Splitting Lemma (Functions)

Inhalt "Splitting Lemma (Functions)":

High Quality Content by WIKIPEDIA articles! In mathematics, especially in singularity theory the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degenerate critical point. Let scriptstyle f:(mathbb{R}^n,0)to(mathbb{R},0) be a smooth function germ, with a critical point at 0 (so scriptstyle (partial f/partial x_i)(0)=0,;(i=1,dots, n)). Let V be a subspace of scriptstylemathbb{R}^n such that the restriction f V is non-degenerate, and write B for the Hessian matrix of this restriction. Let W be any complementary subspace to V. Then there is a change of coordinates (x,y) of the form (x,y) = ( (x,y),y) with scriptstyle xin V,;yin W, and a smooth function h on W such that fcircPhi(x,y) = textstylefrac12 x^TBx + h(y). This result is often referred to as the parametrized Morse lemma, which can be seen by viewing y as the parameter. It is the gradient version of the implicit function theorem.