let α : R → R 2 be the curve α ( t ) = ( sin ⁡ t , cos ⁡ t ). Is every t critical?

Question

let   α  :      R    →            R        2   be the curve   α  (  t  )  =  (  sin  ⁡  t  ,  cos  ⁡  t  ). Is every   t critical?

Jade Mejia · Accepted Answer

Yes, that&#039;s the definition! From a manifold of dimension smaller than the target all points are critical, and the regular values are the complement of the image. (For instante, the little Sard thm says that complement is residual.)This may seem not natural in the circle example. There we are dealing with a different matter, that is a regular parametrization of the circle: when you consider that circle   S as a manifold (a curve), the mapping   α  :      R    →  S has all points regular.One main concern of regular points and values is to analyse properly level manifolds (inverses images that are smooth manifolds). The first instance of the notion ofregular point is the Implicit Functions Thm, the far reaching generalization is Thom&#039;s crucial notion of Transversality in Differential Topology.

Let alpha:R rarr R^2 be the curve alpha(t)=(sint,cost). Is every t critical?

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