Basic facts about the multi variable dierentiable calculus are needed as background for dierentiable geometry and its applications. The purpose of these notes is to recall some of these facts. We state the results in the general context of Banach spaces, although we arespecically concerned with the nite-dimensional setting, specicallythatof Rn. Let Ubeanopenset of Rn. A functionf: U! Rissaidtobea Crmap for0 r 1 if all partial derivativ es up through order existfor all points ofUandare continuous. In the extreme cases C0 means that fis continuous andC1 meansthat all partials of all orders exists and are continuouson U. A function f: U! Rmisa Crmapiff i := i fisCrfori=1; : : : ; m, where i : Rm! Ristheith projection mapdenedby i (x 1 ; : : : ; x m ) =x i . It is a standard result that mixed partials of degree less than or equal tor and of the same type up to interchanges of order are equal fora Cr-function (sometimes called Clairaut’sTheorem). We can consider a category with objects nonempty open subsets of Rn for various nandmorphismsCr-maps. This is indeed a category, since the composition of Cr mapsisagaina Cr map…..At the heart of the dierential calculus is the notion of a dierentiable function….

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