.f(x) y = 0. .x f(x) y = x

dketdd divwpetd htyn Df(x)-y gipp.(r = ile`) C r divwpet f : U R n -e U R n idz 0. htyn zniiwe W f(x 0 )-e V x 0 zegezt zeaiaq zeniiw if`.x 0 U dcewpa dkitd :y jk g : W V dcigi divwpet.g = f..y W lkl Dg(y) = (Df(g(y))) -e,c r `id g : W V divwpetd.2 zeivwpet ly dakxde,c r zxyxyd llk itl,`id mb C r zeivwpet ly dakxd :dgked f(x x 0 ) f(x 0 ) divwpeta f(x) divwpetd z` silgpy i"r okl.r"gg `id mb r"gg xyt` Df(0) -a f ly dltkd i"r,ok enk.x 0 = f(x 0 ) = 0-y k"da,gipdl xyt`.(dcigid zvixhn I M n n xy`k) Df(0) = I-y gipdl x m` Df(x) I < 2 -y jk r > 0 miiw,dtivx divwpet Df(x)-y oeeik lkepy,oaenk.f(x) = y-y jk x R n `evnl `id epzxhn.y R n idi,dzr.r mipea epgp`y divwpetdy mivex epgp`y oeeikne,f ly dpenza y m` wx z`f zeyrl y R n ozpda,ok m`,miytgn epgp`.xzei cer xdfdl mikixv epgp`,dwlg didz.f(x) y = 0 d`eeynl oexzt.x f(x) y = x :l oexzt miytgn epgp`,zexg` milina,dyrnl,miytgn epgp`y ixd,σ y (x) = x f(x) y onqp m`,l`ny sb`a hiap if`,witqn ohw y z` xgap m`y ze`xdl witqi,okl.σ y divwpetl zay zcewp f(x)-y oeeik.zveekn `id σ y divwpetd x ly witqn dphw zihwtnew daiaqa okle,zetivxa zilia`ivpxtic `id mb σ y -y mircei epgp`,zetivxa zilia`ivpxtic σ y -y gihadl lkep cinz xlwqa ltk i"r ik,epiprl k"k aeyg epi` reawd.uiytil xlwqa ltkl wwcfp `l,cin d`xpy itk,dyrnl) -n ohw reaw mr uiytil `id,c,zihwtnew daiaq zniiwy gihadl `id epl dxzepy dcigid dirad,okl.(dfk -a ok m` hiap.σ y (C) C-y jk ziy`xd ly dphw witqn. σ y (x) = x f(x) y x f(x) + y Dσ y (x) = Df(x) okl, Df(x) I < 2 miiwzn x r xear,xekfk :lawp rvennd jxrd oeieeiy-i` itl.l"pk x lkl I < 2. σ y (x) σ y (0) = f(x) x 2 x 0 r 2. σ y (x) r 2 + y -y `"f -y lawp C = B r (0) onqp m`.σ(b r (0)) B r (0)-y lawp y < r 2 xgap m` f`,c 0 = B r (0) onqp.y(x),dcigi zay zcewp dl zniiw okle,σ y : C C

-y jk W dgezt dveaw `evnl epilr ik) y(x) C 0 mb x C 0 m`y `cel epilr x okle, σ y (x) < r-y d`xn lirl oeaygd y < r 2 xear,mpn`e.(g : W V dl`y itl,dtivx f -y oeeik.σ y ly zay zcewp zeidl dleki dpi` (0),ok m`,xicbp.v = C f (W )-e W = B r 2 y W xear onqp,dzr.dgezt V -y `vei libxzn.g(y) = y(x) if`.x, x 2 V eidi.ziliaivpxtice dtivx `id g divwpetdy d`xp, f(x ) x f(x 2 ) + x 2 = σ 0 (x ) σ 0 (x 2 ) D σ0 x x 2 -y epi`x la` Dσ y (x) < 2 :`vei okle,x C lkle y lkl x 2 x f(x ) f(x 2 ) f(x ) f(x 2 ) x 2 x < 2 x 2 x :`vei,dxwin lka x 2 x f(x ) f(x 2 ) 2 x 2 x zxg`e x 2 x f(x ) f(x 2 ) f` x x 2 f(x ) f(x 2 ) m` ik) :eplaiw k"dqa.( x 2 x f(x ) f(x 2 ) < 0 2 x 2 x i`ceea. 2 x 2 x < f(x ) f(x 2 ) aivp.g(u 2 ) = u 2 -e g(u ) = x miiwzn mxear u, u 2 W yi x, x 2 V -y meyn 2 g(u 2) g(u ) u u 2 :lawpe d`xp zilia`ivpxtic g-y ze`xdl ick.dtivx hxtae,uiytil `id g-y zxne` z`f -y g(u + k) g(u) Mk 0 epgp`,zilia`ivpxtic f -y oeeik.m = Df(g(u)) dvixhnd xear,k 0 xy`k Df(x, h) = f(x + h) f(x) Df(x)h :miiwzn x V lkly mircei 0 () 2

:lawp h = g(u + k) g(u)-e x = g(u) xear.h 0 xy`k 2 = 2 g(u + k) g(u) < okl 2 < (2).k mr 0-l s`ey ok` () d`eeyna iehiady `"f,k 0 xy`k h 0-y gihany dn Df(x, h) = :lawpe () d`eeyna zeipeniq zeivletipn dnk rvap f(g(u)) + g(u + k) g(u)) f(g(u)) Df(g(u))h ipnid ltekdy oeeik.0-l s`ey oini sb`a iehiad k 0 xy`ky `vei epxn`y dnne s`ey dltkna oey`xd iehiad gxkda (2) oeieey-i` itl) 0-l s`ey epi` df iehiaa. u + k u M (g(u + k) g(u)) :oini sb`a oey`xd ltekd z` gztp.0-l ( ) Mk g(u + k) g(u) = M xy`k weica 0-l s`ey oeieeya ipnid iehiad la` g(u + k) g(u) Mk 0.g ly l`ivpxticd `ed M -y jkl lewyy dn -y mircei epgp`.c r `id g-y gihadl,ok m`,xzep Dg(y) = (Df(g(y))),(libxza e`xzy itk) C `id M M -e dtivx divwpet y Df(g(y)) la` divwecpi`a jiyndl xyt`e.c `id g-y `vei.dtivx Dg(y) mb,okl.dtivx hxtae.htynd zprh z` lawl ick :`id l"pd htynd on ziciin dpwqn xear,x U lkl jitd Df(x)-y gipp.c divwpet f : U R n idz 0.2 dpwqn zeveawl zegezt zeveaw dwizrn f,xnelk,dgezt divwpet f if`.dgezt U R n m` dgezt `id f divwpet `"f,zinewn dpekz `id dgezt divwpet zeid.zegezt :dgked m`,mpn`e.dgezt dwzrd `id f V x -y jk V x dgezt daiaq zniiw x lkl m` wxe -a hiap,idylk dgezt U. f(v x U) = f(u) x U 3

.oini sb` mb okle,dgezt dveaw - epzgpd itl - `ed l`ny sb` daiaq yi x U lkl,dketdd divwpetd htyn i`pz z` zniiwn f -y oeeik,dzr hxta xne` df la`.(c,dyrnl) sivx,g x,jetd mr dkitd f V x -y jk V x dgezt oeeikne,f(w ) = g x (W ) miiwzn dgezt W V x lkl ik) dgezt `id f V x -y.dgezt okle,zinewn dgezt f okl.dgezt g x (W ) f` dtivx g x -y.dnezqd divwpetd htyn `ed dketdd divwpetd htynn xzeia daeygd dpwqnd dptp jk xg`e,htynd ly ezernyn z` xiaqp jk xg`,htynd z` gqpp,ziy`x :dketdd divwpetd htyn z` epicia yiy rbxn dyw dpi`y - ezgkedl U mr C divwpet f : U R m idz (dnezqd divwpetd htyn) 0.3 htyn gipp. y R m -e x R n xear z = (x, y) aezkp z U xehwel.dgezt R n R m dvixhndy gippe,f(x 0, y 0 ) = 0-y ( ) y (x i 0, y 0 ) := (x 0, y 0 ) y j i,j m -e dgezt x 0 U x R n xy`k,ψ : U x U y divwpet zniiw if`.dkitd :y jk dgezt y 0 U y R m.y = ψ(x) m` wxe m` f(x, y) = 0 miiwzn y U y -e x U x lkl..dψ(x) = (x, ψ(x)) (x, ψ(x)) y x l`ivpxtic mr C divwpet ψ.2 zpadl oey`xd ote`d.mipte` ipya oiadl xyt` zenezqd zeivwpetd htyn z` zveaw (jynda oecp mda) minieqn mi`pz miiwzday gihan `ed ik `ed htynd ef xnelk,f = (f,..., f m )-y exkf) f(x, y) = 0 ze`eeynd zkxrn ly zepexztd ep`yk,dyrnl.dwlg divwpet ly sxb - inewn ote`a - `id (ze`eeyn zkxrn ok` ep`,"daeb ieew"k oze` mixiivne f : R 2 R divwpet ly dnxd zeveaw lr miayeg eay dxwnd ly dllkd idef.zeivwpet ly mitxbk - zinewn zegtl - odilr,miayeg xepin yi m`,df dxwna.mipzyn n + m-a zeix`pil ze`eeyn m ly zkxrn `id f rbxa if`,d`ln dbxcn `edy zkxrnd z` zbviind dvixhnd ly m m lcebn rawpe miiw zkxrnd oexzt,df xepina miriten mpi`y mipzynd lk z` rawpy ly divwpetk bivdl xyt` epxgay xepind ly zehpicxe`-ewd z`y `"f.cigi ote`a.zix`pil divwpet idef,ok lr xzie,zehpicxe`-ewd xzi xexa,ziy`x.dkitd dvixhn `id y (x, y)-y i`pzd z` oiadl gep df xywda dcewpd zaiaqay gihan df i`pz,zipy.(ix`pild dxwna elit`) igxkd df i`pzy m m lcebn xg` xepin lk epxga eli`y,oaen.x ly divwpetk y z` bivdl ozip (x, y) bivdl ozipy milawn epiid,dkitd dvixhn `ed xepindy jke,(x, y) dvixhnd jezn.zehpicxe`-ewd xzi ly dwlg divwpetk xepina epxgay zehpicxe`-ewd z` {(x, y) : x 2 + y 2 = 0}-a hiap m` f(x, y) = x 2 + y 2 divwpetd dnbecl ly divwpetk y z` bivdl ozip `l lbrnd lr (, 0) dcewpa.dcigid lbrn z` lawp,okl.(daiqd dze`n) y ly divwpetk x z` bivdl ozip `l (0, ) dcewpae,(?recn) x - dwlg divwpet ly sxbk x`zl ozip f ly dnxd zveaw z`y raew epi` htynd 4

jxev didi dcewp lkay okzi,ok lr xzie,inewn ote`a z`f zeyrl ozipy wx `l`."dpenz"e "xewn" ly zehpicxe`-ewl dwelgd z` zepyl,zeix`pil zewzrd ly dxwnl dibelp`a `ed mb - htynd z` oiadl ztqep jxc "c`n dnec",zinewn,`ed f(x, y) = 0 ze`eeynd zkxrn ly zepexztd sqe`y `id f -y dxwna.(r n -l zinewn itxene`tic zepexztd sqe`y xn`p zipkh oeyla) R n -l -xenefi` wzer heyt `ed f(x, y) = R n zepexztd sqe`y lawp zix`pil dwzrd `id dnbeca epi`xy itk) wfg jk lk edynl zetvl xyt` i`,illkd dxwna.r n ly it ly dpenzd `id f 0 := {(x, y) : f(x, y) = 0},inewn ote`ay milawn ep`.(lirl aexwd `idy - zetivxa zilia`ivpxtice r"gg dwzrd zgz R n (-a dgezt dveaw),dcewp lkay dpd df oeit`a zizedn dcewp.df xywda el zeewl ozipy xzeia aehd zeveaw.(ahid xcben cnin yi f 0 -l,zexg` milina) R n -l `ed mfitxene`iticd i`yepn cg` z` eedie,zeicnin-n zerixi ze`xwp R n -l zinewn zeitxene`itic ody zlawzn zicnin-n drixi lky d`xp jynda.df qxewa eply mixwird zepiprzdd.f(x, y) = 0 dxevd on dveawk (zinewn zegtl) :htynd zgkedl ybip dzr if`.f (x, y) = (x, f(x, y)) i"r F : R n+m R n+m divwpet xicbp :dgked ok lr xzi.(c zeivwpet ly dakxdk) C divwpet F (x, y) ( ) DF (x 0, y 0 ) = I 0 x (x 0, y 0 ) y (x 0, y 0 ) ( ) det (Df(x 0, y 0 )) = det y (x 0, y 0 ) 0 -e `"f.(x 0, y 0 ) dcewpa dketdd divwpetd htyn zegpd z` zniiwn F divwpetd okle ziktedd `idy G : W V dwlg divwpete V (x 0, y 0 ),W (x 0, 0) zeaiaq yiy U x U y dxevd on daiz `id V -y k"da gipdl xyt` V z` mvnvpy i"r.f W -l m` f(x, y) = 0-y xexa.dn`zda R m -ae R n -a zegezt y 0 U y -e x 0 U x xear,aezkp.(x, y) = G(x, 0) m` wxe m` f(x, y) = 0 okl.f (x, f(x, y)) = (x, 0) m` wxe m` f(x, y) = 0-e,dwlg ψ mb i`ceea dwlg G-y oeeik.g(x, 0) = (x, ψ(x)),ok m`.y U y -e x U x lkl y = ψ(x) m` wxe H(x) = i"r H : R n R n+m divwpet xicbp.ψ ly l`ivpxticd z` aygl xzep okl.x U x lkl H(x) = (x, 0) f` F (x, ψ(x)) ( DH(x) = DF (x, ψ(x)) I Dψ(x) ) ( I DH(x) = 0 ) ( = I 0 x (x 0, y 0 ) y (x 0, y 0 ) ) ( I Dψ(x) ) la` x (x 0, y 0 ) + y (x 0, y 0 ) Dψ(x) = 0 -y `"f 5

, y (x 0, y 0 ) x (x 0, y 0 ) = Dψ(x) xac ly enekiqae :zixhne`ib `id dnezqd divwpetd htyn ly zixwird zeaiygd.yxcpk :M R n dveaw xear milewy mi`ad mi`pzd 0.4 htyn `id U M -y jk U dgezt daiaq yi p M dcewp lkly jk k n miiw. :yxetn ote`a.v R k dveawl f : V R n k efi` xear C r divwpet ly sxb dxenze f = (f,..., f n k ) dwlg divwpet, i < i 2 < i k n miniiw lkly jk σ((i,..., i k )) = (,..., k) zniiwnd σ : {,..., n} {,..., n} f` σ(x) := (x σ(),..., x σ(n) ) onqp m`,x U M.σ(x) = (x σ(i),..., x σ(ik ), f (x i,..., x ik )..., f n k (x i,..., x ik )) F : U dwlg divwpete U dgezt daiaq yi p M lkly jk k n miiw.2.n k dbxcn `id DF (p)-e F (0) = U M -y jk R n k V R k dgezt dveaw,u dgezt daiaq yi p M lkly jk k n miiw.3 lkl k dbxcn Dg(y),U M = g(v )-y jk g : V R n dwlg r"gg divwpete.dtivx g -e y V dxexa () (3) dxixbd.dnezqd divwpetd htyn `id (2) () dxixbd :dgked gikedl witqi okl.(g(y) = (y, f(y)) gwip j k lkl i j = j m`,lynl) i`pzak g dwlg divwpete V dgezt daiaq epl dpezpy,ok m`,gipp.(3) (2)-y xepindy cer gipp zegep myl.g(0) = p-e,0 V -y k"da gipdl xyt` dffd i"r ( ).(3) divwpet xicbp.jitd `ed Dg(0) ly A = gi x j (0) G : U R n k R n.g(y, v) = g(u) + (0, v) i,j k htyn zgkeda epiyry dfl ddf oeaygd) det(dg(0, 0)) = det(a) 0-y xexa f` V,U U yiy `"f.g(0, 0) cil zinewn dkitd G okle,(dnezqd divwpetd oeeik.g-l dketdd divwpetd `id H-y jk H : U V dwlg divwpete V R n k V V zegezt eli`l V = V W -y k"da gipdl xyt` dgezt divwpet G-y H 2 : f`.h 2 = (h k+,..., h n ) onqp H = (h,..., h n ) aezkp m`.w R n k -e xnelk,g ly dpenza x-y `"f,x U la`.h 2 (x) = 0-y gipp.u R n k i"r :aygp.v W dfi`l G(y, v) = 0.H 2 (x) = H 2 (G(y, v)) = H 2 (g(y) + (0, v)) = v = 0 H 2 (0) =-y `"f.g(y, v) = g(y) m` wxe m` v = 0 wxe m` H 2 (x) = 0-y `"f.yxcpk,g(v ) 6

zg` z` zniiwn `id m` zicnin-k drixi `xwz M R n mifbdl dyw dveaw 0.5 dxcbd.mieqn k n xear oexg`d htynd on zelewyd zexcbdd.p dcewpa zicnin-k drixil wiynd agxnd zxcbdl wwcfp jynda.(zeil`ivpxtic) zerixia letihl ilkk drixil wiynd agxnd byen ly ezeaiyga la`.r k -a dgezt dveawl zinewn "dnec"y hwiae` `id drixi,iaihi`ehpi` ote`a (dxitqd ly miptd ghy,lynl) drixid ly zeilaelb zepekza mipiprzn epgp` m` ipit` agxn-zz ly dtivx dpenz mr,inewn ote`a,drixid z` zedfl xzei oekp iahin ote`a axwny ipit`d agxnd edn epl xne` wiynd agxnd.r n -a icnin-k :yxetn ote`a.dcewpe dcewp lka drixid z` sxbd `ed p dcewpa M -l wiynd agxnd.r n -a zicnin-k drixi M idz 0.6 dxcbd divwpet ly sxb `id M U dveawd U a witqn dphw daiaqa xy`k,df(a) ly.v R k efi`l f : V R n k,c daiaqde) f divwpetd zxigaa ielz epi` `"f,ahid xcben wiynd agxnd 0.7 dnl.(v :d`ad dprhd on ziciin zraep dnld :mieey md mi`ad mix`pild miagxnd 0.8 dprh.p dcewpa M drixil wiynd agxnd. divwpet dfi`l M U = F (0) miiwzn p cil zinewn,xy`k ker DF (p).2.n k dbxcn DF (p)-y jk F : U R n k dgezt V R k,r"gg C divwpet g : V R n m` Dg(g (p)) ly dpenzd.3.k dbxcn Dg(a)-e U dgezt efi`l g(v ) = M U -y jk agxndy meyn,mieey md (3)-ae (2)-a mixcbend miagxndy ze`xdl witqi :dgked eze`n mixehwe miagxn ipy el` zegpddn.(3) ly ihxt dxwn `ed ()-a xcbend okl,f g(v ) = 0 la`.ipyd z` likn mdn cg`y gikedl witqi okl,cnind didy dn dfe,im(dg(a)) ker DF (p) hxta.y V lkl DF (g(y))dg(y) = 0.gikedl enk f lkl if`,dprhd ly (2)-a enk F rawp?dnld zgkedl dwitqn dprhd recn.f -a ielz epi` okle,ker DF (p)-l deey (df alya f -a ielzd) wiynd agxndy ()-a 7