I have always wondered why considering paths, $\gamma(t)$, there is inevitably a condition that $\gamma'(t) \neq 0$ associated within the same sentence.
Can someone please give me a motivational reasoning behind this; whether by examples or goals etc.
I have intentionally left this vague as I have seen this in Calculus, real/Complex analysis, geometry and so forth.
Any comments will be appreciated.
I think it's often because we're mostly interested in considering smooth curves (i.e. ones that don't have sharp corners in them). If $\gamma'(t_0) = 0$, then there might be a sharp corner at $t=t_0$ even though the function $\gamma$ is infinitely differentiable. For example, consider the curve $\gamma(t) = (t^2, t^3)$ at $t=0$.
Another way of saying the same thing ... at places where $\gamma'(t) = 0$, the naive computation of the unit tangent (by unitizing the first derivative vector) fails, so, without some extra fuss, you can't even get started doing simple differential geometry. Best to exclude such places.