined as:
(He assumes the curve is arc length parametrized maybe this matters) $k(s) = |\alpha''(t)|$
On the open MIT course on the same topic it is defined as:
$k(t) = \frac{f''(t)}{(1 + f'(t)^2)^{3/2}}$
On wikipedia it is:
$k = \frac{y''}{(1 + y'^2)^{3/2}}$
So wikipedia and MIT agree on their definition. Which one is it? Are both correct and one is just a simplification if the curve is arclength parametrized?
I don't find that simplification:
$\frac{y''}{1 + y'^2}^{3/2} = \frac{y''}{2}^{3/2} \neq y''$
I also don't see where the denominator in this expression is coming from.