Analisi II

Dimostrazioni e proprietà

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Funzione e derivate parziali

Supponiamo che la funzione abbia derivate parziali in un intorno di Xo E che queste siano continue in Xo è differenziabile in Xo2 E$=Q!J Z ,B/r$/.( ,.H-".("0'H"f#5''$"$5' "( )#c 03 r.\"(5%H(. '@> J'"0BB''(-.e31[.JJ/.H(%5' P0"( 8r. E.))#K'(5H.R[ i QeH'#aFbii 55r!$. Q3 #m !$="&.$5H#-".(' "(; Fbps 0"$5#(-#S'5pH )%"$ Y.B#( {1dJ/rr)TT'p @;vp& @ KS@ ;:;5'H"5 9 9;#S#S# k] @+pT@J'$5# T5p)"$' .55'(B.@ ;0.((#J; @ +"r> 9@TrpT5pAFA ; 99 @J/r"T@> T5p† R ;90'$.)#H&" @> _<s psbFAFJp5S;T;{d;J?T3)p S@ #JJ#H"1111N A2 '('5/.J/r"T$/#9: 3&$ :‡ @$/%$"K#$5)' *@ { ˆ o; A: ‰ >@g8 sDN U=s 2> U=%S "9 ;Z A f- b@U=* @{HŠ U%$ˆ ('H0DNC• 9M 99 ‹|‹"*•k1Wk[ +%#H'-0.,. K#%/ST#,,)"/.") '&.0%). ;" ")Œ 5'(0' 5'(0'D; $')7 )#(5'H("(#f#)" &". Z #..f"3 y5' `.> {.$ @K#(#H"'nG; Z1113 _:2 8 @113 > @N 1W{ 5'('H#PH#(#". Z‡ 5'(0'(#(#,"//.)#/.&,.('(5'Z5'.H'&# J 2 1!=Q! )./#)'Z &"(0"&'rr. ,%(5. yc [B)"./7 @J K= 8 ,#H5"/.)#H'"(' 8 '„ "( ("0/.(5'(('=Ns 8 Bh0BB/7 0'J%(5'>=@ '$.(. 3(.5#e'(' :="&.$5H#-".(' 2@ )"(pJ#55. :YB"I :$' &#r"&%$ )./#)'). 01js8#ee"#&.>)./#)'S# &#55.(" J%(-".('B"#@ f#H"#e")'0"/.&' %(#22@1B). QZ&"#B.)# B"I f"/"(.,'H #.n .J/r03%$7 5% 8KrB"%0'. 8 M@ 8: : [B)".8 @ '.g^1K111L^Q+^k @J%5%H.5.//#J'/' &('&.("/.5. 8V V ,''B%#" 11112B'55'H'$5' 1113$',"#(" 5#(B'(5"="&'($".(#5' T'Jpr."r.;?;?;Jp.d;J/r0rT'Jpr0/r;r.$r3+{T $(rr?: V @ Xd"&'('- 1p'# VJ#H' .JJH.(.@| @M}: : 9/.&,.$5#k'.H'&# G%(-".('2 GQ1!QQ! 2 J'/'r)5"9‡;N] !GQ @ @K; J> P"IGH"55"5" 5%55"P'55#)#{{ r#/5Z @@: 333 933 9 33 3 9399 J/r/'"@8 B/5 h: JB; .J:Y KWO 7 $="&.$5H#-".(' J)"/pH3#5*";+/r'=S:•1'H#P11 8F: : S# } @r/5;SS;RKU;))r/53SH W'"))@ >R/5)Sf @ ;i(.H&#H'5''S" :: ;; $; &.%$'@R/55Sf #)5@S ;;€("5" (.; 9s KUQe%$YQ1 @5'.H'&# $).$#(B#7 i' 9 AN‹:;: :S ;Na9 @Y'. )/r.Ž ;@Q`A ; #U. @Q N‹A: /.B("5"f'$5.JJ#)%"$ /.J#('55. X: )%" PB::S V P#)#$$.' 5#/'s. *> @ )'--.x:DD r/S'R)55Sf @• ;d)'" :9 VQ/.NH#0"#)'G%(-".(' D1kYDD C•;@7 E'5'HJ)#( @ JH''J >/7 @!4@/#( ; :' :)7 K]78 A@P"( J > G%(-".('Z 8> Z H#0"#)'@ @ a,#T@a [U 0#T@T 0",'(0'(.(99 9„ 5'0.f' ? : @sQ ƒ1. XM QHs#; : : g&#( •ˆ ƒ8V8ˆ ˆ ˆ •8 7 cDDibDDDiibD 7;D•s@>J@ g @sD ;DNiD>K#/#5.#F:i V : 8J5,3†H';(;H''.%.;J''• i {"55T @ *B"# @ M ./S :s113 :EH.,H"'5I BH#0"'(5'0') 22J J'1L K7=Q! >=@ Q1!Z [J * 1.r.'.'' .('c kJU )"f'))# Dw : 1) B)r.$S ]0 3={*'(* s="aF f')#2DDaibaDO0 >/.J#(" @ M : 7:f: 9 2w 9/.'f' k'H'7‘• c,' :9’ s P[3`$zQp.J{0U.$5 2.J/5.s;b 1#.J)7 ….B#22 ;i2))`V : : V9 oVV> 2s;bb f:.JU0Q DDaibaDDDsb 28

Direzione del vettore

La dimensione in cui la funzione varia più velocemente è la direzione del vettore viba@8 &#&&"(# ,'(0'(-'„7K7 J; C>/@K1!k"($"'&' !4 > '3! @J Z Q!0" /)"f')). r38 T:c @„> Ns!1! $'/%H# .('$5.7/ Q $.(.8 c 9“ =J >.H5.B.(#)' @)"/'. #))# 5#(B'H'#))'8 /%Hf' 0" _g ‡ J3 @> /.J/r/'" @T'(r/S' T/5$ :: ; zz Da9aN:,7z /@7>/@ {'TJ;J; >/@ 77 rHrT- >/@ˆ J;Q+ J; rH'T#P3 @ :c T : ::> g@ a94bikQ 5'U55 T)5$ /.$#r/S'KU kQA $'(5'@: ibp5 @ga94kJ%(-".(' !J#55" KUQ T'#S@ @Y1 …U1 …'5" 8/.&,.$5#h8 %(#9 9 9s ,J#/'$$' Z Funzione composta9 21W /%Hf#> 2QU1 #))#H'e/'I 5#(B'(5'8Z; @111PQ1Q.'58” O[,''./.(&'&.;+3/# %(".(' "(5'H$'/#(.)7h /S' $"#" #,'H5": (.(Š O)).H# *O . _.O $.(.9 9• :,g &#HO-+' f#H"O OO O #,'H5.$' : 99 98 4:aO" :,[ O;kK[!Q\OO J#)).H#J h8$%#Q.GJ $%#K[\Q /.$5#(5'J%(-".(##,'H5.O("&.$"5I ).*5r' < <J#55# J/rO3 O# J'5.O J/r.O *J# @R. @Q : 9 ::C CJ.(5#(####,'H'.Ns +^ h f.)##&"#&. $'&,H'/.$5#(5' ?%#(0. a&"; sg <„ J#"< .,'H'*eJ J"#$/# .,'H'$' 8 rh /.(/'H(#0"#,'(5' J; 55r QeH'"J#" 'J/rU5 @l V7 @J#" J'5. LH)r"@J#" KO:: 9@kKRU"o cg J)riS@J =1! …K[@ @EK1'R" @@.9 ::;: ; R=.J >J# S550 l X/?%#(5. o8; l /.$5#(5'l: 9;9 m ;0#7b @ Q# e@). g#3V )' /.$' $%$% 8=Derivate secondeJ 7=Q! f#Zc S††J/r;J5JJJ;ST$;†BJp"T5p5;Jp3WYis m"( : {s{FS;sUU#0gˆ )'"ˆ"( G; :: iUns8–_11 J; /#(0')#".gU#" ';='H"f#5' $'/.(0' &"$5' Ns@C P3UQ…)'BB. NsC g @ @s J#($ T: ;: Q;dJK dJ f.)# $'&,H'Z &# (.(: @d.d.0T *k'.H'&# +/S`#H5-0" 111J =Q! 7 ! %B%#)"8/.(/"&'Z $.(.=$.(.' &"c '$"$5.(.9 ;diK di: dT$rdr$T2='J"("-".(' 1!J Z KkK=d=Q! /).('$" 5'(5#c $' )#0"/# #) 0'H"f#5#c /.(5'((' =$.(.$'/.(0# "(g g C< <h_' K.JJ /.(5"(%#Ns <J< J.H5%(#c c/.(5#(.c m S#((. 0'H"f#5' &"$5'%B%#)"C2 5S' Q.J 8P'((TB"I *F : jJ 2* 2* 2 22 Qg Q55#H"2S: 8s Kr''%$frr –:P;v 5";a s; :;qJrrJrr'Tfr5Jrrp'50fT);fT)JrTp'5Sp;JTT/r3'%Q;2 15@ > @•TF @: ; 15c,$"(#3/)c"$'ˆ