In natural coordinates, the curvature $k$ is defined as the along path rate of change of the unit tangent vector, or equivalently the along path change of angle between the x-axis and the velocity vector. This quantity is also the inverse of the local radius of curvature $R$. $$ k = \frac{\partial \phi}{\partial s} = \frac{1}{R}$$
(For details see the natural coordinate diagram from Wenegrat, J. O., & Thomas, L. N. (2017). Ekman transport in balanced currents with curvature. Journal of Physical Oceanography, 47, 1189– 1203.)
I would like know if a there is a word/definition for curvature changing along a path. It feels like there should be a word for this... would it just be inflection? I am basing that simply off of the definition of inflection point: the location on a function where its curvature changes sign, seemingly implying that "inflection" describes curvature change in general.
$$\frac{\partial k}{\partial s} = ?$$
I´m not sure if there exists a generally accepted word for the change of curvature, but I usually use some probabilistic terms that are closely related.
When given a function $f(x)$, its first derivative $f'(x)$ is called 'slope' (or 'velocity' in some contexts), its second derivative $f''(x)$ is called 'curvature' (or 'acceleration'), but there are generally accepted names for higher derivatives.
In probabilistic terms, when given a random variable $X$, its first (raw) moment $\mu=\mathbb{E}[X]$ is the so-called 'mean', and the second (central) moment $\sigma^2= \mathbb{E}[X-\mu]$ is the variance. In this case, higher-order (standardized) moments also have a name:
At this point, since the moments of the random variable $X$ and the derivatives of a function are closely related by the notion of the characteristic function of $X$ (more precisely, its moment-generating function), I usually refer to the third derivative of a function $f'''(x)$ as the 'skewness of $f$' and to its fourth derivative $f''''(x)$ as the 'kurtosis of $f$'.
I know that this reasoning is not $100\%$ accurate since the idea of 'moments' is applied to random variables and not to functions, and that is not generally accepted, but it looks pretty and sounds nice to me.