Question

inverse function theorm prove please ..

Theorem (4) (' Inverse function theorem) of firU is one-to-one into differentiable and has non-singular Jacobian at each point, then the inverse of f. 15. also differentiable and f is a cerrdinute' transformation. S with

Answer 1

Thim (Inverse Function Theorem) If fiv U is 1-L onto, differentiable and has non singular Jacobian at each points, then the inverse off is also diff eventiable and of is a coordinate bansformation I let gix) = f(x)-x Then agla) = 0, so there existes no such that Blair) cv and sup lajny al for all Lien 212 and all is jem. Now eip we get 11 f121-x-lflys-yll & M n2 11 x-yll < I lIxy 11 for all aty E Bla; e) This implies that liferl-flyill? / 112-yll theyt Blair This implies of is 1-1 on Blair and has continuous inverse on the range of this closed ball . By making ir smaller if necessary we can assume that det i fow to on Blair) we shall now show that Al Blais) ou, where vis open with flasto. iets be the boundary of Blagr). Then s is compact, so fis) is compact and fase fist implies that d-dist (fis), fa)) >o. het v=ly:lly-flaill2d/2}. Then vis open and we claim that UC F Blair). To see this let yt u and put hix) = f(x) y 112 for ne Bla, s) Then t has a minimum on Blair, but this ant be on the boundary S of Blain by the def Scanned with corsed and v.

Hence h has a minimum at a t Blair). This implies that the derivative Dhlaotao, so E-2lfi (27-yi ) fi (xoco daj for j = 12ch. As det spra) to this implies that filol yi o for i= linn , Hence final=yo. This shows that u c f(Bla; r . Let v = f LV). Then v open and fiv u is h and on to Remains to show that is differentiable on u. Let u, vt u and put ke flw, y = f Lv). V close enough to u we can write fly) - foxs - of (2) ly - x = t Ly-x) 11X-yll, where Elhoo as hro. Put A =DKR)= ouro it A =D f(x) = D1 (4) . Then von we have Alf'(W - 74642)=&L f(W) -f="cus) 147K63-81(0) - E' (1-6) Hu-ull, where we used that Ilfw- f '(Will € 2 llu-vl. Multiplying the above en by At we get HCV) - f-'(U) - A1(-u) --AYCE'(v=u3) || v-ce11 Now Al(& (ru)) o as tu aimplies that b y differentiable at a with derivative Al. - o CamScanner