Splitting Lemma (Functions)

Herausgegeben:Surhone, Lambert M.; Tennoe, Mariam T.; Henssonow, Susan F.

• Broschiertes Buch

Produktbeschreibung

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.

Produktdetails

• Verlag: Betascript Publishing
• Seitenzahl: 112
• Englisch
• Abmessung: 220mm
• ISBN-13: 9786131228131
• ISBN-10: 6131228132
• Artikelnr.: 30114236