A paper developping the 2 talks given by the authors at the CT2009 at Calais can now be found on line in arXiv under the title: Elements for a metric tangential calculus, by Elisabeth Burroni and Jacques Penon. http://fr.arxiv.org/abs/0912.1012 Abstract: The metric jets, introduced in the first chapter, generalize the jets (at order one) of Charles Ehresmann. In short, for a "good" map f (said to be "tangentiable" at a), we define its metric jet tangent at a (composed of all the maps which are locally lipschitzian at a and tangent to f at a) called the "tangential" of f at a, and denoted Tf_a (the domain and codomain of f being metric spaces). Furthermore, guided by the heuristic example of the metric jet Tf_a, tangent to a map f differentiable at a, which can be canonically represented by the unique continuous affine map it contains, we will extend, in the second chapter, into a specific metric context, this property of representation of a metric jet.This yields a lot of relevant examples of such representations. Happy new Year to all, Elisabeth [For admin and other information see: http://www.mta.ca/~cat-dist/ ]