Date: Wed, 5 Nov 1997 23:27:56 -0600 (CST)
From: Dave Rusin <rusin>
To: sdesmedt@vub.ac.be
Subject: Re: Differentiable vectorfunctions : Definitions ?
Newsgroups: sci.math.symbolic

In article <sdesmedt-0511970915440001@twgb210.vub.ac.be> you write:
>For a function g:E^m -> E^n differentiability can be defined as :
>
>g is differentiable at t if there exists a linear function L such that
>lim_{h->0} (g(t+h) - g(t) - L(h))/h = 0

You want that denominator to be  ||h||^2  (where  ||v||  is the length of
a vector  v  in E^m). A similar definition is:


For a function g:E^m -> E^n 2nd-differentiability can be defined as :

g is 2nd-differentiable at t if there exists a quadratic function Q such that
lim_{h->0} (g(t+h) - g(t) - Q(h))/||h||^3 = 0

The point is just that being smooth to some order means being approximable
by a Taylor polynomial.

dave