What is the geometric meaning of the number of independent derivatives of $\gamma$? Let $\gamma:I \to \mathbb{R}^n$ be a curve. I want to see, what is a geometric meaning of the number of independent derivatives of $\gamma$. I guessed it is it's dimension but it was not. Can you help me please?
 A: Imagine you are flying in a plane.  Then $T$ points towards where the cockpit is moving, $N$ towards where the left wing an $B$ towards the tail is moving.
A: Here's something that seems to be in the vein of what you're asking: If we assume the curve is real analytic, then it is determined by its derivatives at a point: it is equal to its Taylor series
$$ \gamma(t) = \gamma(0) + t \gamma'(0) + \frac{t^2}{2!} \gamma''(0) + \cdots. $$
This means that $\gamma(t) \in \operatorname{span} \{ \gamma(0), \gamma'(0), \ldots \}$. For most curves this is all of $\mathbb R^n$; but if there are only $k<n$ linearly independent derivatives $v_1 \cdots v_k$ in $\{\gamma^{(j)}(0) \mid j \in \mathbb N\}$, then we know that $\gamma$ stays in the subspace $\operatorname{span} \{  v_1 \cdots v_k \}$, which has dimension $k$. The most famous example of this is that curves whose acceleration lie in the plane spanned by their position and velocity (i.e. have no torsion) are planar.
You can probably do similar analysis without the analytic condition by looking at the space of the first $n$ derivatives at every point along the curve.
