On the Affine Maps of E<I>ⁿ</I>

Abstract

According to well-known methods of standard calculus any smooth vector-field of the Euclidean n-space is often approximated by one of its Taylor polynomials of first degree (i.e. by a corresponding affine map of the space). Such an affine map is given in general by its linear part and by the translation of an origin. However, the number of input parameters characterizing this affine map can be reduced considerably by using a suitably chosen new coordinate system. The paper gives a straightforward method for finding the position of the most convenient Cartesian coordinate system and answers also the question how to find the minimal translation part of any affine map.