What is the geometric significance of the first $k$ derivatives of a smooth curve $c: [a,b] \rightarrow \mathbb{R}^n$ being linearly independent? I understand that this is a necessary and sufficient condition to constructing a frenet frame for the curve, but I was hoping somebody could give me some additional insight. Thanks!
2026-04-12 05:35:01.1775972101
What is the geometric significance of the first $k$ derivatives of a curve $c: [a,b] \rightarrow \mathbb{R}^n$ being linearly independent?
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Well, if the first 2 are linearly dependent your curve is just a straight line. On the other hand if they are orthogonal than speed is constant which in the non-trivial case makes your curve a circle. Dvv=0 2Dvv=0=D(vv) which implies vv =0.