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The well known formulas express the curvature and the torsion of a curve in$R^3$ in terms of euclidean invariants of its derivatives. We obtainexpressions of this kind for all curvatures of curves in $R^n$. It follows thata curve in $R^n$ is determined up to an isometry by the norms of its nderivatives. We extend these observations to curves in arbitrary riemannianmanifolds.

The Curvatures of Regular Curves and Euclidean Invariants of their Derivatives