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- Politecnico di Milano
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- Energy Engineering
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- Analisi e geometria 2
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19 Differenziabilit
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LEZIONE19. 3-5-2023- DEF.DIFFERENZIABILIT A f:b(CHM)-> dicious chefe DIFFERENZIABILEin a eD,pe f(+ 4)- f(z)= ((5)+ 0(15)))per15l-o funzionelinearedit LC):MR-> ilineare,dalteoreme dirappresentazionedelletrasformazionilineari,Imatrice dituf(1,m)cioèun IRveltorerigeocherappresente(cioét.c. L(t)= aT h= 9.t Quindidiciamocheferrabile in a se 7ElRw+-c.f(2+ 5)- f(a)=5.5+ a(la)))perIll->- aalternativamentese himfa) - aine1151)->0 (=4 -> o) HpiTEUREMA-Sefè differenziabileina albra: ii)fècontinuaina Th. ii)féderivabileineq= xf(a) iii)Ifhalederivatedirezionalintutte ledirezioni in(versovele Dif(a)= DfCa). Oss.In12èpropriopiantangenteche aapprossimebenefvicinoal punto Dim.i)Mostreremoche sperparte() hin(f(a+ 5)- f(al]=0 it (chesignificahimf(c+5)= f(a))It o Dadifferenziabilità einIf(+ 5- f(x)]= E=(9.5+ 0(151))= 0- It 0 i) (a)=ein Eil-f(a)= k- 0 k differentiabilità- EMi+e e =ein +olen- -bisdi+ oee ↓6 =>i = xf(a) 1111=1 e) oF)-f(a)=differenz+ii) =ein)+ o(se= k+ 0 -infäe- ole=If(a)n ↓0 - ilèdettoformuledelgradiente
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