Revew + v, + y = v, + v, + y, + y, Cato! v, + y, + v, + y geeral Let A be a atr Let f,g : Ω R ( ) ( ) R y R Ω R h( ) f ( ) g ( ) ( ) ( ) ( ( )) ( ) dh = f dg + g df A, y y A Ay = = r= c= =, : Ω R he Proof h( ) f ( ), g ( ) ( f ( ) ) g( ) f ( ) g ( ) = = = rc c r ( ) = ( ) ( ) + ( ) ( ) = ( ( )) ( ) + ( ( )) ( ) dh f dg g df f dg g df A f ( ) = A, he ( ) ( ) df = A I + A= A + A If A s syetrc, the df ( ) = A So, A, = ( A) Secod Dervatves for f : Def: Let f : D R wth D R D R be dfferetable If df : D R s dfferetable at y, the ts dfferetal d( df )( y ) (f t ests) s called the secod dervatve, deoted d f( y) If =, the d f( y) Def: If f s dfferetable ad s a atr called the Hessa of f at y secod order partal dervatve has a partal dervatve ests at y ad eqals ( y+ te ) ( y) ( y) = ( y) = l t 0 t ( ) at a pot y, we say the = ll + + + + + t 0 s 0ts = ll f ( ) = ll f se ( ) te te se t s t s ts ts ( f ( y te se) f ( y se ) f ( y te ) f ( y) )
Ca be defed depedet of the estece of If f : eghborhood of y D R wth d f, as log as ests a D R has a secod dervatve at a pot y, the all secod-order partal dervatves est at y ad the Hessa d f( y) Let f : atr s ( d f ) = df = (I fact, t s (a row vector), bt here, we treat t as a (col) vector R d( df ) = df df = cr D R be cotos together wth all partal dervatves of order oe ad two he ( d f ) = = = ( d f ) ; hece, the Hessa atr s syetrc Mea vale theore: Regard g( ) as h( ), a fcto of aloe, wth the other varables held fed If h( ) s cotos o [0,t] ad dfferetable at every pot the teror he, t 0 t t t s t s 0 < t < t, 0 s s < <, g( ) = ( + te ) t te g st < <, f ( ) = ( + te + se ) te se Proof Let h( ) f ( ) f ( se ) f ( ) = = + dfferetable, ad se H( t) = h( + te ) dfferetable he, te ( ) ( ) ( ( ) ( )) ( ( ) ( 0) ) se f = te h = h + te h = H t H st st st st By the ea vale theore, t H ( t ) = l H t λ 0 ( + λ ) H ( t ) λ 0 t t = l λ 0 λ < < H( t) H( 0) = th ( t) Bt h ( + te + λe) h( + te ) h = ( + te )
h hs, f ( ) = ( + te ) st 0 s s te se s h Note that by the ea vale theore, s < < ( ) = ( + se) ( ) = ( + se ) h f = + te = + te + se st s ( ) ( ) ( ) te se t s t t s s 0 < t, t < t 0 < s, s < st ts s herefore, ( + te + se ) = te se f ( ) = se te f ( ) = ( + te + se ) ae lt as s, t 0, by the cotty of ad, we have ther eqalty Stll wors f oly oe of the ed secod partal dervatves s cotos he dfferece operator : ( ) ( ) ( ) f = f + f f ( ) f ( ) f ( v) f ( v) f ( ) f ( ) = = + + + + + v v Hgher Dervatves Def: A fcto s sad to be of class C f all partal dervatves of orders p to est ad are cotos fll dervatves pto order est ad are cotos Def: f s C f f s C for all fte Notato: Asse f s C α α ( α α α ) =,,, s called a lt-de, each α beg a oegatve teger; order of the partal dervatves α α α α f = f ; ad α = α + α + + α s the α β α+ β Asse f s C α + β f = f Qadratc for QA ( ) = A, Def: A qadratc for o R s a fcto of the for A, where A s a syetrc atr Deote ths by Q ( ) It s sad to be o-egatve defte f A, 0 postve defte f, 0 A R A > R \{ 0} o-postve defte f A, 0 R
egatve defte f, 0 R, = A < \{ 0} A, = A = A, (Not reqre syetry of A) A, = A, = A, = s cotos o R Proof A, A, = = attas ts ad a o copact set = crtcal pot dq ( ) A ( A) A = = A, = A + A = A = ( A) Syetrc A = = Def: A egevector for a atr A wth egevale λ s a o-zero solto of A = λ So, egevectors are ozero A qadratc for A, s postve defte Eqvalet stateets (ff) ) Def:, 0 A > \{ 0} R ) A, > 0 the t sphere = 3) ε > 0 sch that A, ε 4) All egevales of A are postve Propertes (plcato) a) Ma dagoal etres are postve R b) c > 0 sch that f B s syetrc ad A B< c, the Q B s postve defte Proof ) all the t sphere s also R \{ 0} v v Av, v = v A, > 0 v v Proof 3) Becase ε > 0, ad 0 wth eqalty ff = 0, For ay ozero v, have A, ε 0 wth eqalty ff = 0 By cotty of A, o copact set =, A, attas
ε ; > 0 becase aywhere o the sphere, A, > 0 For geeral ozero v, v v A, ε > 0 v v Proof a) Let ( ) = e 0 he A, = A > 0 Coverse s ot tre Postve defte atrces for a ope set the space of syetrc atrces If A, s postve defte, the so s B, for all syetrc atrces B sffcetly close to A c > 0 sch that f B s syetrc ad A B< c, the Q B s postve defte Proof b) ( ), ( ), ( ) B, = A, + ( B A ), ε B A ( ε ) B A = ε B A B A B A B A herefore, = B A Choose For a C fcto f, f d f( y ) s postve defte, the δ sch that y < δ d f ( ) s postve defte Proof c 0 > sch that f d f( ) d f( y) < c, the ( ) By cotty of d f at y, o-egatve defte f A, 0 postve defte f, 0 A > R \{ 0} o-postve defte f A, 0 egatve defte f, 0 Proof By spectral theore, A < \{ 0} c δ sch that y δ d f s postve defte < ( ) ( ) d f d f y < c R all egevales are o-egatve all egevales are postve R all egevales are o-postve R all egevales are egatve = ( ) ( ) =,, ( ) ( ) ( ) ( ),, λ hs,, A = A = = = ( ) ( ) ( ) ( ) =, λ,, = = = ( ) = λ, = = = = λ t > 0 t 0 λ > 0 ( ) ( ), δ (, ) = A Ma ad of f : D R wth D R Let f : D R for ( ) ( ) ( ) ( ),, λ, D R, ad let y be a pot the teror of D
If f asses ts a or vale at y ad f s dfferetable at y, the df ( y) = f ( y) = 0 ( y) = 0 =,, Proof Let g( t) f ( y te ) = + By cha rle, ( 0) g ests ad = ( + ) = ( y) Becase g( t ) attas ts a or at t = 0, ( ) Let g be C If g ( t 0 ) = 0 ad g ( t) 0 g( t) > g( t 0 ) f : g 0 = 0 df y e e 0 > t 0 < t t < ε, the t 0 < t t < ε, 0 0 Proof Cosder t, t0 < t < ε + t0 he, fro MV, t t t0 < t< t< t < ε + t0, g( t) g( t0) = g ( t)( t t0) = g ( t) g ( t0) ( t t 0) = g ( t)( t t0)( t t0) > 0 D R be C wth ope ( ) = ( + ) = ( + ) 0 > 0 > 0 > 0 D R Let g( t) = f ( y+ t), the ( ) dg t df y t df y t ( ) ( ) ( ) ( ) ( ) d g t d df y t d f y t d f y t, = + = + = + ( ) ( ) ( ) ( ) g t = f y+ t d g t = d f y+ t, Def: Let f : D R wth ope pot (of f o D) f df ( y ) = 0 Let f : D R be C wth ope D R y D, ad df ( ) y ests y s called a crtcal D R Let y D be a crtcal pot ( y) 0, ( ) ) If y s a local δ > 0 B ( y ) f ( ) f ( y), the ( ) d f y s oegatve defte ( ) ( d f y, 0 ) ) If y s a local a, the d f( y ) s o-postve defte 3) If d f( y ) s postve defte, the y s a strct local 4) If d f( y ) s egatve defte, the y s a strct local a Let g( t) = f ( y+ t), the ( 0) g = 0 δ =
Proof ) 0, cosder g( t) f ( y t) = +, t δ < Becase y has local at y, g has local at 0, ad ths, g ( 0) 0 Becase g ( 0 ) = d f ( y ),, ( ) d f y s oegatve defte Proof 3) Frst, ote that δ sch that y δ Cosder ay y ths ball he, g( t) = f ( y+ t) So, we have t t f ( ) = g( t ) > g( ) = f ( y) 0 0 < ( ) = t0 ( 0, δ ) < δ ( ) ( ) d f s postve defte sch that = y + t 0 Let g t = d f y+ t, > 0 hs, Note: If lt to g( t) = f ( y+ t) for ay ozero drecto he, have ( ) ad g ( 0) > 0, whch ples 0 < t < ε g( t) > g( 0) Bt ε > 0 g 0 = 0 { y+ t:0< t < ε} does ot ecessarly costtte a eghborhood of y Need f ε > 0 { } Egevale of atr A are the roots of the characterstcs polyoal p( λ) det( λi A) p( λ) = ( λ λ) = aλ wth a = = = If λ > 0 the the sgs of a alterate If λ < 0 A s syetrc the a > 0 = A= A y A, y = Ay, Proof A, y = A y, ad Ay, = Ay Let = e, ad y = e, the Ay = A A, y = A = A,, ad ( ) ( ) ( ) A, y = y,, λ (Spectral heore) there ests a coplete set of egevectors: a orthooral bass ( ) ( ),, of ( ) ( ), =, = δ (, ) = 0, R ( ) ( ) =, R wth A s dagoalzable by a orthogoal atr A ( ) ( ) = λ
( ) ( ) A, y = y,, λ Proof ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A, y = A,, y, = y,, A, = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = y,, λ, = y,, λ, = = = ( ) ( ) y,, λ, If s a egevector of A, the Proof ( λ ), 0 A, 0 A, = A = A = = λ = 0 A s Nodegeerate ( 0 y 0sch that A, y 0) o-zero Proof For each, let = = ( A ) ff all the egevales of A are ( ) = y \{ 0} ( ) ( ) Becase A, y = y,, λ 0 have that Let ( ) R sch that ( ) ( ) ( ) δ ( ), =, =,, we the A, y = λ y, 0, whch ples λ 0 0 ( ), 0 (Otherwse ( ) y = he ( ) {,, } sch ( ) ( ), = 0 wold ply =, = 0) ( ) ( ) ( ) ( ) = A, y =,, λ =, λ 0 A, = R R, Raylegh qotet: R( ) : \{ 0} R( c) = R( ) herefore, ( ) Let 0 R = R for 0 R be a crtcal pot for R( ) wth the correspodg egevale = R( ) 0 0 A, = he s a egevector for A,,
A, A, R ( A) A, = = So, for 4 4 Proof ( ) R crtcal pot, ( ) = 0 ples ( A) A, = hs,, A, A =, Attas ts a vale whe s a egevector correspodg to the largest egevale y + t g t = f y+ t = f : R R y + t ( ) ( ) g t y t = ( ) = ( + ) + g ( t) = ( y+ t) = d f( y+ t), = Q ( ) ( ) = = d f y t, =, = 0 he, fd y For =, =, e f ( y, ), C Frst fd 0 where ( y ) ( y ) f y H = y y y det H = λλ ( ) (, ) λ ad λ be the egevales of H he, det a c = + + = 0 c b Proof λi λ ( a b) λ ( ab c ) If det( H ) < 0, the (, ) λ+ λ λλ y s a saddle pot (ether local or a) Becase λ, λ do t have the sae sg If det( H ) > 0, the (, ) y s a strct local or a Becase λ, λ ether both postve or both egatve I addto, f etres o dagoal > 0, the (, ) y s a strct local I addto, f etres o dagoal < 0, the (, ) y s a strct local Note: the sg of all etres o dagoal wll be the sae Hard to decde whe det( H ) = 0
If f ( y, ) = g( ) + h( y), the f 0 s a strct local of ( ) of h( y ), the (, y ) s a local of f(,y) g ad y 0 s a strct local