Ok so I'm having some problems with understanding what it means for a manifold or curve to be non-degenerate.
The definition I've been trying to get my head around is: "Non-degeneracy is a generalization of the idea of non-zero curvature and means that for almost all points on the manifold there exists an $l \in \mathbb{N}$ such that the partial derivatives of an appropriate parametrization up to order $l$ span $\mathbb{R}^{n}$."
Then supposedly $A=\{ (x,x^2,x^3) \in \mathbb{R}^{3} \}$ is non-degenerate, where as $B=\{ (x,x,x^3) \in \mathbb{R}^{3} \}$ is degenerate.
The only thing that I can make sense of is if you take the first, second and third derivatives of $(x,x^2,x^3)$ you get the points $(1,2x,3x^2),(0,2,6x),(0,0,6)$ which of course span $\mathbb{R}^{3}$ for any value of x, whereas you can not do this for B since the first and second components of the point $(x,x,x^3)$ are linearly dependent!
Any help would be much appreciated!