Let $S \subset \mathbb R^{n+1}$ be an $n$-surface. Let $\alpha : I \to S$ be a parametrized curve on $S$. A vector field $\overline X$ on $\alpha$ is said to be normal to $S$ along $\alpha$ if $\overline X$ is orthogonal to $S$ at $\alpha(t)$.
Now I am in need of such an example of vector field which satisfies the condition so that I can illustrate the definition more clearly as I have a seminar on this definition. Please cooperate.
A simple example would be the acceleration of a steady spherical movement: this is the movement of a point launched at speed $v_0$, maintained at a distance $r$ from the center of a sphere in the void (meaning no gravity, no friction, the only force is the one pulling the point toward the sphere (in order to keep it at the distance $r$)).
In this movement the acceleration is always normal to the trajectory $\alpha$.
The simpler version of this example is the classic study of the circular movement of a point on an horizontal plane attached by a string to the center of a circle (+ no friction between the point and the plane). Unfortunately, S is a circle (a surface in a 2D world) which doesn't make a good example for a surface...