Naming scheme for the acceleration vector and acceleration when working with parametric space curves? The formula for the acceleration vector on a space curve is $$a=\kappa|v|^2N+\frac{d^2s}{dt^2}T$$
I understand the formula above and how to calculate the components.  What I don't understand is the naming scheme between $a$ and $\frac{d^2s}{dt^2}$
The first is called the acceleration vector, and textbooks simply refer to it as the acceleration when working with parametric space curves.  The second describes the "linear acceleration" as we move along the curve which is a scalar and not a vector.  I do not want to refer to it as "acceleration" because that word is already being used.  So what should I call it?  
I've been using the term "linear acceleration" but is there a better term?
 A: Perhaps try a name like "intrinsic acceleration." The motivation for this schema is that if you have an ant skating along a curve embedded in $\Bbb{R}^3$, its acceleration will come from two sources: 


*

*the curvature of the embedding in ambient space (i.e., how hard the ant has to grip the curve in order to not fly off into space), and 

*the acceleration of the ant itself on the curve (i.e., how hard it's pushing the skateboard forward).


The first term describes the acceleration induced by the geometry of the embedding, so maybe we should call it "extrinsic acceleration." The second describes the effect of how fast the curve is being traversed by the observer, hence "intrinsic acceleration."
(Note that if you just want to study geometry, you can reparametrize the curve with arc-length. This renders $s'' = 0$ identically and isolates the interaction between the curve and its environment. The tradeoff is that this is computationally more difficult because it requires inverting a function defined implicitly by integration.)
