Matematika | Analízis » Túldeterminált parciális differenciálegyenletek


     
    
      
 !"# $ %&)(+*-,/.10/24573 6 8:9;9:<>=@?A8B=9;CD= Tartalomjegyzék ?:=E?:=GFIH:JLKEMOPRN Q KAH>N SBTUVXN WZY PRN [ S PW T]M V JH^ `H;a PRWZbcPR[ =)=)=)=)=)=)=)=)=)=)=)=)= ?:=d8B=fe7Khg VjiLVjk1Q KlPN WZV a:m Vjk1WEV J Vj[ SBM Vj[ JnH iLboVXp H ^ [ S1K k J V a i PN W a1Hq inr VjN [ =)=)=)=)= ?:=utD=wv Vx[ JLH iLboVjp H ^ [ K V a:m VXkVXy zN JBVXN ynV =)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)= ?:=|<>=wv Vx[ JLH iLboVjp H ^ [cynp K b~}W JPN kc[ K V a:m VXkVXy zN JBVXN ynV =)=)=)=)=)=)=)=)=)=)=)= ?:=d€B=fe7K ynp JLK r }N Q KAH N [ SDK k J V a i PN WEy H [:PRy P N a;H [ =)=)=)=)=)=)=)=)=)=)=)=)=)=)=)= ?:=uCD=f i H r‚VXk K }y JBV N J VXW =ƒ=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)= 1. PDE és geometriai interpretációjuk ? 1 s {  ?A8 ?X< 8B=E?:= „ k1PW KEJLK [D}y L} q a;a:M†VXN k m Vj[ =)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)= 16l? C 8B=d8B=

„ˆ‡ P}QŠ‰ m]‹ŒŽH PWEV  y‘[ mcJDV N J VXW’b PN y H Y>iLVXkY }I^ “ e7”4‹ iLV =)=)=)=)=)= A? s 2. Cauchy-Kowalewsky tétel tD=E?:= ” y‘pj[ Hq p Hq [IP~“ e7” WZV zN i PN y PN ‰ H p =)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)= tD=d8B=f•’K kV PN i K y–“ e7”— `H iLb PRN W K y K k J V a i PN WE‰P J1H N y P N a P =)=)=)=)=)=)=)=)=)=)=)= tD=utD=fe7Khg VjiLVjk1Q KlPN W H:T Vxi P N JLH iUy‘p K b˜r H N WE}1bcP =)=)=)=)=)=)=)=)=)=)=)=)=)=)= tD=|<>=GŒ™HAM PRi KlPN ky–YVji KEMOPN W PN y =)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)= tD=d€B=fš ‰V ‡ PRi J Pk ‹›ŒœPq ‰WZVji J ‰V H iLVjb =)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)= tD=uCD= “ VXN WEYP]†W K k1V PRN i K yO[ H kkVxž KlHŸN =)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)=)= 3. Formális integrálhatóság :8  t8 t;t t:< t;C t;{ 22 1. 1.1 ¸¬¹Lº»¹Š¼ ÆAÑ PDE és geometriai interpretációjuk Motiváció, példák, alpvető fogalmak ¹¾

¿²À]ÁEÂùLÄx¹Š¼ŊÁ ÆA¾ ÇZÈÆ ¿ ɘ¾ ÊÌÍ Ë º;º1Î ¹Š¾ ¼;»Ï ¹Š¾ Ð ½ ¹LÄj¹ŠÐ‘Ð Í Ë ½ ÆlÑ ¿ ½ ƒ À]ÁEÂùLÄx¹Š¼ŊÁ ÆA¾ ÇZÈÆ ¿ Ƀ¾ ÊÌÍ Ë º;º1Î ¹Š¾ ¼»@¿›ÏÒI¹ Ç »>Äx¹ Ó ?AÔ 1 Példa f, g : U (⊂ R2 )−−R α : U−−R  ∂α   (x, y) = f(x, y),  ∂x ∂α   (x, y) = g(x, y).  ∂y ” p°PRp£V a:m VXkWZV J iLVjk1YynpjVji PRN W J PW PN rPRk™k1Vjb H WEY1‰1P JH N boV a>S boVji J Pp f VXN yÃP g n} q a;a:M†VjN k m Vj[ [ Hq p H:qS JnJœH;‹ a:mPRbo VXkk N mDK r‚H VXk =ÖWFIV N J KEVjM p K [GP VXW`PmDY1KP J kVj[ bo V a;H WZY J PN yLP ‹LM S PR[B[ H i a;HqH ynynpjN V L} q S a;a¬VXN y [VXWEW ‰ VXkk PRWuÕ k VjWOPoVXkk r‚Vjk§P VjW`PY1P k1PR[ PRkfboV WZY PyLP PplP W V N J VXp K [×Pp α(x, y) Y KEg VjiLVXkQ KXPN WZ‰1P J1H)N L} q a;a:M†VXN k m PRb K iLV J VXWu՛VXy }q WZkVj[×P VXk JnK V a:m VXk ‹ WZV J Vj[ S P[B[ H i Ó 8;Ô ∂g

∂f (x, y) = (x, y) V a:m Vjk1WEV J r K p JLH yLPk J VXWu՛VXy }q W S ‰ ∂yK y‘pXVXk#Pp α(x,∂xy)‹ iŠP J VXWu՛VXy }q W’P ∂2 f ∂2 f (x, y) = (x, y) ∂y∂x ∂x∂y V a:K@m J Vjk1WEV Jl=£” p J VXJ ‰ P N J K¯S P+boH;a:V m a;H WEY1‰1P JH N y P N a V a:m ynp }q [By V N a VXy VXW JBV N J VXWZV =´” kkVj[cPWZP T Õ PN k [ }1Y;՛}>[˜ÕØVjWZVXk VXk ‰ PRp K H f = x y k y, g = yQ y x MOPN W`Pynp JPN yÙboVXWEWZV J Pp Ó ?AÔ V a:m Vjk1WEV J kVj[wk K kQXyœboV a;H WEY PRN yLP S ‰ K y‘pXVXkGP Ó 8;Ô) VXW JDV N J VXW Vjrr‚Vjk Pp×VXynV J J r‚VXk SkVXb J VXWu ՛VXy JB}q N WJ =ÚFIJ KEJ)J }q Y}1k>[FIb K H J kYPRk J K PÚN boH ^ V a;H WZYN J‰1P K+J1H N `y H;P Na a i H N W PRrÓrPkÛPpwVjynV r‚Vjk ‰1P—P VXW V VXW VXWu՛VXy }WÝÜ k Pp [ Vjy rrÞW Ó P k ÕØ}>[ P 8;Ô V a:m VXkWZV J kVXboQjyLÓ PR[!ynp }q [By V N a VXy S YV VXWZV a VXkY Hf^ VXW JBV N J V K yÚPp ?AÔ boV a;H WZY ‹ ‰1P J1H N y P N a´PN k1PR[ S

PplPp+P 8;ÔUMOPN W`Pynp JPN yßboVXWZWZV JnJ P VjW`PY1P J boV a;H WZY‰1P J1H>N = Geometriai interpretáció   VXW }1q WEV J V J [VjiLVXy }1q k>[ S PoVXW m V J Pp α a i P N ÕØP)‰1P J>PN i H p-boV a>S4VXN yUP VXW }q WZV J b K k ‹ YMVXk JLTH H H k J Õ PRN rJ>PkœN HP VXWN }q WZV J)VjN a>i K=4k „ JH ^ ՛VOK Pm p (1, 0,q f(x,J J y)) SlM K WZWZV Jnq M V†KP (0,J a 1,N g(x, Vx[ i [c‰1P PRi pXÓ p P[oboV p W Vjk VXW }1WEV V kV VXpjp }>[ k V i PW VXW }q WZy))V J‘‹ kVj[ =´„ iLVXkYynpXVji J VjWu՛VXynVXk Ô°K k J V a i PN WE‰P J1H>N S ‰PÙb K k1YVXk TH k J iŠP K WZWZVXy‘pj[VXY K [ V a:mÖK k J V a i PN W VXW }q WZV Jl= 3 v PW H N rPk S PRboVXkk mDK r‚VXk M Pk K W m VXk M VXW }q WZV JXS P[B[ H i-P π : M R ‹°T i H ՛Vj[Q KlH N Ó PRp†VXWEy H ^ [ V N J [ H]H iLY K k P N JPRN iLP ÔK k M Vxi JPN WZ‰1P JH>N S boVji J P π V a:m ynpX}>r1boVjiLp KAH>N S Pp K b T W K Q KhJ JBV N J VXW

VjN i J VjWZb VjN r‚VXkIPp M‹ Vjk#P z [ H]H iLY K k P N J P+[ KZ V‘Õ›VXpX‰V JH ^ Pp (x, y)‹ k1PW S]J VX‰ P N J Pp H´^ N S K J a:m α : R R L} q a;a:M†VXN k mGa i P N ÕØP =„ p M T PRiLP V N J VjinVXp VXN y V N J M VjW PWZW b k V PY;ÕØPÕßP ϕ : (x, y) (x, y, α(x, y), ) VXN y J V J ynp H ^ WZV a VXy T‚H k J rPk#Pp M VjN i K k JH:^ J VjinV P VXN y†P S PpXPp-PRp (1, 0, ) VXN y°P (0, 1, ) VXN y°P M Vx[ JLH i H [ VXy‘p zN JLK [˜[ K¯=4š VX‰ P N J T‚H JLH H S T M = D k yLPkIPR[B[ i ‰1P 2 2 ∂ϕ ∂x ∂ϕ ∂y ∂α ∂x ∂α ∂y ∂α   ∂α  , 0, 1, ∂y ∂y = p  1, 0, ” p T VjY KEa~T‚H k JLH yLPkIPR[B[ H i J VXW|ÕØVjy }q W S ‰1P ∂α = f, ∂x  (1, 0, f), (0, 1, g) ∂α = g. ∂y ª ¹nº» Í Ë ½ Ê ¹ Ç Ï ÈÉ º» f, g : R R ¹ Ç ¹LºD¹¿£¿›¹ŠÐ уÆàâáã ¹Lº»¹Š¼ Ç ¹¿â¼‚¹ ½ Æ 0 = (0, 0) ¹Lº» ½ É Ë ¹LÄLº¼» »¹ Ñ É Ë¹ÄL¿ ¼;¹R¾ »ä¹ ¹LÑ ¼Ã¹å

¿ ¹R¤ ä½]¹Š½ ¼É Ï ÄÙȂ¿É ¹º¿æ» Ð Ñ@ɐç Ç ¹LºD¹ŠÐ z ∈ R ¹LÐx¹¿ ¹Š¾ ¼ Ç ¹¾ ¿›¹ Ñ Á ½ ÉÇ » Æ ¼ α ÌÊ Í Ë º;º1Î ¹Š¾ ¼;» Æ ½ ¾ 0 1 Tétel. 2 0  α(0, 0) = z0      ∂α =f ∂x    ∂α   =g ∂y Žè é`êÌë;ì‚í°ïæî ð ñÌî òRó • V a:m Vjk α(0, 0) = z S bcP՛YYVjô1k KlPN Wu՛}>[ Pp VXWE‰PRynpXk PN W PN y P N M PRW@Pp α(x, 0) n} q a;a:M†VjN k mBJ  0 bcP՛YÚY1Vxôk KXPRN Wu՛}[oP α(x, y) := α(x, 0) + α(x, 0) := z0 + Zx ∂α ∂x =f V a:m VXkWZV J f(t, 0)dt 0 Zy g(y, t)dt = z0 + ŒŽK ‰1Pynpjk PRN W M P Ó 8;Ôõ‹¯J Pp J [:P T ՛}>[ S ‰ H;a:m Pp 0 y−− Zx f(t, 0)dt + 0 ∂α (x, y) − f(x, y) ∂x Zy g(y, s)ds 0 P p H k H ynPk)K k]JL}K WZW`P =Ô KEMOp£N } J a:m N J–Pk 9 K y‚k mBH N K W MOK PN = k EK a P p´Pp y = 0 boVXk JBVjN k SlJLHAMOPN r1r P°N [ K ynp PN b zN JnM P Pp y ynpXVji k

YVji PW Õ P PY Y [ ® Ñ ¹ ÇAɐç Ñ@Éwç ö ¹ ¾ Ç À Æ ¾ ¿ ɐ¾ Ç ÆA¾ Ç ¿ ÆlÇ Æ ¾ ¼ É Ð Æ ää÷¹ŠÐx¹¿¹¿§ÎRÁ Ñ Ð‘º ¹ ¾ ÇZÈÆ ¿ Í ¼ ½ Ï È‚Æ÷ÆAÑ f, g : U (⊂ ¹¿ÚÀ]ÁE’¹LÄj¹Š¼ŊÁ Æl¾ ÇZÈÆ ¿ Éø¾ ÊÌÍ Ë º;º1Î ¹Š¾ ¼»Ï ¹Š¾ Ð ½ ¹LÄx¹ŠÐnÐ Í Ë ½ ÆAÑ ¿ ÆAÑ α : U−−R R )−−R ½ƒ¾ ½ƒ¹¾ ¿nÎ Æl¾ Ç ¿ ÉÑ É ¾ ЎÀ]ÁEÂùnÄj¹Š¼ŊÁ ÆA¾ ÇEÈ‚Æ ¿ É)¾ ÊÌÍ Ë º;º1Î ¹Š¾ ¼»@¿›ÏÒI¹ Ç »DÄj¹  ∂α   (x, y) = f(x, y, α(x, y)),  Ó tÔ ∂x 2 Példa 3 ∂α   (x, y) = g(x, y, α(x, y)).  ∂y „ b PN y H YinVXkY } ^ M V a:m VXy T PRiLQ KXPRN W K yUYVji KEMOPN W J P[cV a:m VXkW H ^ y V N a/VxN r H ^ W’PY H N Y K [ S ‰ H;a:m ∂α2 (x, y) = ∂y∂x || 2 ∂ α (x, y) = ∂x∂y ∂f ∂f ∂α (x, y, α(x, y)) + (x, y, α(x, y)) (x, y) ∂y ∂z ∂y ∂g ∂g ∂α (x, y, α(x, y)) +

(x, y, α(x, y)) (x, y) ∂x ∂z ∂x IF KEM VjW KhJnJ b V N a kVXbQjyLPR[Pp f K WZWZV JnM V˜P g ynpXVjinV T VXW S VXp~b V N a k1Vjb J}1N W4‰1Py‘pXk H y Vjrr‚VjkcP `H iLb PN rPRk =Ãv™K ynp H k J PRp Ó tÔ VjiLVXYV JLK V a:m VXkWZV J Vj[V JU VjWZ‰1PynpXk PN W M PƒPY H N Y K [ S ‰ H;a:m PoVXkk mBK r‚VXk M Pk boV a;H WZY PN y S PR[B[ H i-Pp ∂f ∂f (x, y, α(x, y)) + (x, y, α(x, y)) · g(x, y, α(x, y)) ∂y ∂z ∂g ∂g = (x, y, α(x, y)) + (x, y, α(x, y)) · f(x, y, α(x, y)) ∂x ∂z ù P˜Pp U b K kYVXk (x, y) T‚H k J ÕØP˜VXynV JBVXN k#W V N J Vjp K [cboV a;H WZY PN y S PR[B[ H i-Pp Ó <ÌÔ V a:H~^ m Vj k1WEJBV NJ J kVjJl[ S@J N VXa:Wum ՛VXy }q H WZk K VßHq[VXM WZW =´J ” pŽq } J1H N rr K H ^b H PN i°QXynJPRH ^ [o=´Pp ” f VXN yO P gJDN LJ } q a;a:M†ÓVXN tk Ô m Vx[Ba:miLV zN i‹ VXW VjW V VjW z PRp [o[ p V WZVXk }W@VXWZWZVXk i pj‰1V [ p+P VXW V VXW@P V Vjk WZV J iLúVXkYynpXVji²boV a;H WZY‰1P J1H N y P N a´PN

k1PR[×y‘p }>q [Dy V N a VjyŽV a:m ynp }>q [Dy V N a VXy VXW JDV N J VXW V N J PY;ՑP = Vjk1Vxi K [D}yLPRk MOPRN W`Pynp JnH:JnJ‚ L} q a;a:M†VXN k m Vj[BinV°VXp J VjiLb VjN ynpXV J VjynVXkÙkVXb J VjWu՛VXy }q W = “ VXN Y PN }W Pp f(x, y, z) = xyz, g(x, y, z) = x y z MOPN W`Pynp JPN yboVXWZWZV J‘J Pp Ó tÔ V a:m Vjk1WEV J iLVXkYynpXVxiLkVj[Þr K p JLH yLPkˆk K kQXy M PW`PoVXkk mDK Ó <ÌÔ V a:m VXkWZV J kVXb J VXWu՛VXy }q W =£ŒÖVjN inY VlN û ‹ K H ^ a;H N S K (x, y) iŠP WZWZVXy‘pj[VXY boV WZY PyLP ‰ ynpXVXkÖPp PoVjk1k mDK rVXkIPRp f VXN y g L} q a;a:M†VXN k m Vj[V J P ∂f ∂f ∂g ∂g + ·g = + ·f ∂y ∂z ∂x ∂z 4 4 4 f(x, y, z) = ., b H N Y H k UM P N W`PRynp J ՛}>[ S PR[][ H i M Pk ‹ V)boV a;H WZY PN y = g(x, y, z) = . ¬¸ ¹Lº»¹Š¼ ¼» ü ¾ Ç ¿›Ï ÆlÈ‚ÉÇ U Æ 0 ∈ R ¹Lº»I¼» ü ¾ Ç ¿ ½ É Ë ÄL¼;»¹ Ñ ¹¿›¹jÏ à¯ÆAÑ ¹L¼ Æ ½ ÉBÉ ÄxÀBÁE¼ Æ ¾ ¿ Æ ¾

½ Æ ¿ t , x ý ÎR¹ Çþ ¹ ÇRÉË Çÿþ‚Í Ë ½ ã Ï ¹Š¾ Ð Ç ¹nº»¹Š¼¹ ½ f : U × V R ý ÐnÁEÒ ÆGà ãcÊÍ Ë º;º1Î ¹Š¾ ¼»¹ ½ ÒoÁE¼‚À¹Š¼ i = 1, ., mý Äj¹å ¤ ½]½ É ÄcÒoÁZ¼À¹Š¼ x ∈ V ý Äj¹ ö£É º;¼º1¿ Î É ¹LÐ ¼Æ »¼ Ï Æ ½B½ ºÉ » Ä Ç ¹¾ ¿¹ Ñ Á ½ ÉÇ » Æ ¼ α : W−−V Æ 0 ∈ R ½ É Ë Än¼»¹ Ñ ¹A¿ ¹R¾ 乊¼ À¹ 1¼1Á ÆA¾ Ç ¿ ÊÍ Ë ¾ ÈÉ  2 Tétel. U × V ⊂ R n × Rm Rn × R m C∞ ÈÆ (0, x) n i j n i
m  α(0) = x,  ∂α (t) = fi (t, α(t)), ∂ti ý ¼¹ ½ Î Æ ¼ ÉÇ » Æ ¼ ½ É Ë ÄL¼»¹ Ñ ¹¿›¹XÏ ÈÉ º» Æ n n ∂fj ∂fi X ∂fj k X ∂fi k − + f − f = 0, ∂ti ∂tj k=1 ∂xk i k=1 ∂xk j i, j = 1., m Ê ¹ Ç ¿ ¹¾ ¿›¹ Ç ¿›¹ Ç þ ¹ŠÐ Í Ë Ç å èŽé`êÌë;ì‚í°ïæî ð ñÌî òRó ûBp }q [By V N a VXy‘y V N a r K p H k m‚zN J>PRN yLP MDK W P N a;H

y =£”£a:m†VjN i J VXWEb }1^ y V N a PRY H N Y K [ÚP W V N J VXp VXN y–r K p H k mzN JPN y PRN r H N W =¬„ W V N J VXp VXN y–r K p H k mzN JPN y PN ‰ H p J Vj[ K k J y }>q [IP α(0, 0, ., 0) = x ∂α (t, 0, ., 0) = f1 (t, 0, , 0, α(t, 0, 0)) ∂t1 Y KEg VjiLVXkQ KXPRN WZV a:m VXkWZV J iLVXkYynpXVji JXS P K k1Vx[˜ynV azN J y V N a¬V N M VXWboV a PY;՛}[ Pp n} q a;a:M†VjN k mBJX=°š Vx[ K k J y }>q [ÚVXW H ^ ynp Hq i-P α(t, 0, ., 0) β1 (0) = x β10 (t) = f1 (t, 0, ., 0, β1(t)) [H Hq p HNq ky V N a VXyŽY KEg VjiLK VXkJ Q KXPNM WZV a:m VXkHWZV J =¬V Jl” =†” kH kVj[ÖWZV a:m VXk WZY PynP~P |t| < ε k Vji PWZWZ}b k [B[ i 1 ” pX} JPN k J Vj[ K k J y }>q [ÖP . α(t1 , 0, ., 0) = β1 (t1 ) β1 : (R, 0) (Rn , x) boV a‹ β2,t1 (0) = α(t1 , 0, ., 0) 0 β2,t (t) = f2 (t1 , t, ., 0, β2,t1 (t)) 1 Y KEg VjiLVXkQ KXPRN WZV a:m VXkWZV J V Jl=ޔ kkVj[ WZV a:m VXk β : (R, 0) (R , x) boV a;H WZY PN yLPøP K k J Vji M PWEWZ}b

H k =Ž” W V N a [ K QXy K iLV MUPN WZPynp JnM PcP t ‹ V J7M PRk H W m PRk ε S ‰ H;a:m |t| < ε K J M H ^ Hq K =°• V a:m VXk |t| < ε k Vji PWZWZ}b kcb }>[ Y [cP β n 2 2,t1 1 ” [B[ H i 1 2 2 . α(t1 , t, 0, ., 0) = β2,t1 (t) α(0, ., 0) = β2,0 (0) = β1 (0) = x ∂α (t1 , t, 0, ., 0) = f2 (t1 , t, 0, , β2,t1 (t)) = f2 (t1 , t, 0, , α(t1, t, 0, , 0)) ∂t2 v™K pXy a´PN Wu՛}>[oP t y‘pXVji K k JLKT PRiLQ KXPN W K yOYVji KEMUPN W J P Jl=4” ‰‰VXp M VjpXVXyny }>q [Úr‚V™i H;q a p zN J V JnJ t VXy‘V JBVXN k P Ó €;Ô ∂α (t , t, 0, ., 0) − f (t , t, 0, , α(t , t, 0, , 0)) g(t) := ∂t n} q a;a:M†VjN k mBJÖVXN y–b } J Py‘yn}>[cboV a>S ‰ H;a:m g(t) ≡ 0 =£„ t = 0 VjN i JBVjN [Vx[IboVXWEWZV JnJ 1 1 1 1 1 1 1 ∂α (t1 , 0, 0, ., 0) − f1 (t1 , 0, , α(t1, 0, 0, , 0)) ∂t1 = β10 (t1 ) − f1 (t, 0, ., 0, β1(t1 )) = 0 g(0) = š/HAMOPN r1r P N J V J ynp H ^ WZV a Vjy t‹ iLV g 0 (0) = ∂2 α ∂f1 (t1

, t, 0, ., 0) − (t1 , t, ., α(t1, t, 0, , 0)) ∂t2 ∂t1 ∂t2 n X ∂f1 ∂αi − (t1 , t, ., α(t1, t, 0, , 0)) · (t1 , t, ., 0) ∂xi ∂t2 i=1 „! `H iLb~}W P N J V a:m ynpXVxi }>^ r1rVXk VXW zN i M P Ó Pp α PRi a }1boVjk J }bcP-P (t , t, 0, ., 0) SlK WZWZV J‘M V P f VXN y-YVji KEMUPN W J ÕØP K k1PR[oPRp+PRi a }1boVjk J }bcPÙP (t , t, 0, ., α(t , t, 0, , 0))Ô  1 1 1 n ∂2 α ∂f1 X ∂f1 ∂αi g (0) = − − · ∂t2 ∂t1 ∂t2 i=1 ∂xi ∂t2   n ∂ ∂α ∂f1 X ∂f1 ∂αi = − − · ∂t1 ∂t2 ∂t2 i=1 ∂xi ∂t2 0 (5) ∂f2 = ∂t1 + n n X ∂f2 ∂αi ∂f1 X ∂f1 ∂αi · − − · ∂x ∂t ∂t ∂x ∂t2 i 1 2 i i=1 i=1 n n X  ∂f1 X ∂f2  i ∂f1 ∂αi (5) ∂f2 i = + · g (t) + f1 − − · ∂t1 i=1 ∂xi ∂t2 i=1 ∂xi ∂t2  ∂α ∂t2 = f2 =  n n n   ∂f2 ∂f1 X ∂f2 i X ∂f1 i X ∂f2 i .  = − + · f1 − · f2 + · g (t) = ∂t1 ∂t2 i=1 ∂xi ∂xi ∂xi i=1 i=1 = „ n X ∂f2

∂xi · gi (t) J VX‰ P N J˜J VXWu՛VXy zN JLK P VXk JnK W K k1V PRN i K yœY KEg VjiLVXkQ KXPRN WOV a:m VXkWZV J V Jl=f” kkVj[fboV a‹ H ZW Y PN ynP MBK ynp H k JlS Pœ[VXpXYV JLKà VXW JBV N J VXWZVj[ÚPWZP T Õ PN k S P g(t) ≡ 0 = 2 i=1 g(t) ú V H ob V J i K P KK k J Vji T iLV J>PN Q H N  „ p Rn+m ‹ r‚VXk boV ˜a M P k#PY M P n Y>r M Vj[ JLH iLboVXp H ^  i . 1 Xi = (0., fm), 1, .0}, f|i , , | {z {z i} n m XMV N y~J‘PJ p J N [K VjN JRiLH ^YVXpjp N }>qT1[ T S ‰ H;a:møM Pk ‹ V H W m Pk VXW }q WZV JlS P K k1Vx[ J V J ynp H ^ WZV a VXy T‚H k J Õ PN rPRk V Vxi k ÕØV V VXk PRp D =. {X , , X } M Vx[ JLH iLboVjp H ^ [ PN W J PWZWß[ KZ VXy‘p zN J V J‘J n‹ Y K boVXkp KlH N yƒy zN [ =o„ boVjk1k mDK rVXk M Pk K W m Vjk S PR[B[ H i π : M R T i H ÕØVx[Q KAH N Pp VXWEy H ^ m PR[ JLH i‘iŠP N T V a:mÚN K kJlSM Vji J>PRNK WZ‰1P J1H N WZVja [ V NN T VXp VXN y SUVXN y²Pp K k M VjiLpjV)boV a ‰1P J>PRN i H pŽV a:m

α : R R WZVj[ V VXp VXy P kVXbP i P ÕØP~Pp M  1 n n n m Rn −−− M ⊂ Rn+m ” [B[ H i)b K k YVXk (t , ., t ) T PRiŠPRb V N J jV iLkVj[ boV ;a XV WZVXW H ^ H {V , ., V }, P‰ W (t1 , ., tn) 1 1 (t1 , ., tn, α1 , , αm) n p∈M T‚H k J rPk n . Vi =  ∂αm ∂α1 , ., 0., 1, 0, ∂ti ∂ti i ” p T VjY KEa~T‚H k JLH yLPkIPR[B[ H i–[ H:q M V J [VXp K [Úr‚V S ‰1P ∂α = fi , ∂ti  . i = 1, ., n Tp M = ù P§V a:m y H [:PRy P N a Ó VXW }q WZV JŠÔ b K kYVXk V a:m VXy T‚H k J Õ PN ‰ H p#‰ H pXp PN iLVXkYVXW }q k>[!V a:mS P T‚H k J r H N WB[ KEK kY1}W HŽN M Vx[ JLH i JXS P[B[ H i°Pp J b H kY՛}>[ S ‰ H;a:m boV a PY J }k[œP7y H [:Py P N a;H k Ó VjW }q WZV J VXk Ô V a:mcM Vj[ JLH iLboVXp H:^ J  1.2 Differenciálegyenletek, vektormezők, integrálgörbék • V a:m VXk X ∈ X(M) V a:mfM Vx[ JLH iLboVjp H"^ VXN y γ : (−ε, ε) M V a:mfa1Hq inr‚VÚPp M y H [Py P N a;H k Ó VXW

}q WZV J VXk Ôx=O„ γ‹æJ Pp X K k J V a i PN W a1Hq inr VØN Õ VjN k1Vx[ÖkV M VjpXp }>q [ S ‰1P~b K kY1Vjk JBN S PpXPpÙP γ b K kYVXk T k J Õ PRN rPk PRp VjN i K k JH:^ M Vj[ JLH iŠP t ∈ (−ε, ε) VXynV Vjk γ(t) = X boV a V a:m VXp K [ÚPp X PY H:JnJßT‚H k J rPk VXW M V JnJÖVxN i JBVjN [ V N M VXW¯  γ(t) @• H [ PN W K y-[ HBH inY K k P N J PRiLVXkYynpXVxi J ‰1PynpXk PN W M Po‰1P X = X S P[B[ H i)P a1Hq inrV T‚H k JLH yLPkIPR[B[ H i²WZVXy‘pÙPp X K k J V a i PN W a1Hq inr V‘N ՛V S ‰1P i ∂ ∂xi γ(t) = xi (t) xi = Xi (x(t)), P pXPp°‰1KlP–N P γa:m [ H b T‚J H kVXky L} q a;a:M†=¬VXN k ” m V K boV a;H WZT Y PN N yLP K V a:KEm g VXWEy H ^ iLVXKlkN Y }+^ a:m [ Hq p Hq k1J y V N a VXy¬Y NKZ E‹ JDN H ^ VjinVXkQ PWZV VXkWZV iLVjk1YynpjVjiLkVj[ kkVj[ƒPW`P Õ PkƒPUY VjinVXkQ PWZV VXkWZV Vj[ŽVXWZb VXWZV Vjr W K y‘boVji J PpƒVxž K ynp J VXkQ K P VXN y²}k K Q KEJPN yßP˜boV a; VXWEVXW H ^ [VXpXYV JLK7VjN i

JBVxN [Vj[IboVjWZWZV JnJX= „ p X M Vx[ JLH iLboVjp H ^ ‰ Hq p J Pi JLH p H÷N `H W m Pok1PR[ÛkV M VXpXp }q [ P ϕ : (−ε, ε) × NT N S m B(x , a) − M WEVj[ V VXp VXy boVXW iLV X 0 d dt ϕX (t, x) = X(t0 , x). XV W Hq W VXN a:y‘mr‚VXk)P TϕH JL(t,H:J x)H;q a = N ϕJ;q (x)S `H iLb~H }W P N JK y’PWE[PT‚WZbcH PJLpXH pj}[ =’” pXVx[)K y‘pXJ Vji a K k J N Pa1boHq VXkN k mBKh‹ r‚Vjk×V x k i p z }k[ PR[B[ i°P ϕ (x ) k ynPk×Pp-Pp k V i PW inr V‘Õ›V Pp Xa:‹ km Vj[ S P K P t = Lq 0a;a:T M†PRN iŠPmBb J V N J VjiLk VXN WPp x ‹õH ko‰1PK WZPJ Y [VjinKEMOVXynN p J:}q K¯SW =£” pXH Vj[PW` `P H T m Õ PN k S ‰1P)V f : M R } VXk PR[PRiL}k>[Pp X ynpjVji k YVji PWZk PR[][ ißP W P ynV a:mBJ y V N a¬V N M VXW¯ K f(ϕ (q)) − f(q) (Xf)(q) = X f = W b h  X t=t0 X t 0 t 0 0 h q h0 1.3 Vektormező kiegyenesı́tése ¸¬¹Lº»¹Š¼ L¹ º» БÁZÒ Æ ÎR¹ ½ ¿ É ÄLÒI¹ Ñ@É ç ÏÒI¹ Ç »>Äj¹ X(p) 6= 0 å

¤ ½B½ É Ä Æ p ½ É Ë ÄL¼;»¹ ý Ñ ¹A¿ ¹R¾ 乊¼ Î Æ ¼ ÉÇ » Æ ¼ (U, x) ½ ÉBÉ ÄjÀ]ÁZ¼ Æ ¾ ¿ Æ Äj¹Š¼À]Ð Ñ ¹LÄjÏ ÈÉ º» X = å 3 Tétel. X∈M ∂ ∂x1 èŽéZê]ë;ì‚í†ïæî ð ñÌî òRó „ [VXpXYV JLK (t , ., t ) [ H]H iLY K k P N J PRiLVjk1YynpjVji×WZV a:m VXk H W m Pk S ‰ H;a:m ‹ H KEa£H>N S X ‹ k1Vx[ T VjY KEa P VXWZVjWÃboV a>=´š Vj[ K k J y }>q [ÖP p kVj[ÖPp i 1 p n ∂ ∂t1 0 χ(a1 , ., an) = ϕa1 (0, a2, , an) WZVx[ V N T Vjp VXN y J P [ Hq iLk m VXpXV JBVxN r‚VXk =O” [B[ H i 0 ∈ Rn   ∂ ∂ χ∗ f= (f ◦ χ) ∂t1 a ∂t1 a 1 (f(χ(a1 + h, a2 , ., an)) − f(χ(a1 , , an))) = h0 h 1 = (f(ϕa1 +h (0, a2, ., an)) − f(χ(a1 , , an))) h0 h 1 = (f(ϕh ϕa1 (0, a2, ., an)) − f(χ(a1 , , an))) h0 h 1 = (f(ϕh χ(a1 , a2 , ., an)) − f(χ(a1 , , an))) h0 h = (Xf)χ(a) . WKb WKb WKb WKb û T VjQ KlPN W K yLPkÖP 0‹ rPkIVXpjVj[ ynpjVji K k J χ∗  ∂ ∂ti 0  χ∗

 ∂ ∂t1 0  = ∂ ∂t1 . 0 ùP i>1 S P[B[ H i ∂ (f ◦ χ) ∂t1 0   i 1 f(χ(0, . h, , 0)) − f(χ(0, , 0)) = h0 h   i 1 = f(0, . h, , 0) − f(0, , 0) h0 h ∂f = i . ∂x 0 f= WKb WKb 1.4 Vektormezők szimultán kiegyenesı́tése ÆÄ Ò ¹ ¿›¹LÄj¹LÐ)¿ Ä ¼Ð ÄL¹nÒ º»ˆÅŠÁ À]ÁEŊÂÃÐ ¹ É Ò ÄAÉ ¿ Ä Ñ Ò Í Ð Ï ¹Š¾ Ð X Ä ∈ X(M) Æ ÒoÁZ¹¼¹ ½ ÆAÑ n¹ º» /ö Æ ý Æ ¾ Æ ÑXÊæÉ Æ ¾ É ¾ Éxö´É þ ÆœÆ ϕ Æ ½]½ É ÆlÑ X := α Xý ¾ Æ 4 Lemma  α : M M   t ∗  = α ◦ ϕ ◦ α−1 ϕ t èŽéZê]ë;ì‚í†ïæî ð ñÌî òRó (α∗ X)q f = (α∗ Xα−1 q )f = Xα−1 q (f ◦ α)
1 = f ◦ α(ϕh (α−1 )) − (f ◦ α(α−1 (q))) h0 h
1 = f ◦ α(ϕh (α−1 )) − f(q) h0 h
1 = f(α ◦ ϕh ◦ α−1 (q)) − f(q) h0 h
1  (q)) − f(q) = f(ϕ h h0 h WKb WKb WKb WKb ¹Lº»À]ÁEÂù É Ò É Ä Ñ Ò Í Ð ¹Š¾ Ð α X = X(M)Ï Æ

½B½ É Ä α ◦ ϕ = ϕ ◦ α å 1 Definı́ció ® Ñ Y ∈ X(M) ÎR¹ ½ ¿ É ÄLÒI¹ Ñ‚É ç X ∈ X(M) Ð Ñ ¹LÄLÁZ¼¿æÁ¸@Áâ¹ ý À¹LÄLÁõÎ ÆA¾ Ç ¿ þ Æ ¾ ¼ Æ ½ ¼¹ÎR¹ ÑXÑ´Í Ë ½ ÆAÑ
. K (L Y)p = W b Y − (ϕ Y) ÎR¹ ½ ¿ É ÄLÒI¹ Ñ‚É ç ¿nå ¤ Ñ ¹ ½ ΐÁõÎ ÆlÇ ¹Š¼Ð Æ (L Y) := [X, Y] À¹ 1¼ ü ¾ ŊÁ É ¾ Î ÆAÇ Ï ÆlÈ‚ÉÇ ÁZÐ Æ 5 Következmény  α:MM  ∗ X h0 p X h∗ p
X p p . [X, Y]f = X(Yf) − Y(Xf). ¬¸ ¹Lº»¹Š¼ ÆAÑ X ∈ X(M) ÊæÉÇ » Æ Ò Æ)Æ ϕ ¹Š¾ Ð ÆlÑ)ÆAÑ Y ∈ X(M) ÊæÉÇ » Æ Ò Æ)Æ ψ å ¤ ½]½ É Ä ö£É ¼¿ É Ð Æ ¼ ÒoÁEÆ ¼‚½]À½ ¹ŠÉ ¼ Ä [X,¹ŠÐ Y] Äj=¹å 0Ï ÈÆ ÆÚÊæÉÇ » Æ Ò É ½½ É ÒoÒ Í ¿ ÆA¾ Ç ¼ Æ ½ Ï ÆlÑRÆAÑ ϕ ◦ ψ = ψ ◦ϕ s ¾ tý èŽéZê]ë;ì‚í†ïæî ð ñÌî òRó P Ô ù P ϕ ◦ ψ = ψ ◦ ϕ S PR[B[ H i (ϕ ) Y = Y SUVXN y zN a:m [X,

Y] = L Y = 0 = r Ô֚ V a:m/}>q [ VXW S ‰ H;a:m [X, Y] ≡ 0 =†” [][ H iŽb K kYVXk q‹ iŠP
K (L Y)q = W b Y − (ϕ Y) = 0. 6 Lemma t s t s t t s s t ∗ t X h0 q X X h∗ q s š Vj[ K k J y }>q [ Tp M WKb K =W b K =W b ‹ r‚VXk#P . c(t) = (ϕt )∗ Yp a1Hq inr V N Jl= 1 (c(h + t) − c(t)) h0 h
1 ((ϕt+h )∗ Y)p − ((ϕt )∗ Y)p h0 h . −t p
q=ϕ 1 (ϕt )∗ (ϕh∗ Y)ϕ−1 − (ϕ ) Y −1 = t ∗ ϕt p t p h0 h  
1 = (ϕt )∗ (LX Y)q = 0 = (ϕt )∗ (ϕh∗ Y)q − Yq h0 h c 0 (t) = WKb Vjk1kVj[—PW`P T Õ PN k c(t) = c(0) S PpXPpI[ H ky J Pky S×VXN y [ H bob~} JPN WPp Y `H W m P P N M PW = (ϕt )∗ Y = Y S×VXN y×P ϕt ¸¬¹Lº»¹Š¼ ÐnÁZÒ Æ Ï ö´É ¼¿ É ¼ ½ƒ¹L¾ ¼¿ Ç ÁZ¼¹ Æ ¾ ÄLÁZÐ Æ ¼ ÊÍ Ë º;ºD¹¿ Ç ¹Š¼ ÎR¹ ½ ¿ É ÄLÒI¹ Ñ@É ç Æ ½ É Ë ÄL¼;»¹ÄjÑÀ]¹ÁZ¼ ¿ ¹¾ ä¿ ¹Š¼Äj¹ŠÏ†¼ÒIÀ]Ð ¹ Ç »¹LÄ ¹ ½ Äj¹œÁEº

Æl¼Ñ Ï ¿ ÈÉ ºÄ » Ï º» å ¤ Ï ½]½ É Ä Ç ¹¾ ¿›¹ Ñ Á ½ ÉÇ » Æ ¼ ½ ÉBÉ Æ ¾ Æ Ñ Æ ö´É ½ É Ë Í Ë Ç ÈÉ èŽéZê]ë;ì‚í†ïæî ð ñÌî òRó „ [VXpXYV JLK (t , ., t ) [ H]H iLY K k P N J PRiLVjk1YynpjVji×WZV a:m VXk H W m Pk S ‰ H;a:m ‹ H KEa£H>N S X (p)‹ kVj[ T VXY KZa P VXWZVXW’boV a>=£š Vj[ K k J y }>q [ÚP p kVj[ÖPp i 7 Tétel. (Vektormezők szimultán kiegyenesı́tése) X1 , ., Xk ∈ M p [Xi , Xj ] = 0 (U, x) p Xi = ∂x∂ i i = 1.k 1 n ∂ ∂ti 0 i χ(a1 , ., an) = ϕ1a1 ϕkak (0, , 0, ak+1, , an) ZW Vx[ V N T Vjp VXN y J P 0 ∈ R [ Hq iLk m VXpjV JBVjN r‚Vjk =£F V a ՛V a:m VXpXp }q [ S ‰ H;a:m P M Vj[ JnH iLboVXp H ^ [ •@K V ‹ p PN i H N ՛VXWZV£Pp H k H yLPk)kB}WZW`P S‚VjN y zN a:m P `H W m P H [²[ H bob } J>PRN WZk1PR[ =’FIK kYVXk 1 ≤ i ≤ k‹ iŠP n χ∗  ∂ ∂ti a  ∂ (f ◦ χ) ∂ti a 1 = (f(χ(a1 , ., ai + h, , an)) − f(χ(a1 , , an)))

h0 h
1 = f(ϕ1a1 .ϕiai +h ϕkak (0, , 0, ak+1, an)) − f(χ(a1 , , an)) h0 h
1 = f(ϕ1a1 .ϕih ϕiai ϕkak (0, , 0, ak+1, an)) − f(χ(a1 , , an)) h0 h
1 = f(ϕih ϕ1a1 .ϕiai ϕkak (0, , 0, ak+1, an)) − f(χ(a1 , , an)) h0 h
1 f(ϕih χ(a1 , ., an)) − f(χ(a1 , , an)) = h0 h = (Xi f)χ(a) . f= WKb WKb WKb WKb WKb û T VjQ KlPN W K yLPkÖP 0‹ rPkIVXpjVj[ ynpjVji K k J χ∗  ∂ ∂ti 0  χ∗  ∂ ∂ti 0  = ∂ ∂t1 0 . ùP i>1 S PR[][ H i ∂ (f ◦ χ) ∂t1 0   i 1 f(χ(0, . h, , 0)) − f(χ(0, , 0)) = h0 h   i ∂f 1 = f(0, . h, , 0) − f(0, , 0) = i h0 h ∂x 0 f= WKb WKb ” pXVj[ynpXVji K k J χ Pp 9R‹ rPk K k M Vxi JPN WZ‰1P JH>N S1zN a:m P χ K k M Vji J>PN WZ‰1P J1H N P 0 ∈ R V a:m [ K y [ Hq iLk m VXpjV JBVjN r‚Vjk =†„ χ K k M VjiLpXV)Pœ[ zN MOPN k J [ HBH iLY K k P N J PRiLVXkYynpXVji J PRYÕØP = ∗ n 1.5 Disztibúciók, integrálsokaságok å ® Ñ Ð É ½ Æ Ð Æ ¾ º É ¼

ÒI¹Lº Æ À É ¿n¿ ký À]ÁZÒI¹Š¼ Ñ Á É ¾ Ð#À]ÁZÐ Ñ ¿æÁä Í ¾ ŊÁ É ¾ ¼ Æ ½ ¼¹ÎR¹ ÑXÑ4Í Ë ½ Æ Äx¹Š¼À1¹ Ç ¹Š¾ Ðl¿›Ï ÈÆ ÒoÁZ¼À1¹L¼ p ∈ M ¹ŠÐx¹¿ ¹Š¾ ¼ ∆ ⊂ T M ¹nº»  È  É X Ñ Ñ Æ ¾ ý À]ÁZÒI¹Š¼ Ñ Á É ¾ Ð ÆAÇ ¿ ¹L¾ ÄAå á å ®¹LÐxÑ ¹¿ Ò ¹Š¼ É ¼À þ Í ½ Ï ÈÉ º» ÆlÑ å X ÎR¹ ½ ¿ É ÄLÒI¹ тÉ+ç Æ ∆ý Æ ¾ ä Æ ¼#Î Æ ¼Ï È‚Æ ä Æ ¾ ÄLÒI¹ Ç » p ∈ M ¾ Æ X(p) ∈ ∆ å ® Ñ Ò É ¼À þ Í ½ Ï ÈÉ º» ÆAÑ X , ., X ÎR¹ ½ ¿ É ÄnÒI¹ Ñ@É ç ½ ÆlÑ U ⊂ M ¼;» ü ¾ Ç ¿ ÈÆAÇ Ò ÆlÑRÉ ¼ ºD¹L¼‚¹nÄ ÆA¾ Ç þ Æ ¾ ½ Æ ∆ý ¿Ï ÈÆ ä Æ ¾ ÄnÒI¹ Ç » p ∈ U ¹ŠÐx¹A¿ ¹Š¾ ¼ Æ X (p)ϘåEåhå Ï X (p) ºD¹L¼‚¹nÄ ÆA¾ Ç þ Æ˜Æ ∆ ý ¿‘å å ® ∆ ¹Lº» ÐnÁEÒ Æwà C É Ð Ñ ¿ ÆA¾ Ç » Í ¾ ã À]ÁZÐ Ñ ¿ ÄLÁä Í ¾ ŊÁ É ¾ Ï

ÈÆ ä Æ ¾ ÄLÒI¹ Ç » ö£É ¼¿ ½ É Ë ÄL¼»¹ ý Ñ ¹¿ ¹R¾ ä¹L¼IºD¹Š¼¹LÄ ÆA¾ ÇEÈ‚Æ ¿ É ¾ БÁZÒ Æoà C É Ð Ñ ¿ ÆA¾ Ç » Í ¾ ã ÎR¹ ½ ¿ É ÄLÒI¹ Ñ@É ç ½B½ ¹ Ç å å ® ∆ ký À]ÁZÒI¹L¼ Ñ Á É ¾ ÐÙÀBÁEÐ Ñ ¿æÁä Í ¾ ŊÁ É ¾ ÁZ¼¿›¹Lº;Ä ÆA¾ ÇEÈ‚Æ ¿ É ¾ Ï ÈÆ ÒoÁZ¼À¹Š¼ p ∈ M ¹ŠÐ ¹¿ ¹Š¾ ¼ Î Æ ¼ ÆAÑ Mý ¼¹ ½ ÉÇ » Æ ¼ pý Äj¹†Á ÇZÇ ¹ŠÐ Ñ ½ ¹XÀ É ç ký À]ÁZÒI¹Š¼ Ñ Á É ¾ Ð N =. N Ä ¹Š¾ Ð Ñ Ð É ½ Æ Ð Æ ¾ º Æ Ï ÈÉ º»IÒoÁZ¼À¹Š¼ q ∈ N ¹LÐx¹¿ ¹Š¾ ¼ Ó CÔ T N=∆ . ® ÒI¹Š¼¼»Áõ乊¼ ÆAÑ ÁE¼ ½ Ç Í ¾ Ñ Á É ¾ ¿ i : N Mý ÒI¹ ÇBþ ¹ ÇRÉË Çÿþ‚Í Ë ½ Ï Æ ½B½ É Ä à jã´þ ¹ Ç ¹Š¼¿ ¹Š¾ Ðx¹ 2 Definı́ció M p ∆p k p  p p  1 k  1 k p  k k  p q q  ‹ Y K boVXkp KAH N y£Y K ynp JLK r }N Q KAHƒN

K k J V a i PN WE‰P J1H>N S ‰1P)b K kYVXk p ∈ M VKXy‘V JBVXN k M KAH>PN kS HH;Wa:m m P k ‹ Y K boVjSOk1N p KAH N y K N y H [:Py P N a P VXN y JDH N W m Pk f : N M K k;ÕØVx[ JAzN M boboVxiLp ‰ b f VXy–b kYVXk q ∈ N VXynV VXk Megjegyzés. „ i∗ (Tp N) = ∆q . ∆k k p∈ f∗,q (Tq N) = ∆f(q) . ” [B[ H i7} a:m Pk K y7Pp f [ V N T V˜Pp M‹ k1Vx[ k‹ Y K boVXkp KAH N y–i VXN ynpjy H [:PRy P N a PoWZVXynp S P K iLV+P €˜ VXW JBV N J VXWZV K‚J VXW|ÕØVjy }q WZkVj[ = “ VXN WZY1P ?:=-• V a:m VXk f, g : R R [ V N J y K bcP n} q a;a:M†VXN k mS²VjN y J Vj[ K k J y }>q [ P R ‹ rPRkP K J K N KlH:N JlS P‰ H W ∆ Y ynp i r }Q 2 . ∆(x,y,z) = 3 1, 0, f(x, y)), (0, 1, g(x, y) ” pÙPœY K ynp JLK r }N Q KAH˜N T‚H k JLH ynPk PR[][ H i K k J V a i PN WZ‰1P J1H>N S ‰P~P ∂f ∂g = ∂y ∂x K k J V a i PN WZ‰1P J1H N y P N a;K VXW JBV N J VXW J VXWu՛VXy }1q W =
“ VXN WZY1P B8 =-• V a:m VXk f, g : R R [ V N J y K

cb P n } q a;a:M†VXN k mS²VjN y J Vj[ K k J y }>q [ P R ‹ rPRkP K J K N KlH:N JlS P‰ H W ∆ Y ynp i r }Q 3 . ∆(x,y,z) = 3 1, 0, f(x, y, z)), (0, 1, g(x, y, z) ” pÙPœY K ynp JLK r }N Q KAH˜N T‚H k JLH ynPk PR[][ H i K k J V a i PN WZ‰1P J1H>N S ‰P~P
∂f ∂f ∂g ∂g + g= + f ∂y ∂z ∂x ∂z K k J V a i PN WZ‰1P J1H N y P N a;K VXW JBV N J VXW J VXWu՛VXy }1q W = ¸¬¹Lº»¹Š¼ ¹Lº» ÐnÁZÒ ÆoÇ ¹ ½ƒ¹ ¾ ö ¹ Ñ ¹Š¾ ÐxÏ ¹Š¾ Ð Ç ¹Lº»¹Š¼ X ∈ X(M)Ï Y ∈ N ½ƒ¹¾ ¿ ÎR¹ ½ ¿ É ÄLÒI¹ Ñ‚É ç å ® Ñ ¿Ò É ¼‚À þ Í ½ Ï ÈÉ º» ÆAÑ X ¹Š¾ Ð Y ¹nº»Ò Æ ¾ Ðn¼ Æ ½ f ÒI¹Lº Ê ¹ ÇlÉ ç Á à|þ ¹ ÇRÉË Ç ¹Š¾ ÐAä>¹Š¼ Ï ÒoÁE¼‚À¹Š¼ p ∈ N ¹ŠÐx¹A¿ ¹Š¾ ¼ X ∼ Y ã ÈÆ 3 Definı́ció f :N M f f∗p Xp = Yf(p) . ® Ñ ö£É ¼¿ É Ð Æ ¼ Æ ½B½ É Ä fý ÒI¹nº Ê ¹ Ç ¹ ÇAÉnç þ ¹ ÆlÑ Y ý ¼ Æ ½ Ï ÈÆ Ò Í

¼À¹Š¼ 8 Állı́tás X š V a:m/}>q [ VjWÃ} a:m Pk K y S ‰ H;a:m (Yg) ◦ f = X(g ◦ f). X ∼f Y =°” [B[ H i-b K kY1Vjk ¹ŠÐ ¹¿ ¹Š¾ ¼ Ó s;Ô JBN (M) VXynV Vjk g ∈ C∞ (M) g ∈ C∞ . (Yg)f(p) = Yf(p) g = (f∗p Xp )g = Xp (g ◦ f) = X(g ◦ f)p . F V a; `H inY zN JnM P S ‰1P Ó s;Ô°KZa Pp S Pp T‚H k JnH yLPkÚPp J ՛VXWZVjk JlS ‰ H;a:m M Vj[ JLH iLboVXp H ^ [ f‹ inVXW PN Q KlH N rPRk M Pkk1PR[ = ¬¸ ¹Lº»¹Š¼ ÈÉ º»IÒoÁZ¼À¹Š¼ ¹Lº»ÐnÁZÒ Æ ÁEÒoÒI¹LÄ Ñ Á É ¾ Ï ¹Š¾ Ð Ç ¹Lº»¹Š¼ ¹ŠÐ ¹¿ ¹Š¾ ¼ 9 Állı́tás f:NM p∈M f∗p Xp = Yf(p) Y ∈ X(M) S PplPRp)P ÏÌÒI¹ Ç »DÄj¹–¿›¹ Çÿþ ¹LÐ Í Ë Ç Ï Yf(p) ∈ f∗p (Tp N). ¤ ½B½ É Äƒ¹Lº» ¹L¾ Äl¿›¹ Ç Ò Í ç ¹Š¼Î Æ ¼ ÉÇ » Æ ¼ X ∈ X(M)Ï ÈÉ º» X ∼ Y å mBK W MOPRN k#Pp X VjN i JBVxN [V K V a:m†VjN i J VXWEb }1^ Vjk#boV a ‰1P JPRN i H p H:JnJ PR[ =U„ p J

[VXWEW/QXyLP[Ir‚V ‹ W HP N J kBN }K k>[ S N‰ H;N a:m N PRp MzHN a:m J YVjH ô1k HN KlPN JBW NJ J X V a:mJ y K bcH P M Vj[ JnK¯H S iLboH;VXa:p m H ^ WZVXy‘K p =)„ p K boboVjiL‹ p KlH N [ W [ PW y+WZV z i PRy PRiŠP k1P [ p V VXWßPp b kYÕØPI[ ‰ b kYVXk p ∈ N k1Vx[ M Pk H W m PRk (U, x) VXN y†Pp f(p) ∈ M‹ kVj[ M Pk H W m PRk (V, y) [ HBH inY K k P N J PR[ Hq ink m VXpXV J V S ‰ H;a:m f  y ◦ f ◦ x−1 : (a1 , ., an) = (a1 , , an, 0, , 0) ” p J P˜[ ]H H Li Y K k P N J PRiLVjk1YynpjVji J ‰1PynpXk PN W M P ùP Y = Y i ∂y∂ i f∗ S PR[][ H i  X = Xi ∂x∂ i ∂ ∂x i  S P‰ H W p FIKEM VjWÃPRp Y n} q a;a:M†VjN k m Vj[Öy K b PRN [ S‚zN a:m Pp = ∂ ∂y i . f(p) K yßy K bcP = Xi (a1 , ., an) = Y i (a1 , , an, 0, , 0) i Xi = Y i ◦ f ¹Š¾ Ð X ∼ Y Ï Æ ½B½ É Ä [X , X ] ∼ [Y , Y ] å Æ Žè éZê]ë;ì‚í†ïæî ð ñÌî òRó • V a:m VXk g ∈ C (M) S PR[B[ H i 10 Állı́tás  X 1 ∼f Y1 2 f 2

1 2 f 1 2 ∞ ” kk1Vx[IPRW`P T Õ PRN k (Yi g) ◦ f = Xi (g ◦ f).
([Y1 , Y2 ]g) ◦ f = Y1 (Y2 g) − Y2 (Y1 g) ◦f

=X1 (Y2 g) ◦ f −X2 (Y1 g) ◦ f

=X1 (X2 (g ◦ f) −X2 (X1 g ◦ f) = [X1 , X2 ](g ◦ f), P K PRp PN WEW zN JPN y JUKZa Pp H W|ՑPœP {GPRN WZW zN J>PRN yUPW`P T Õ PN k T‚H k JLH ynPk Pp J ՛VXWZVXk JLK¯S ‰ H;a:m = [X , X ] ∼ [Y , Y ] 1 2 1.6 ® f 1 2 Frobenius tétel À]ÁZÐ Ñ ¿âÄLÁä Í ¾ ŊÁ É ¾ ¿-ÁE¼¿›¹Lº;Ä ÆA¾ ÇZÈÆ ¿ É ¾ ¼ Æ ½ ¼ ¹ÎR¹ 4Ñ Í Ë ¼ ½ Ï ÈÆ ÒoÁZ¼À¹Š¼ å 4 Definı́ció ∆ [X, Y] ∈ ∆ X, Y ∈ ∆ ¹ŠÐ ¹¿ ¹Š¾ ¼ ¬¸ ¹Lº»¹Š¼ ÆlÑ U ⊂ M ¼;» ü ¾ Ç ¿ ȂÆlÇ Ò ÆAÑRÉ ¼ Æ ∆ ÆAÑ X , ., X ÎR¹ ½ ¿ É ÄLÒI¹ Ñ@É ç ½ ÆA¾ Ç ¿ ÆAÇ ºD¹Š¼¹LÄ ÆA¾ Ç Î Æ å ¤ ½]½ É Ä Æ ∆ ö£É ¼¿ É Ð Æ ¼ Æ ½]½ É ÄÁE¼¿›¹Lº;Ä ÆA¾ ÇZÈÆ ¿ É ¾ Ï ÈÆ Î Æ ¼¼ Æ ½

ÉÇ » Æ ¼ C ÊÍ Ë º;º1Î ¹L¾ ¼»¹ ½ Ï ÈÉ º» [X , X ] = C X . èŽé`êÌë;ì‚í°ïæî ð ñÌî òRó „ VjW JBV N J VjW™k mDK W MUPN k ynp }q [By V N a VXy = „ p VXWZV a VXkY H ^ y V N a ‰VXp WZV a:m VXk =–„ boVXkk mBK r‚VXk X, Y ∈ ∆ S PR[][ H i X = f X VXN y Y = g X =-„ •’K V X, Y ∈ X(M) p PN i H N ՛VXW }>q [V J boV a:MDK pjy a´PN W M PœPp)PY H N Y K [ S ‰ H;a:m 11 Állı́tás 1 k l ij i j l ij l α α β β [X, Y] = [fα Xα , gβ Yβ ] = fα (Xα gβ )Yβ − gβ (Yβ fα )Xα + fα gβ [Xα , Yβ ] = JBP bN K Pp J b~} J P J ÕØP ‹ S ‰ H;=a:m Pp×VxiLVXYb VXN k m r‚VXkkV M PRk Pp Vxinr‚VXk#PRplPp)P ∆ rPk = fα (Xα gβ )Yβ − gβ (Yβ fα )Xα + fα gβ Clα,β Xl X1 , ., Xk P N W J PW a Vjk1Vxi PN W J ¸¬¹Lº»¹Š¼ ¹Lº» ¹ŠÐ ¹¿ ¹Š¾ ¼ Ç ¹A¾ ¿›¹ Ñ Á ½ ý » À]ÁZ¼ÒI¹Š¼ Ñ Á É ¾ ÐUÁE¼¿›¹Lº;ÁZÄ Ð ÆA¾ ÇZÈÆ ¿ ÄjÉ À]¾ ÁZÀ]¼ ÁZÐ Ñ ¿

¿âÄLÄjÁ¹Šä ¼Í À]¾ ŊРÁ É ¾ ¹Lå ÄxϤ ½B½ ºÉ » Ä-ÒoÁZ¼À1¹L¼ ÉÇ Æ ÇâÉ ½ÙÆA¾ Ç ½ ÉBÉ Æ ¾ Æ Ñ È‚É 12 Tétel. (Frobenius tétel) ∆ k (x, U) x∈M x(p) = 0 x(U) = (−ε, ε) × . × (−ε, ε), ¹Š¾ вÒoÁZ¼À¹Š¼ÚÄ É Ë º Ñ ü ¾ ¿›¹¿‘¿ a åhåEå a L¹ ¾ ÄA¿ ¹ ¾ ½ ¹ ½ Äj¹XÏ ÆAÈÉÇ |a | < εÏ i = k + 1.nÏ ÆAÑ k+1 n i .  Nak+1.an = q ∈ M | xk+1 (q) = ak+1 , , xn(q) = an ý ä¹ Ç Á@Ä ¹Š¾ Ð Ñ Ð É ½ Æ Ð Æ ¾ ºI¹Lº» ÁZ¼¿›¹nº;Ä ÆA¾ Ç Ð É ½ Æ Ð Æ ¾ º Æ~Æ ∆ À]ÁZÐ Ñ ¿âÄnÁä Í ¾ ÅLÁ É ¾ ¼ Æ ½ å èŽé`êÌë;ì‚í°ïæî ð ñÌî òRó š V a:m/}>q [ VXW S ‰ H;a:m V a:m [ HBH iLY K k P N J PRiLVXkYynpXVji J PWE[PWZbcPp M P†VjW VjN i J:}>q [ S ‰ H;a:m x(p) = 0 SOVXN y M ∂ ∂ , ., k 1 ∂x 0 ∂x ∆0 = ” pIV a:mfJ V J ynp H ^ WZV a Vjy [ HBH inY K k P N J PRiLVXkYynpXVxinr H ^ Wß[ KZK

kY}W M P S bcP՛YfV a:m VjW JLH W PN y J VjN y `H i a P JPN y J PWE[PWZbcPRp M PÖVXW VjN iL‰V JRH>^ = ” pXVx[ } JPN k J Vj[ K k J y }q [fP π : R R T i H‹ ՛Vj[Q KlH:N J PpÖVXWZy H ^ k [ H]H iLY K k P N JPRN iLP = FIKEM VXW π : ∆ πR K p H b H pjô1pXb~}y S²zN a:m P 0 V a:m [ Hq iLk m VjpXV JBKVjN rH VXk H K y K W m VXk=-WZVX” ynp S N PRJ plPp K Pp˜V~[ K Hq m iLk m VXpj‹V J r‚VXkøa:m†W V NN MÃJ H ^ q T^ H k JLH N []J rPRK k P p b pjô1pXb~}1y p Vji b k1YVXk W VXk q iŠP×V Vji VXWZb }VXk#W V VXp [ H W mπPRk : ∆ πR X (q), ., X (q) ∈ ∆ ‰ H;a:m   0 n ∗ ∗ 0 k 0 k 1 π∗ ” [B[ H iŽPp k Xi ∈ X(Rn ) k ∂ ∂xi q = ∂ ∂ti M Vj[ JLH inboVXp H ^ π‹ boV a; VXWZVXW H ^ ÕØV7P q . π(q) ∂ ∈ X(R)k ∂ti SUVXN y zN a:m   ∂ ∂ π∗ [Xi , Xj ]q = , = 0. ∂ti ∂tj IF KEM VjW [X , X ] ∈ ∆ VXN y+P π K p H b H inô1pXb~}1y S´zN a:m [X , X ] ≡ 0 =×FIKEM VXW†PÚ[ H b ‹ b~} JP N JLH i H [ VjN i

JBVjN [V)p VjN iL}y SÃzN a:moM Pk H W m PRkI[ HBH iLY K k P N J PRiLVXkYynpXVji S ‰ H;a:m i j q ∗ q Xi = i ∂ , ∂xi j i = 1, ., k ” [B[ H iIP i H;q a p zN J V J‘J a =h=E= a VjN i JBVjN [Vj[BinV[P T‚H:JnJ i VXN y‘pXy H [Py P N a P ∆ V a:m K k ‹ J V a i PN WZy H [Py P N a P˜WEVXynp S ‰ K ynpXVXkIb K kY1Vjk T‚H k J Õ PRN rPk Pp VjN i K k JH:^ J VjinV+P = X M Vj[ ‹ JLH inboVXp H ^ [cboV a; VXWEVXW Hw^ VjN i JBVxN [ V N M VXW a VXkVji PN W Jl= k+1 n ∂ ∂xi i ý 2. 2.1 Cauchy-Kowalewsky tétel Analitikus függvények ÊÌÍ Ë º;º1Î ¹Š¾ ¼»@¿ Æ ¼ lÆ ¼Ç Áõ¿æ¿æ¼ Á ½ Í Ðn¼ Î Æ ¼½ ¼ ¹» ÎR¹ ¼ jÑ Ñ´Í Ë ½ nÄ ¼Æ»¹ D¹A¿›¿¹XÆ Ï ÄA¿ É Ò Æ ¾Ð ¼» ÄÉ ä ¼Ï ¹ Ȃ¿ Æ ö£É Æ ½ Æ ÉÇ Æ ½ É Ë Ñ ÆAÈÉÇ É ÆAÊ þ ý ®Ñ ÒoÁZ¼À¹Š¼ È ¹¿ É ç Ï ÆlÑRÆAÑ 5 Definı́ció u(x1 , ., xn) p = (ξ1 , ., ξn) ∈ D u(x1 , ., xn) = ∞ X i1 .in ai1

.in (x1 − ξ1 )i1 · · · (xn − ξn )in :? =²„ boVXkk mDK r‚VXk M PWZP K W m VXk x = ξ VjN i JBVjN [Vx[BiLV+[ H k M Vji a VXky-P y H i S PR[][ H i™Pp JDN a N N J H M a S°N a:m J H M a´N Ó Ôx= |x | < xi V W PkcPry‘pXW } [ k Vji VXky VXyUV VXkWZV VXy‘VXkÚ[ k Vji PWP ?? 8B=²„ pŽPk1PW KhJLK [D}y n} q a;a:M†VjN k m k mDK W MUPN koP[ PRN iL‰ PN k m ynp H iOY KEg VxiLVXkQ KlPN WE‰P J1H N VjN yOP)YV ‹ i KEMUPN W J ÕØP K y–Pk1PW KEJLK [D}y =¬„ YVji KEMOPN W J ÕØPB Megjegyzés. i i i i   νj ∞ X  Y  ∂k u i1 in  (ij + µj ) ν1 νn =   ai1 +ν1 .in +ν1 x1 · · · xn ∂x1 .∂xn µ =1 i1 .in j tD=²„UJnJH N W S ‰ H;a:m M PRW`P K>M†V N a:J VXWZVXkynpXVxiUY KEg VjinVXkQ KlPN WZ‰1P JH>N S b V N a kVXbPk1PW KEJLK [B}1y = “´W’Pp  j=1.n 1 e − x2 , . f(x) = ha x 6= 0 n} q a;a:M†VXN k m kVXb PRkPRW KEJLK [B}yœP 0 [ Hq in0,k m VXpXhaV JDVjN r‚VXxk@= b 0 K k1YVXkfYVji

KEMOPN W J ÕØP 9 P 9R‹ rPk S Y1V)P L} q a;a:M†VXN k m bcP a PÙkVXb p VjN iL}y = <>=²„ kPRW KEJLK [B}y L} q a;a:M†VXN k m Pk1PW KEJLK [B}1y L} q a;a:M†VXN k m V K y²Pk1PW KEJLK [D}y = “´W¯ f(x) = ∞ XM Mr xn , = r − x n=0 rn PRkPRW KEJLK [B}y L} q a;a:M†VXN k m r‚Vƒr‚VX‰VXW m V J‘J VXy zN J Õ }>q [ÚPp |x| < r PRkPRW KEJLK [B}y L} q a;a:M†VXN k mBJlS PR[][ H iŽPœr K k H b KlPN W K y JBV N J VjWÃPRW`P T Õ PRN kIP u(x) = x1 + . + xn ∞ XM Mr = (x1 + . + xn )n r − (x1 + . + xn ) n=0 rn y H i²[ H k M Vji a VXky-b K k1YVXk = ∞ X i1 .in =0 M ri1 +···+in |x1 | + · · · |xn | < r (i1 + . + in )! i1 x1 · · · xinn i1 ! · · · i n ! jV N i JBVxN [BiLV = ý ¿nÎ Æ ¾ ¼»Ð É ÄA¿ Ò ÆXþ É Ä ÆA¾ ÇÿþxÆ Æ  È Æ Ï ÈÆ ¿›¹ Ç þ ¹ŠÐ Í Ë Ç Ï È‚É º» ® 6 Definı́ció (Majoráns sorok) P p = ai1 .in xi1 xin p<P ÈÆ ¿nÎ Æ ¾ ¼»>Ð É ÄjÏ ÆAѐÆAÑ P

= P αi1 .in xi1 xin ||ai1 .in || ≤ αi1 in 2.2 A Cauchy-Kowalewsky tétel másodrendű PDE-re š Vx[ K k J y }>q [ÚP Ó {Ô F(x , ., x , u, p , p , p , , p ) = 0 b PN y H Y>iLVXkY }^ “ e7”4‹æJl=´„ [VjpXYV JLK–VxN i JBVjN [ VXW JBV N J VXW J Pp S Pk1PW KEJnK [D}y VXW }1q WEV JnJ VXW zN i¯Õ›}>[ WZV  S = (x , ., x ) | f(x , , x ) = 0 , P‰ H W P ( ) 6= 0 S°VXN yßVXpƒP+[VXpjY1V JLK™VjN i JBVjN [[ H b T P JLK r K W K y£Pp Ó {Ô V a:m VXkWZV JnJ VXW S PpXP P VjÓ W }q WZV J VXk P boV a; VXWZVXW H ^ b PN y H Y>iLVXkY } ^ YVji KEMOPN W J PR[Ök K kQXynVXkVj[IVXWZWZVjk J b H kY PN y‘rPk PJRi K {Ô JnKV a:m Vjk1 `H WEV JnJ VXN W =´N š V a:m }>[ VXW JLHAMUPRN rr P N S ‰ H;a:m Pp S V a:m p T‚H k J Õ PRN rPk×P)[:PiŠPR[ ‹ Vxi ynp [D}y iLb PxÕ PRiŠP 1 n 1 1 n n 11 1 nn n n ∂f i ∂xi X ∂F ξi ξj 6= 0. ∂pij ij š V a:m¬}>q [ VXW S ‰ H;a:m SUVjN y²r‚V M VjpXV J Õ }>q [ PRp };N ÕU[ HBH

inY K k P N JDVjN [P J  1. Kezdeti érték transzformálása ∂f ∂xn y1 = x 1 , „ PQ H r K b P N J i K ž‚ 6= 0 . yn−1 = xn−1 , yn = f(x1 , ., xn)   1  ∂(y1 , ., yn)  0 = ∂(x1 , ., xn)  ∂f ∂x1 ··· ··· ∂f ∂x2 ··· ===  0 0     ∂f ∂xn P Jnb J K iLV a ‹ }W PRN i K y = N a:m Pp (y , .,JLKŽy N )JBr‚N V M `H;VjpXa V J ‰V a;J H ^ b K k KæJw=4š};N Վ[ NH]JOH JiLY K k K P N J J PRiLJVjHk1^ YynpjVja i = PRp S kVj[ÚPp y = 0 [VXpXYV Vji Vj[ boV VXWZVXWEk VX‰ P Vj[ k ‰V bcP P Pœ[VXpXYV JLK iLVXkYynpXVxi K W m Vjk1kVj[ =†„ [VXpXYV JnK7VjN i JDVjN [Vj[ÖboV a PY PN yLP S‹ VXk@   0 1 1 n n u = φ(x1 , ., xn−1) „ p S‹ VXk P “ e7”  pn = ψ(x1 , ., xn−1)   ∂φ ∂φ ∂2 φ ∂2 φ ∂ψ ∂ψ F x1 , ., xn−1, 0, φ, , , ., , ψ, , ., , , ., , pnn = 0. ∂x1 ∂xn−1 ∂x1 ∂x1 ∂xn−1 ∂xn−1 ∂x1 ∂xn−1 ý „ p~PRi a }1boVjk J }b˜rPRk#b K

kY1VjkPRY H:J‘JlS [ KEM†V N M V P V a:m Vjk1WEV J r H ^ W‚[ KZ V‘Õ›VXpj‰1V JH ^ VXp S PpXPp pnn =œ VXW J Vjynynp }>q [ S ‰ H;a:m P XV k JLK ∂F 6= 0. ∂pnn 2. Az egyenlet+kezdet érték kvázi-lineári rendszerré alakı́tása ± ¹LÒ ½ Æ Ä Æ ½ ¿¹LÄLÁZÐ Ñ ¿æÁ ½ Í Ð ½ ¹ Ñ À¹¿æÁ ¹L¾ ÄA¿ ¹ ¾ ½ ö Ä É ä Ç ¹Š¾ Ò Æ Äj¹XÀ Í ½ÙÆA¾ ÇZÈÆ ¿ É ¾ ¹Lº» ½ Î ÆA¾ Ñ Á ý Ç ÁZ¼¹ Æ ¾ ÄLÁZÐ ¼¹ŠÒ ý ½ Æ Ä Æ ½ ¿›¹nÄLÁZÐ Ñ ¿âÁ ½ Í Ð ½ ¹ Ñ À¹¿âÁ ¹L¾ ÄA¿ ¹ ¾ ½ ö Ä É ä Ç ¹Š¾ Ò Æ ¾ Ä Æ å èŽé`êÌë;ì‚í°ïæî ð ñÌî òRó š Vj[ K k J y }>q [ÖPp u, p , ., p ‹ iLVÙPœ[ H:q M V J [VXp H ^ “ e7”4‹æJ  Ó Ô ∂u =p ∂x Ó ?l9Ô ∂p =p ∂x Ó ?;?AÔ ∂p ∂p = i = 1.n − 1, k = 1n ∂x ∂x ! X ∂F X ∂F p Ó ?A8;Ô ∂p ∂F 1 F + =− p + p + ∂x ∂u ∂p ∂p ∂x „ ‰ H W’Ppƒ} JLH WZy H N V a:m VXkWZV J Pp Ó {Ô

x ynpjVji K k KEJLK YVji KhMUPN W PRN ynyLPW[P TJ }>[ S ‰ K ynpXVjk 13 Tétel. 1 nn n n i in n ik kn n i n n nn n ik ∂F ∂pnn xn n kn k=1 k k=1 ik n n n F xn n X ∂F X ∂F pik ∂F ∂F pnn + pn + pkn + + =0 ∂u ∂pk ∂pik ∂xn ∂pnn ∂xn k=1 k=1 „ “ 7e ”¬‹ ‰ VXp J PRi JLH p H N [VXpjY1V JLKƒVjN i JDVjN [Vj[IPp xn = 0 u = φ(x1 , ., xn−1), ∂φ pi = ∂xi pn = ψ(x1 , ., xn−1) y zN [ H k’ i<n ∂2 φ i≤k<n ∂xi ∂xk ∂ψ = i<n ∂xi : F(x1 , ., xn−1, 0, φ, , pnn) = 0 pik = pin Bm K W MOPN kÚPp)VxiLVXYV JLK Y KEg VxiLVXkQ KlPN WhiLVXkYynpXVjißV a:m boV a;H WEY PRN yLP)VXkkVj[ÖPÙiLVXkYynpXVjink1Vx[ P˜boV a;H WZY PN y P N J PY;ÕØP =4v™K pXy a4PRN Wu՛}[cboV a>S ‰ H;a:mÖ `H inY zN JnM P˜b K P˜‰1VjW m pXV Jl= pnn  δ i = p i − u xi , δij = pij − uxi xj . „ p Ó Ô b K P JnJ δ ” p VjN i J b K kYVXk ý n =0 SOVXN y zN a:m δnn = 0  Ó ?ltÔ ‹ iLV

δnn = pnn − uxn xn = (pn )xn − uxn xn = uxn xn − uxn xn = 0. „p i<n Ó ?X<ÌÔ ∂δnn ∂δin = (pin )xn − (uxi xn )xn = (pnn )xi − (uxn xn )xi = =0 ∂xn ∂xi xn = 0 VXp VjN i J [VXpXYV JLKƒVjN i JBVjN [Vx[IboVXWEWZV JnJ = ù PRy H kW H N Pk S ‰1P δin = pin − uxi xn = pin − (pn )xi = ψxi − ψxi = 0, δin ≡ 0 i<n ∂δ = (pi )xn − uxi xn = pin − uxi xn = δin = 0 ∂xn „ [VXpXYV JLK™VjN i JBVjN [Vj[ÖboVXWZWZV J‘J δ = p − u i i xi = φ xi − φ xi = 0 S’zN a:m δi ≡ 0 =£vIV N a‚}1q W ∂δik = (pik )xn − uxi xk xn = (pin )xk − (uxi xn )xk = (δin )xk = 0 ∂xn ob Vji J P δ ≡ 0‹¯J b PRN i×rVXS†W P N N Jna:J m }[ =ۄ [Vj=pXY„ V JLK#‹ VjN i JBVxN [H Vj[ H boVXWZWZV J‘J N δ S£N a:=m p ‹ − P u =φ −φ =0 z δ ≡0 δ [øN Pp k yLPkøkB}WZW PR[ z p[ ‰VXW m V JnJ VXy zN J ‰V JH ^ [ÚPp u boV a; VjWZVXW H ^ YVji KEMOPN W J ÕØP KhM PW = a:m Ó ?X<ÌÔ V a:m VXkWZV J r H ^ W in xi xk

xi xk ik xi xk ik ik  F(x1 , ., xn, u, ux1 , , uxn , ux1x1 , , uxnxn ) = 0 „ÛMDK pjy a´PN W J V a:m Vjk1WEV J iLVXkYynpXVxil 3. A kvázi-lineáris egyenletrendszer speciális alakúvá transzformálása X ∂qi = ∂xn Airl ∂qr + Bi ∂xl Ó ?A€;Ô i = 1.N „ [ VXpXYV JÖVjN i JBVjN [Vx[ Ó ?lCÔ q (x , .x ) = φ (x , x ) „ p PN W J PW PN k H yUVXynV J r‚VXk#PRp A L} q a;a ‰V J Pp x , ., x , q , , q ‹æJH ^ W = H H;a¬VXN kk1Vx[§PpXPpokB}WZW P N MUPIN PRW`PR[ zN J Pk K P ê ð é Az első transzformáció ~‰ b î ð î ð iLVXkYynpXVji V N Jl= N Õ MOPN W JLH p H N [ Q = q − φ =w” [B[ H i Pp x = 0 ‰ KET VjiLy zN [ H k P Q = 0 J VXWu՛VXy }q W = N a:m PW`P[ zN J ՛}>[ÚPœiLVXkYynpXVji JXS ‰ H;a:m PRp A VXN y B Második transzformáció  } boVjk1k mDK y V N a kV L} q a;a Õ Hq k#Pp x ‹¯JRH ^ W = ” ‰‰VXp `H iLb PRN W K yLPkcr‚V M VjpXV J Õ }>q [#Pp x ‹¯J b K k Jß n} q a;aH^ MOPN W JLH p H:N

JlS PplPpƒr‚V M V ‹ pjV J Õ }>q [—JLK7P N QJBN J L} q a;aHø^ MOPRN W JLH KEp T H:N J P N H = = 1 V a:m VXkWZV J V J P Q = 0 [VjpXYV Vji Vj[V Pp x = 0 ‰ VjiLy z [ k r=1.N l=1.n−1 i 1 n−1 irl i 1 1 n−1 n 1 N $#&% ( % %) %*#&%#&% + i i i n i irl n n ∂QN+1 ∂xn N+1 n N+1 i ý  VXW zN i¯Õ›}[™PkkP[ VXW JBV N J VXW V N JXS ‰ H;a:m Pk1PW KEJLK [B}1yboV a;H WZY PN yW V N J VXp K [ =4FIK k1YVXk J ‰P JnMUPN k m y H i‘rP zN i M PB 4. Megoldás létezésének bizonyı́tása qi = Airl = ∞ X cii xi |i|=0 ∞ X = i1 .in =0 Bi = cii1 .in xi11 · · · xinn ∞ X i j airl ij x q = |i|+|j|=0 ∞ X ∞ X i1 .in−1 =0 biij xi qj = j1 .jN−1 =0 ∞ X „ q ‹ kVj[ x ynpXVji K k JLK YVji KEMOPN W J Õ PRN iLP |i|+|j|=0 r i1 .in−1 =0 j1 .jN−1 =0 ∞ X il cri1 .in xi11 · · · xill −1 ” pXVj[ÖynpXVji K k J P Ó 8:9Ô V a:m Vjk1WEV J 1 .in =0 · · · xinn = ∞ X i1 .in =0

(il + 1)cii1.il+1 in xi11 · · · xinn (in + 1)cii1 .in+1 xi11 · · · xinn =  jN X j1 X X X in−1 i1 1 i1 N iN = airl x · · · x c x · · · c x i1 .in−1 j1 jN−1 1 i1 iN n−1 r,l X i n−1 bii1 .in−1 j1 jN xi11 · · · xn−1 X i · (il + 1)cri1 .il+1in xi11 · · · xinn X  j1 X jN iN c1i1 xi1 ··· cN x iN VXkY1Vjp M VÙP˜boV a; VXWZVjW H ^ ‰P nJ MUP N k mH [:P J PY H N Y K [cV a:m + , i n−1 bii1 .in−1 j1 jN xi11 · · · xn−1 qj11 · · · qjNN l ∂qr = ∂xl i X i j1 jN i1 n−1 airl i1 .in−1 j1 jN x1 · · · xn−1 q1 · · · qN X X V a:m Vjk1WEV Jl= FIKEM VXW)P Õ H r1r H WZY1PW H k PN WZW H N [ KZ V‘Õ›VXp VXN yØr‚VXkÛPp l ≤ n J VXWu՛VXy }q W S PRp Pp H k H y-‰P JnMUPN k mH [IV a:m¬} q JnJ ‰1P J1H;N K kP[ Hq y‘ynpXVX‰1Py H kW zN JPN y PRN r H N WÃPY H N Y K [ÚV a:m iLVj[B}>iLp zN M `H iLb~}W`PB (in + 1)cii1.in+1 xi11 · · · xinn = Pii1 .in (airl , bi , ckl1 ,,ln ) xi11 · · · xinn n

(in + 1)cii1 .in+1 = Pii1 in (airl , bi , ckl1,,ln ) 5. A megoldás konvergenciájának bizonyı́tása „ ob V a;H WEY PRN y)[ H k M Vxi a VXkQ KlPxN Õ PN k1PR[ør K p H k mzN JPN yLPcr K p H k mzN JPN y PN ‰ H p×boV a PY}k>[øV a:m H W m PRko“ e7”¬‹¯JlS boVXW m bcPÕ H i PN W|ՑP™PRp-VjiLVjY1V JLKEJlS£VXN y M PkcPk1PW KEJnK [D}y´boV a;H WZY PN yLP =4F P ‹ Õ H i PN ky²“ e7” V a:mÚH W m PRk X Ó ?As;Ô ∂σ ∂σ = A +B i = 1.N 1.lépés: stratégia, majoráns PDE fogalma i ∂xn ∗ irl r=1.N l=1.n−1 r ∂xl ∗ i ý V a:m Vjk1WEV J iLVXkYynpXVxi+P σ = 0 Pp x y zN [ H k S PR‰ H W A < A VjN y VXN y 0 ≤ |b | < b Ô S>JLHAMUPRN rr P N M PRk 0 ≤ |a | < a i ijk ∗ijk i σi = n ∗i X ∗ ijk ijk Bi < B∗i S PplPRp γii1 .in xi1 xin a:P k1m PW KEJLK [B}1SÃy-N a:bom V a;H WZY PN yLP =ÙFIKEM VXW£PcboV a;H WZY PN yŽy H i PN k VXWZW V N T4H ^ P ‚T H W K k H b H [ÖboV a V ‹ VXpjk1Vx[ z (in +

1)γii1 .in+1 = Pii1 in (a∗irl , b∗i , γkl1 ,,ln ) ” [B[ H i cii1 .in+1 = Pii1 in (a, b, ck) ≤ Pii1 in (|a|, |b|, |ck |) ≤ Pii1 in (a∗ , b∗ , γk ) = γii1 in+1 , P pXPpwPpøPk1PW KEJLK [D}ycboV a;H WZY PN ycbcPÕ H i PN WZk KŽ `H;a ՑP PpøVjinVXYV JLK V :a m jV k1WEV J iLVXkYynpXVxi boV a;H WEY PRN y P N JlSzN a:m VXp+} J1H N rr K@K yU[ H k M Vji a VXkyŽWZVXynp = v#PRN W`Pynp J ‰P J1H N } N a:m V a:m k1P a:m M VXN y–[ K QXy K r S ‰ H;a:m Pp 2. lépés: a majoráns PDE Airl VXN y B ‹ [V J bcPÕ H i PN WuÕØP+P i Mr „ boV a; VXWZVjW H ^ “ e7”  P‰ H W’Pp r − (x1 + · · · + xn−1 + q1 + · · · + qN ) ∂q Mr = ∂xn r − (x1 + · · · + xn−1 + q1 + · · · + qN ) xn = 0 . X ∂qr ∂xl +1 y zN [D‰ H p J PRi JLH p H N [VXpjY1V JLKƒVjN i JDVjN [Vj[ q = 0 S i = 1.N =4„ r,l i qi = Q(x1 + . + xn−1 , xn ) = Q(X, xn ), r‚V M VXpXV JBVjN y V N M VXVXW¬P “ e7”  ∂Q Mr = ∂xn r − X − NQ Pœ[VXpXYV JLK’

VXW JBV N J VXW Q(x, 0) = 0 PRp  i = 1.N ∂Q N(n − 1) +1 ∂X  boVXWEWZV JnJl=/„ boV a;H WEY PRN yX xn = 0 p (r − X)2 − 2Nn Mr xn r−X Q(X, xn ) = − nN nN Pk1PW KEJLK [B}1y°boV a;H WZY PN yßPp = (X, xn ) V a:m [ K ßy [ Hq iLk m VXpXV JBVjN r‚VXk = ! 3. Formális integrálhatóság š ‰V ‡ PRi J Pk ‹›ŒœPq ‰WZVji š ‰V H iLVXb a Vjk1VxiŠPW K pjVXy J ‰V ‡ PR}1QŠ‰ m]‹›ŒŽHA PWZVXyØ[:P š ‰1V H iLVXb S K k J ‰VynVXkynV J ‰P J)J ‰V kB}b~rVji H; V B}1P JLKZH ky K y+k H:J k1VjQXVXynyLPi K W m V ]}PRW JLHIJ ‰1V k]}1b˜r‚Vji H; }k>[Dk HA k }1kQ JnKZH ky S PkY J ‰1P J k H k1V H; ‚J ‰V M Pi K PRr1WZVXy T W`P m P T PRi JLKh‹ QX}W`P„ i–i H  WZV = M K SJ T JLK `H H; £J ‡ m̋›ŒŽH y V~‰P V~yLP Y ‰V PRi QX}W`PRi iLb ‰V P}QŠ‰ PRWZVXy‘[PÚy m y J Vjb K b T W K VXy J ‰P J† `H inbcPWBy H WE} JnKZH ky4PW  P m y4V ž K y JX=´š ‰ K y K y´k H:J4J ‰VßQlPynV `H i†P a

VXkVjiŠPW y m y J VXb# H r1y J iL}Q JnKZH ky QlPRk PRi K ynVÖPkY—r‚VÖVxž T W K Q KEJ W m Q H b T } J VjY = ù HA V M Vji SUKE `H iLbcPRW K k J V a iŠPRr K W KEJmÖH; PkPk1PW m]JLK Q)y m y J VXb K y™VXkyn}>iLVXY SJ ‰V `H iLbcPWÃy H WZ} JLKZH ky Q H k M Vji a V S Py K k J ‰1V ‡ P}Q ‰ m]‹Œ™HA PWZVjy‘[:P~QlPynV = š ‰1Vßy KhJ }1P JLKZH k K y´y K b K W`PRi JLH™J ‰V H kV `H i†y m y J VXboy H; W K k1VXPRi´PW a Vxr1iLP K QOV ]}1P JLKh‹ H kyßPkY J ‰V7Pk1PW H;a:mK yOk H:JßH kW m× `H iLbcPRW = P+W K kVlPRiOy m y J VXb K y K k J ‰V ‡ iŠPoVji `H iLb Ó K¯= V =´J ‰V)k]}1b˜r‚Vji H; V B}1P JLKZH ky K y J ‰V)ynPoV+PRy J ‰VƒkB}b˜r‚Vji H; }k>[Dk H k M PRi K PrWEVXyUPkY J ‰V™bcP J i K ž K yOiLV a }1WZPRi Ô£J ‰VXkÖPÙy H WZ} JLKEH koVxž K y J y–PkY KhJßK yU}k K B}V = >H iƒP a VXkVjiŠPRW¬y m y J VXb SH r1y J in}1Q JLKZH ky™P TT VlPil J ‰1V m QlPk r‚V H r J P K kVXY r m Q

H b ‹ T } JLK k a˜J ‰V ;QŠ‰1PRiŠPQ J Vji K y JnK Q)YV J VjiLb K k1Pk J y Ó  ‰ K QŠ‰#P H }k J y JLHa;KEMBK k a PWEW J ‰1V W K k1VXPRi†inVXW`P JLKEH k1y£r‚V J VXVXk J ‰VŽV ]}1P JLKZH ky Ôx= ‰VXk J ‰VXynV™Q H b T P JLK r K W KEJm Q H kY KhJLKh‹ H kyUPRinVŽyLP JLK y‘ô1VXY SÌJ ‰VŽy m y J Vjb QlPkor‚V T } JUK k J ‰1V ‡ iŠPRboVji `H inb ŽKEJ ‰oy H boV iLVjV T PRiŠPRboV J VjiLy =´š ‰VXkÖP T PRiŠPoV J i K pXVXY PRb K W m H; y H WZ} JLKZH kyOQlPkcr‚V H r J P K k1VjY S PkY J ‰VƒkB}b˜r‚Vji H; ÃJ ‰V T PRiŠPoV J VjiLyßYV T VXkYy H k J ‰VƒiŠPRk[ H; ’J ‰V)y m y J VXb =†>H i J ‰1V y m T y J VXboy H; H “ H eƒ” H J H;‰a:V7m y KEJ H }1P JLKEH T‚k H K yUy KJLb H K J WZPRi =¬ š ‰JÙV J H r1y J+J iLJ }Q JLKEH k1yßPi K J y K k K aÙJL K i H b J J ‰1V ‹ û Vjk1QjVji˜Q ‰ b W Q ininVXy kY ‰V PQ ‰1P ‰1VQ ‰1Ó PRiLPQ Vji y Q YV Vji b K k1Pk J y߉1P M V JnH×M Pk K y‘‰ PkY J

‰1V+kB}b~rVji H; ¬T PRiŠPRboV J VjiLy  ‰ K QŠ‰ K k J ‰ K y-QlPy‘V PRinV+PRi‘r KhJ iŠPRi mc }kQ JLKEH k1y Ô QXPkIr‚VƒVxž T W K Q KEJ W m Q H b T } J VXY = • V J }1y–Q H ky K YVji J ‰Vƒy m y J VXb H; ’T Pi JLK PW‚Y KEg VjiLVXk JLK PW‚V B}1P JLKZH ky Ó ?l{Ô F (x , z , z , ., z )=0  ‰1VxiLV ν = 1.p PkY / / /  / / 10 0 32 / / ν zµa = ª È ¹-Ѝ»Ðl¿›¹LÒ ¼ Í ÒIä¹LÄnÐ ∂zµ , ∂xα a µ µ a µ a1.ak zµa1 ,.,ak = ., ∂k z µ . ∂xa1 . ∂xak ÁZÐ Ýë ñ #í éæì ð ñ ZÁ ¼ Æ ¼ ¹ŠÁEº È ä É Ä ÈÉBÉ À ÉÊ x Ï1Á ÊÃÊæÉ Ä Æ ;¼ »cÄj¹ AÆ Ç ÎR¹LÄLÁ Ê »ÁZ¼>º A , A , ., A ∈R Ó ?lÔ F (x , A , ., A ) = 0, ¿ È ¹LÄx¹-¹ ÁZÐl¿æÐ Æ ¼¹ŠÁEº È ä É Ä ÈÉBÉ À U ÉÊ x Æ ¼À Æ Ð ÉÇ Í ¿æÁ É ¼ z (x) À¹ 1¼¹XÀ É ¼ U Ð Í Å È 7 Definı́ció µ µ a 54 "6 87 696 %;:8) =< 6 % µ a1 .ak ν  0 o

µ µ a1.ak 0 µ
¿ ÈÆ ¿   zµ (xo ) = Aµ ,      zµa (xo ) = Aµa , åå š ‰ VoynV J˜H; UJ ‰V (A , A , ., A ) yLP JnK y ZmDK k a Ó ?lÔ)K yÙQlPRWZWZVXYfP k ð ë ë ñ ò ë ð é‘ë;ì Ó H i éæì‚é ð é ñ ñÌð‘ñ P J x Ôx=£š ‰1V-ynV JßH; PWZW J ‰V kJ ‰ H iLYVji `H iLbcPW y H WE} JnKZH kyßP J x K yßk H:J VXY R = k KZH: J ‰Vji ßH iLYy SJ ‰V~“ e7” Ó ?l{Ô-K y Ó W H QXPWZW m>ԖK k J V a iŠPRr1WZV K køPokV KZa ‰Ìr H iL‰ HBH Y H; x ÊæÉ Äc¹ÎR¹Lě» F ∈ ÎRR¹LÄnÁ »ÁZ¼>¿ Ⱥ׹L¿ Äx¹Ö¹²¹ À]ÁEÁZ’Ðl¿æ¹LÐÄj¹ŠÆ¼¿æ¼Á ¹ŠÁhº ¹ È ä É Ä ¿æÈÁ É]¼ É À U Ï]ÉÐ Ê Åx ¿ Æ ¼À ¿ Ê È ÆAÇ Í Æ É à ã Í È ÈÆ f ∈ C (U, R ) µ A       )*B 6 µ a C6ED zµa1 .ak (xo ) = Aµa1 ak 6F ) %) ?>=@ µ a1.ak 0 0 k,x0  0 0 ∞  k,x0 G p 54 0 (jk f)x0 = F0 . š/H J mcJ K J a K

KEJmcš m H K =O„UJ JŽH H]H `H iIP `yH in}bcY PW™y ‰H V WZ} JLk KZH V k iŠzPRr =W (z , P .,W zi²yn)Vji Kæ= VVjy-= QlPPk `H riLV+bc}PynW™VXYynVji K VXyIôynPiLyJLK y ZmBkK V+k afW J ‰1[DVy V ]}1P JLKZH k 1 m / z= X Aα (x − xo )α Ó  ‰VjiLV α K y™P×b~}W JLKh‹¯K kYVxžÖPkY A = (D z)(x )Ôx= “´} JnJLK k a z K k JLH×J ‰VÙV ]}P ‹ JLKEH k S VÙQlPRkQ H b T } J V A r m y H W MBK k a PW a VjriŠP K Q)y m y J VXboy =²š ‰VXk  ÙV QlPRkW HBH [ P JßJ ‰V)Q H k M Vji a VXkQXV H; /J ‰1V `H iLbcPW‚y H WZ} JLKZH k = α α α! α / o α ¸¬¹¿ Í ™Ð Å É ¼ÐnÁâÀ¹LÄ+¿ È ¹ƒ¹ Í Æ æ¿ Á É ¼ 3 Példa G ∂2 z ∂2 z + = 0. ∂x2 ∂y2 2 KEJ ‰ H r MBKZH }y²k H:J P JnKZH ky EK J lQ PkIr‚V  i KEJnJ VXk@ z11 + z22 = 0. š ‰V)Q H V ×Q K VXk J y ;H ÃJ 1‰ V ` H Li bcPW‚y‘Vji K VXy M Vji KZ ZmBK k a J ‰VƒV B}1P JnKZH kIb~}y J yLP LJ K y Zm  H / A11 + A22 = 0. š P[ K k a~

`H iŽVxž>P T WZV˜ A = 1, A = −1 PRk1YIQŠ‰ H]H y K k a  Vƒ‰1P M V+P 2 ‹õH iLYVji-y H WZ} JLKZH k@ (0, 0, 1, 0, −1) = 11 th 22 A1 = A2 = A12 = 0 S U  V  Pk JÙJLH Q H b T } J V J ‰1V H:J ‰VjiœQ H V ×Q K VXk J y ;H OJ 1‰ V `H inbcPW†V ž T Pky KZH k S  V×kVXVjY JnH Y1Vxi KEM V J ‰V V B}1P JLKEH k PkYfy J }Y mJ ‰V ð Xò ð ë ë;ì ñÌð éØë;ì H; J ‰Vƒy m y J VXb#  H  / ?> %JIK) ML ) C6 :   z11 + z22 = 0 z111 + z122 = 0   z 211 + z222 = 0. š ‰V)k]}1b˜r‚Vjiny A , A , A b~}y J yLP JLK y ZmcJ ‰V)y m y J VXb# i ij ijk    A11 + A22 = 0 A111 + A122 = 0   A 211 + A222 = 0.   V J PR[V J ‰V A PkY A PyOPRr HM V S] VŽQlPkor1} K WZYoP 3 ‹õH iLYVjiOy H WZ} JLKZH k  ‰ K QŠ‰ Vxž J VXkYy J ‰V 8 kY H iLYVji-y H WZ} JLKZH kÖPWEiLVXPY mc `H }1kY =†H i²Vxž>P T WZV:  i ij rd A1 = 0, A2 = 0, A11 = 1, A12 = 0, A22 = −1 PkY A111 = 0,

A112 = 0, A122 = 0, A222 = 0. ‰ VXk  V–QlPRini m H } J£J ‰ K y H:T VjiŠP JLKZH k PƒynVjQ H kY JLK boV S: V H r J P K k P7y m y J VXb H; ô M V V ]}1K P JLKZH ky = KZ Z ‚J J ‰ K yOJLH y m y J Vjb K yOQ H ky K y J VXk J°J ‰VXk  VŽô1kY J ‰V 2kY H iLYVjiUy H WZ} JLKEH k1y ‰ Q ‰IPRiLVƒW VjY 4 ‹õH iLYVji²y H WZ} JLKEH k1y S V J Q =h=E= >H iLbcPW K k J V a iLPRr K W KEJm P J x boVlPky J ‰1P J ñ ì‚í kJ ‰ H iLYVji `H iLbcPRWy H WZ} JLKZH kÚP J KZ ZJ K JLH Pk K kô1k KEJ V H inY1Vxi-y H WE} JnKZH k = x QlPRk r‚V)W VXY k „ y K yOyn‰ H kor mJ ‰1V7Pr HM V-VxžP T WZV S]K k H inY1Vxi JLHÙT i HAM V J ‰1P J°J ‰V kJ ‰ H iLYVji y H WE} JnKZH kyUQlPkIr‚V)W KZ ZJ VjY K k K k>ôk KEJ V H iLYVjiŽy H WE} JnKZH ky Ó K¯= V =´J ‰1VxiLV+Vxž K y J y²P `H iLbcPW y H K WE} a JnKZH k ÔxK SD VÙKhJ kVXVXY JLH y J }H;Y mÚJ ‰VÙHAQ  H ky K y J VXkQ mIH; JLPKEH k P=W a VjrH iŠP a K Q)ym m y T J Vjb KQ H a)k J

J P Kh‹ k k PRk k>ô1k VßkB}b~rVji }1k>[Bk ky°PkYoV B}1P k1y } ‰W y VlP[ k ‰1V ‡ PRi J PRk ‹ŒœPq ‰WZVji š ‰V H iLVjb yLP m y J ‰P J-KZ ¬J ‰1V)y m y J VXb K y K k MH WZ} JLKhM V ×PkY iLV a } ‹ W`Pi Ó J ‰VXynV)k H:JnKZH ky ŽK WZWr‚V K k J i H Y}1QjVXY K k J ‰V)kVxž J y‘VXQ JLKZH ky ÔOH kV H k1W m k1VjVXYy JLH y J }Y mcJ ‰VƒôiLy J–T i H W H k a P JLKEH k’ ë;ì ò é ð k ð ë ò í òjð 2 /  th 0 0 , / 0 0
0 N O %) ?> % ?> ) %) %B Fν (x, zµ , ., zµa1ak ) = 0, é é ~ò ò ë ò ð ë ðë é ñ ë ì éØë;ì í k ì ë í ò ë ò ð ð ð ò jò ð PQ> ;7 > ZX %&RT[ ) %) 0 D L=L % % RT)%;: DU6 < S 3 % X  C6ED % ) % ) th @ )  C6ED [ ?> % %B ?> % ñ éØë;ì ó ðé ò ν = 1, ., p ðì ò éí Xò ð æé ì é ò âé ì Dë ë ñ ñ í#ðéâì ð ñ ñ (k+1) ó )VR 1W A 7 A ?> % < % 6 A 6Z6 )*B %B % R YX C6ED @ %;:) =< 6 % th

4 Példa (A Cauchy-Kowalewsky tı́pusú másodrendű PDE) X ∂qi = ∂xn Airl ∂qr + Bi ∂xl i = 1.N, ÆAÈɐǬÆAÑ A ¹L¾ Ð B ÒI¹L¼1¼;»ÁZÐ ¹L¾ ºD¹ ½ ¼¹ŠÒ ÊÍ Ë º;º¼¹ ½ Æ x ý ¿ É ç Ç å „ p N = 1 VXN y n = 3 VXynV JBVXN k PpƒV a:m VXkWZV J r=1.N Ó 8:9Ô l=1.n−1 irl i n Ó 8D?AÔ ∂q ∂q ∂q = A1 + A2 +B ∂x3 ∂x1 ∂x2 PpXPp PWZPR[ } N =4„ [VXpXYV JLKƒVjN i JBVjN [ÖPp P‰ H W’P Ó 8;8;Ô T‚H k J rPkIWZVj‰1V J P q3 = A 1 q1 + A 2 q2 + B x = (x1 , x2 , x3 ) j1 (q) = (x, q, q1 , q2 , A1 q1 + A2 q2 + B), q, q1 , q2 J V J ynp H ^ WZV a VXy-ynp PN b H [ =´„ Ó 8;8;Ô V a:m VXkWZV J–T i H W H;a´PN W PRN yLPÙP Ó 8:tÔ Ó 8<ÌÔ q13 = A1,1 q1 + A1 q11 + A2,1 q2 + A2 q12 + B1 q23 = A1,2 q1 + A1 q12 + A2,2 q2 + A2 q22 + B2 V a:m Vjk1WEVÓ J Vj[][VXW/V a/VjN ynp zN JLK [ K Ó 8;8;ԍ‹æJl=O” kkVj[ V)b PN y H Y>iLVjk1Y }Ö^ boV a;H WZY PN yLP S P K V a:m]‹ r‚Vjk#P 8;€;ÔOT i H W H k

a4PRN W PN yLPÙP Ó 8;€;Ô j (q) = (x, q, q , q , q , q , q , q , q , q , q ), PÓ ‰ H WUP q,Ó q , q , q , q , q J V J ynp H ^ WZV a Vjyœynp PN b H [ S P q , q , q , q T VXY KZa P 8;8;ԜVXN yßP 8<ÌÔ PN W J PWboV a ‰1P J>PRN i H p H:JnJX=´FIK k J Pp™W P N J ‰P J1H N Pp-V a:m VXkWZV J inVXkY1y‘pXVjinr H ^ W S Pp x ynpXVji K k KhJLK YVji KhMUPN W J PR[BiŠP MH k1P J [ H pXk1PR["PpwV a:m VXkWZV J Vj[ = ”£a:m ynp K k J#T i H‹ W H k a´PN W PN yLPR[ H i²PpœPRW`PQXy H k N m PRra:rm iLVXkYJ } ^ YVji KEMOPN W J PR[ÖN PpK }N ÕßV a:Jm VXkWZV J Vj[]r‚K VXk kVXa>b = N ՛a:V m ‹ WZVjk1kVj[+boV a>S P k Y1PRiLPRr };ÕÃV VXkWZV ‰VXp k Y1PRiŠPRr };Õ ynboVxiLV WZVjkŽÕØVjWZVXk [)boV VXpjVj[IkVXb WZVX‰V J k1Vx[ÖVjWZWZVXk J b H kY PRN y H yLPR[ S PpXpXPpÙPœiLVXkYynpXVji K k J V a i PN WZ‰1P JH N WEVXynp = q33 = A1,3 q1 + A1 q13 + A2,3 q2 + A2 q23 + B3 1 1 2 1 11 12 2 3 11 12 13 22 22 23 3 33 13 23 33 3 

¸¬¹¿ Í ™Ð Å É ¼ÐnÁâÀ¹LÄ+¿ È ¹-Л»Ðl¿¹ŠÒ 5 Példa  ∂z   = f(x1 , x2 ),  ∂x1 ∂z    = g(x1 , x2 ). ∂x2 „ ôiLy J²H iLYVji K k KhJLK PWQ H kY KhJLKZH kÖP J ⇔ x = (x1 , x2 )  z1 = f(x), z2 = g(x). QlPkIrV a;KEM VjkIr m j1 z = (x, z, f(x), g(x)). Ó 8:CÔ Ó 8;s;Ô š ‰V T i H W H k a P J VXYÖy m y J Vjb K y z11 = f1 (x), z12 = f2 (x), z21 = g1 (x), z22 = g2 (x). ” pXVj[øynpXVji K k J P VXk JLK j z [VXpXYV JnKVjN i JBVxN [økVXbŸb K k1YVXkVXynV J r‚VXk T i H W H k a´PN WZ‰1P JH>N S PpXPp+P˜boV a;H WZY‰1P J1H N y P N a k1PR[×V a:m ynp }q [By V N a VXy VXW JBV N J VXWZV S ‰ H;a:m P Ó 8:{Ô ∂f ∂g = ∂x ∂x K k J V a i PRN WZ‰1P J1H N y P N a;K’ VXW JBV N J VXW J VXWu՛VXy }q WuÕ Hq k S²VXN yƒPp˜VXYY KZa;K/J PRkB}WZb PN k m P K k>[Br H N W K ynboVji JlS ‰ H;a:m VXpÙP VXW JDV N J VXW’VXWZV a VXkY H ^ K y = 1 2 · L¹ Äj¹LÐnÐ Í Ë ½ ÆAÑ ¿

Æ 6 Példa z(x) = z(x1 , x2 , x3 , x4 ) 2 ÊÍ Ë º;º1Î ¹Š¾ ¼»@¿›Ï Æ ÒoÁEÄj¹ Ó 8:Ô ù PRy H kW H N Pø‰VXW m pXV JXS b K k J Pp#VXW H ^ p Hø^ TßVXN WZY PRN rPk S PplPpIk1Vjb T i H W H k a´PN WZ‰1P J1H N V a:m [VXpjY1V JLK VxN i JBVjN [ S QXyLPR[÷PR[][ H i S ‰1P f = g = ” pPp K k J V a i PN WZ‰1P J1H N y P N a;K- VXW JDV N J VXW MBK ynp H k J QXynPR[ÖP~b PN y H Y K [ T i H W H k a´PN W JOMDK pjy a´PN W`P J PR[ H i²PY H N Y K [ = „Þ€wVjN yŽP s˜TUVXN WZY PRN []rPk J PRW PN W JUK k J V a i PN WZ‰1P J1H N y P N a;K VXW JBV N J VXWZVx[B[VXW Megjegyzés. J PRW PN WE[ H pX}k>[ S PoVXW m PpUV a:m Vjk1WEV J Vj[]r‚VXk J PW PN WZ‰1P JH N V a:m/} q J‘J ‰1P J1H™N L} q a;a:M†VXN k m Vj[BinV MH‹ k1P J [ H p K [ = „ boVXkk mBa;K r‚VX kH™^ VXpXJ Vj[§P LH } q a;a:N M†a;VXH N k m Vj[ Ó Ó P TßTßN VXN WZY N PRN []rPRk§PÓ p 8:CfÔ¬VjNK yœP J gM Ô kÓ 8:VXb Ô iLVjk1YVXWh[VXpXkVj[ P™boV V VXW }W`P՛Y ky P [][:PW

P VXWZY PR[BrPk P WZWZV V V V a:H m VjJk1WEV J Vj[ a:mJ VjWu՛VXy J }q W VXN ynV ÔxS PRJ [B[ NH i£P-J1inH N VXka:Y1m¬y‘pXq VjJni†J k1VjJ1b HœN nK kq Ja;V a:a M†i NPN WZm‰1P J1H>N J =’„ boN VXJLkK k mDK r‚Vjk a;MD Kh‹‹ ynp k Pp7V VXkWZV Vj[Br‚Vjk PW PWZ‰1P V } ‰1P } VXk Vj[ VjWu՛VXy z [oPœboV V WZVjW H™^ K k J V a i PN WZ‰1P JH N y P N a;K VXW JBV N J VXWZVj[V JlS PR[B[ H i£P-iLVjk1YynpjVji T i H W H k a4PRN W PN y P N M PW };N Õ/V a:m Vjk ‹ WZV J Vj[V J QXyLPR[IPœbcP a PyLPr1rÚiLVXkY } ^ YVji KhMUPN W J PR[BiŠP+[P T }k>[ = 7 Példa · ¹LÄj¹LÐnÐ Í Ë ½ ÆAÑ ¿ ÆAÑ z = z(x , x ) ÊÍ Ë º;º1Î ¹L¾ ¼»’¿Ï Æ ÒoÁEÄj¹ Æ À É ¿n¿ A ¹Š¾ Ð A ÒI¹ ÇZÇ ¹A¿n¿ ¿›¹ Ç þ ¹ŠÐ Í Ë Ç¬Æ Ó t;9Ô z − A z = 0, z − A z = 0, . ” pXVXk#V a:m Vjk1WEV J Vj[Bk1Vx[ x ‹ rPk M V JnJ VjWZy H ^ iLVXkY } ^ boV a;H WEY PRN yLP K  Ó t>?AÔ R = {(x , q, q , q ) | q = A q, q = A q, } PpXPpƒP R ‹ r‚VXkÖP

q ∈ R ynp PN by‘plPRrPY H k MOPN W`Pynp J ‰1P J1H>N =4š Vx[ K k J y }>q [cP Ó t;9Ô VXWEy H ^ T i H W H k a´PN W J Õ P N J  z12 = f(x1 , x2 , x3 , x4 ), 34 1 1 z34 = g(x1 , x2 , x3 , x4 ), 12 2 1 1 2 2 2 0 x0 ,1 0 1 2 1 1 2 2 x0 ,1 z1 − A1 z = 0, z2 − A2 z = 0, z11 − A1,1 z − A1 z1 = 0, z21 − A2,1 z − A2 z1 = 0, z12 − A1,2 z − A1 z2 = 0, z22 − A2,2 z − A2 z2 = 0, Ó t8;Ô „ p x ‹ rPk M V J‘J b PN y H Y>iLVXkY }Ö^ boV a;H WEY PRN yLP K P Ó t;tÔ 0 J2,x0 = {(x0 , q, q1 , q2 , q11 , q12 , q22 )} H W m PRkÖi VXN y‘pX‰1PWZbcPplP S P‰ H W’P Ó t8;Ô V a:m Vjk1WEV J Vj[ b K k J W K kV PRN i K yUV a:m VXkWZV J Vj[ q S q S VXN y q K ynboVjiLV J WZ VXkVj[]r‚VXk N N J VXN W|ÕØN VjM y }q WZkVj[ H N =´K„ pƒS V a:H;ma:VXm kWZVa:J m iLVXkYN ynpXH Vji-boV a;^ H WZY PN a;yLPRH [ VXN k N J P V b Py Y>iLVjk1Y } boV WZY Py‘iŠP z − z = 0 VXWZ‰1Py‘pXk PW Py PRy P PW¬PY Y [ ‰ J VjWu՛VXy }q

WZk K VI[VXWEW)P A z + A z − A z − A z = 0 V a:m Vjk1WEV J kVj[ = „ Ó t>?AÔ VXWE‰PRynpXk PN W PN y P N M PRW@P Ó t:<ÌÔ (A + A A − A − A A )z = 0, V a:m Vjk1WEV J PRY H N Y K [ =£” [][ H i²[ V N J VXy‘V J WEVX‰V J y V N a VjyX ?:= A + A A − A − A A = 0 = ” [][ H iP Ó t8;Ô W K k1V PRN i K yÖV a:m VXkWZV J‘‹ inVXkY1y‘pXVjiLkVj[÷PøiŠPk a ՑP €BS Pp K ynboVxiLV J WZVjk1Vx[—ynp PRN bcP C Ó Pp K ynboVjinV J WZVXkVj[ ÔxS Pp)V a:m VXkWZV J inVXkY1y‘pXVji7boV a;H WZY‰1P J1H>N =†„ boV a;H WEY PRN y ‹ q, q , q , q , q , q ‰1PWEbcPpÙboV a PY‰1P J1H N } N a:mS ‰ H;a:m P q‹æJ)MOPN W`Pynp J ՛}[ T PRiŠPRb V N J VjiLkVj[ =œš VX‰ P N J PRp J [:P TJ }>[ =ES ‰ H;a:m b K kY1Vjk R ‹ r‚VXW K VXWEy H ^ iLVXkY }+^ boV a;H WEY PRN y T i H W H k a´PN WZ‰1P JH N  i ij 12 21 1,2 1 2 1,2 1,2 1 1 2 2 11 1 2,1 12 2,1 2 2 2 1 2,1 2 1 1 22 x0 ,1 8B=  Rx0 ,2 = (x0 , q, Ai q , (Ai,j + Ai Aj )q)

| q ∈ R |{z} | {z } qi qij 4= ” [][ H i†P Ó t8;Ô W K kV PRN i K y/V a:m Vjk1WEV J iLVXkYynpXVxi ‹ kVj[ P7iŠPk a ÕØP CD=4„ b PRN y H Y>iLVXkY }^ boV a;H WZY PN y4QXyLPR[ H W m PkWZVX‰V JlS ‰ H;a:m p = 0 J VXWu՛VXy }q W = „ W VXN k m V a ×kVXb T i H W H k a´PN WE‰P J1H N b K kYVXkwVXWZy H ^ iLVXkY }G^ boV a;H WEY PRN y S PRplPp Ó t>?AÔ b K kY1VjkÖVXWEVXboV S QXyLP[ H W m PRk P K inV q = 0 J VXW|ÕØVjy }q W = A1,2 +A1 A2 −A2,1 −A2 A1 6= 0 ÊÌÍ Ë º;º1Î ¹Š¾ ¼;»’¿›Ï Æ ÒoÁEÄj¹ Æ À É ¿n¿ A ý ½ IÒ ¹ ZÇ Ç ¹A¿n¿‚ÒI¹ Ç ý X Ó t€;Ô z − A z = 0, i, j = 1.n ” pXVXk#V a:m Vjk1WEV J Vj[Bk1Vx[ x ‹ rPk M V JnJ VjWZy H ^ iLVXkY } ^ boV a;H WEY PRN yLP K  X  Ó t;CÔ R = (x , q , q ) | q = Aq PTpXPH pƒH P a´RN J ‹N r‚J VXkÖP q ∈ R ynp PN by‘plPRrPY H k MOPN W`Pynp J ‰1P J1H>N =4š Vx[ K k J y }>q [cP Ó t€;Ô VXWEy H ^ i W k PW Õ P  X z − A z = 0, i, j = 1.n Ó ts;Ô · L¹

Äj¹LÐnÐ Í Ë ½ ÆAÑ ¿ ÆAÑ Ç ¹A¿n¿ß¿›¹ Ç þ ¹ŠÐ Í Ë Ç¬Æ 8 Példa (A 7 Példa általánosı́tása) zi = zi (x1 , ., xn) i j i i j j j 0 x0 ,1 0 i i j i j j j j x0 ,1 i j i j j j zijk − Aij,k zj − Aj zjk = 0, i, j, k = 1.n „ p x ‹ rPk M V JnJ b PN y H Y>iLVXkY } ^ boV ;a H ZW Y PN ynP K P i VjN ynpX‰1PWZbcPpXP S P‰ H W’P 0 J2,x0 =  (x0 , qi , qij , qijk ) H W m PRk Ó t;{Ô (Aijk − Aikj + Aik Akj − Aij Ajk )zk = 0, V a:m Vjk1WEV J PRY H N Y K [ =E=h= „ [ VjN inY VjN yŽPp S ‰ H;a:m÷PN W J PRW PN k H yOVXy J r‚VXk ?:= ù H;a:m Pk ‰P JPRN i H pX‰1P J ÕØ}>[øboV a PR[ PN i˜bcP a PyLPRrriLVXkY }§^ YVji KhMUPN W J PR[ô a:m V ‹ WE VXb~JBrN J V M†V N KEJ J VXW V N M VXW PUY KEg VxiLVXkQ KlPN WEV a:m VXkWZV J Vx[ŽboV a;H WZY‰1P J1H N y P N a´PN k1PR[–ynp }>q [Dy V N a VXy VjW V VjWZV 8B= ù H;a:m Pk a:m’H ^ p H ^ Y‰1V J;}q k[~boV a PRini H N W S ‰ H;a:m

P™iLVXkYynpXVxiLkVj[×k K kQXy JHq rr :iLVØÕ ‹ J V JnJ Hq y‘ynpXV n} q a;a/VXN ynV S PRplPp JLHAMOPN r1r K YVji KEMOPN W J PR[ MBK pXy a´PN WZP J PUynVXbˆVxiLVXYb VXN k m Vjp };N ÕØPRrr K k J V a i PN WZ‰1P J1H N y P N a;K VXW JBV N J VXWZVj[V Jl= 0 ]0 3.1 Eszközök a PDE leı́rásához  • V a:m VXk f VXN y g Pp x ∈ R T‚H k J V a:m [ Hq iLk m VXpXV JBVjN r‚VXk YVjô1k KXPN W J [ V N J×M PW H N y S k‹ PY#iLVjk1Y>r‚Vjk x ‹ rPk k‹ PRY inVXkYrVXkY KEg VjiLVXkQ KXPN WZ‰1P J1H N L} q Ó a;a:M†VXN k m=-„ p J b H kY;՛}[ S ‰ H;a:m f VXN y g k‹ PY iLVjk1Y KZa boV a V a:m Vjp K [ÖPp x T‚H k J rPk f ∼ gÔ S ‰P Jk (R, R) 0 0 0 k,x0 f 0 (x0 ) = g 0 (x0 ), f(x0 ) = g(x0 ), fk (x0 ) = gk (x0 ). . „ p zN a:m r‚V M VXpjV J V JnJ ∼ iLVjW PN Q KAH N V a:m Vj[ MBKEM PRWZVXkQ K PÖinVXW PN Q KlH N Pp x V a:m [ Hq iLk m V ‹ pMXV JDHVjNN r‚VX nk q a;Y1a:VxM†ôN k mKXPRN W Jc L} qa:a;m a:M†VXN ‹k m Vj[ K J Vji

VjN r‚Vjk ì = ”´a:m †K W îm ì VXk÷VjÓ[ MBKEa M PRWZVXN kQ K P Ô H ynp M JPN W m]Jq P = PW y } VXk Vj[ V x r‚VXW k ñ y‘} Pi Pk1PR[ kV VXpXp }>[ ð „ p f n} q a;a:M†VXN k m PN W J PW†iLV T inVXpXVXk JPN W J x ‹ r‚VXW K k‹ PYGinVXkY }^ ՛V J V J j (f)‹ W`PW M P a:m ‹ Hq q =4v PW H N y n} q a;a:M†VjN k m Vj[ k‹ PYIinVXkY }×^ ՛V J ÕØV K kVj[ J VjiLV)P j (f) WZPWD՛VXW WuÕ }>[ 0 k,x0 @ 0 ^ ` % ` % )% 3 D$ %*# 0 k k,x0 x0 . Jk (R, R) = {jk,x0 (f) | x0 ∈ R} „ J (R, R) JBVjN i‘r‚VXkor‚V M VXpjV J ‰V JH ^ [ H]H iLY K k P N JPN p PRN y°Pp (x, y, y , ., y ) [ H]H iLY K k P N J P n} q a;a:M VXk m Vj[ ynV azN J y V N a/V N M VXW S P‰ H W K y `H M a:m rPp K y TH k J ) x(j (f)) = x , ( iniLPy − P T‚H J y(j (f)) = f(x ), (QjVXW k ) y (j (f)) = f (x ), == k 1 k,x0 0 k,x0 1 k 0 0 k,x0 0 yk (jk,x0 (f)) = f(k) (x0 ), • V a:m VXk f(x) := x VXN y g(x) := x = ” [B[ H i4P j (f) VXN yÃP j (g) boV a; VXWZVXW H ^

[ H]H iLY K k P N JP N K  S’K JnM = j (f) = (0, 0, 0, 2) WZWZV V j (g) = (0, 0, 0, 0) ” k1kVj[boV a; VXWEVXW H ^ VXk j (f) = j (g) YV j (f) 6= S N J n} q a;a:M†VXN k m Pp x = 0‹ rPk ?x‹ j (g) PplPRpÖP [ V iLVXkY KZa Vx[ MDKhM PWEVXky SMBK ynp H k J b PN y H Y>iLVXkY>r‚VXk kVXb Vj[ MDKEM PWZVXky = J (R , R )  „ VXk JLK YVjô1k zN Q KAH:N JƒJ VjiLb VXN y‘pXV J VXy+b H N Y H kWZVX‰V J [ KEJ Vxi¯Õ›VXynp J Vjk K R R JlzN T }1y } N n} q a;a:M†VjN k m Vj[]iLV:o‰1P f VXN y g PRp x ∈ R T‚H k J V a:m [ Hq iLk m VXpXV JBVjN r‚VXk—YVjô1k KlPN W J [ V N J n} q a;a:M†VjN k mS PR[B[ H i f VXN y g k‹ PYIinVXkY KEa boV a V a:m Vjp K [ÚPp x‹ r‚VXk Ó f ∼ gÔ S ‰P 2 Példa. 2 2 0 2 0 0 2 1 k 3 2 0 n m 0 0 1 0 2 0 0 n m n k,x f(x) = g(x), ∂i fα (x) = ∂i gα (x), . ∂i1 ik fα (x) = ∂i1 ik gα (x) „ p J (R , R ) JDVjN inr‚Vjk—r‚V M VXpXV J ‰V JH ^ [ HBH inY K k P N JPN p PN y Pp (x , y , y , .,

y ) [ H]H iLY K k P N J P L} q a;a:M VXk m Vx[ ynV a>zN J y V N a/V N M VjW S P‰ H W K y `H M a:m rPp K y TH k J–K − VXY K [o[ H]H iLY K k1P J PÕØP ) x (j (f)) = x , ( iniLPy − P T‚H J K HBH iLY K k1P J PՑP ) y (j (f)) = f (x), (QjVXW k α − PY [o[ y (j (f)) = ∂ f (x), == n k i α α i1 k,x m i α α i α i1.ik i α k,x k,x i1 α yαi1 .ik (jk,x (f)) = ∂i1 ik fα (x), = • V a:m VXk VXN y N [ V N J y H [Py P N a>= ù PPp f VjN y g Pp p ∈ M T‚H k J V a:m [ Hq iLk m V ‹ pXV JDVjN r‚VXk÷YVjô1k KXPN W J M N JlzN T }1y } N n} q a;a:M†VjN k m Vj[ Ó S PR[B[ H iÖPp J b H kY;՛}[ S ‰ H;a:m Pp JnM f VjN y g k‹ PY!iLVXkY KZa ‹ boV a V Ha:q m VXm p K [GPJBp N p‹ rVXH k m f ∼ gÔxSK ‰1P JnM M Pk p‹ kVj[ HBSUH K W ‹‹ WZV VPp f(p) = g(p) kVj[![ iLk VXpXV Vjr‚VXk W Pk (U, x) WEWZV V (V, y) Ó [ i Y K k P N J PiLVXkYynpXVjiLV S PoVXW m r‚VXkÚPp y ◦ f ◦ x VXN y–Pp y ◦ g ◦ x n} q a;a:M†VXN k m Vj[ b K k J

JAzN T }y } N } ^ a;a:M†VXN k m Vj[ Ô k‹ PYfinVXkY KEa Vj[ MBKEM PWZVXkynVj[§Pp x(p) T‚H k J rPk = R R „ rV M VXpXV J V JnJ Ó W H [ PN W K y Ô [ H]H iLY K k P N JPN p PRN y†ynV azN J y V N a/V N M VXW‚boV a PY‰1P J1H N P J (M, N)‹ VXk V a:H]m H Ó W K H [ N PRNJW NK Ky Ô (x , y , y , ., yJ ) [ MHBH inY J K k JnP NJ JPN pHBPNH y S KP‰ NH J>WŽN K¯P = j (f) boV a; VXWZVXW H ^ [ iLY k P P PRp y ◦ f ◦ x VXk Vjrr#r‚V VXpXV V [ iLY k P P Jk (M, N) M k,p −1 n −1 m k i α α i −1 α i1.ik k,p f g f(p)=g(p) p M x −1 yofog y yogox−1 ûBp PRN b H W PN ynyLPRW KEa Pp H WZ‰1P J1H>N S ‰ H;a:m Pr‚V M VXpXV J V JnJ Vx[ MDKhM PWEVXkQ K P inVXW PN Q KlH N kVXb L} q a;a Pp x VXN y y [ HBH inY K k P N J PRiLVXkYynpXVxiLVj[ MOPRN W`Pynp J>PN y P N J1H N W = Megjegyzés 1 ûBÓ p KEPRN g b H y VXKXynN V J rVXJ1HÌk N Ô kVX nb q a;a:M†Pp N m†MN N Hq yny‘pXa:VXm y Y N VjiLVXkQ PWZN ‰1M P } H a;VjkH Vjk1Vx[ =

QXyLP[ m V i VjynpX‰1PWZbcPp P PRWÛ[VjWZWÞY W pjkBÓ }k>[ W VXk P ‰VXW m pXV J TUVXN WZY PN }WøPR[B[ H i S ‰1P a W H r PN W K yLPk M P a:m W H [ PN W K ynPk Ô Pp N Pp M × N^ `H inb PRN rPk zN in‰P J1H>N S VXN y QjyLPR[ H W m PRk f : M N L} q a;a:M†VXN k m Vj[V J MBK pXy a´PN WE}1k>[ S P‰ H W@P π ◦ f = id VXW JBV N J VXW J VjWu՛VXy }q W S P‰ H W π : N M P§boV a; VXWEVXW H^ T i H ՛Vj[Q KlH>N = „ p K W m VXk L} q a;a:M†VXN k m Vj[V J boV J ynp VXN ynVj[Bk1Vx[#kV M VXpXp }>q [ =ƒ„ T‚H k J ‰ H p J PRi JLH p H N ôriL}bokP[GkV M VjpXp }>q [ p ∈ M PT‚H p JLH ›Hq W H:q JnJnK T‚NH JXk S JnH [w‰1m PWZbcPp P N JlS Ppl‹æM PRpÖJANPRJBp N H W Nm PRk k [ ‰1PcPp P boVXW Vx[DkVj[÷P V Vjynk VXW+P N[ V T V V N T1T VXk P p = Ó „ p PRN ri PRN k ynplP a;aπP JLH:JnJ–z MH kPRWZW`P ՛VXWZVjp J;}>q [ =dÔ PRN i°VXp7N T PpŽVjN ynVJ™J K y T J VXQ T KlPN W K J>yLPRN r1k1PRJ [ JÌ}^ k K [ S bJ V N a N ynVXb N J

Pp S S ‰ K ynJpXVXKhk MDKlJ N V J K ynp }^ WZH VN a HVXy f : M ‹ J ‰1P i PW yUb Y k Pp N V N WZVj[ V VXp Vjy k Vji iLV PWZ‰1P }k>[ ;boV ynp VXy‘[ Vjk ‹ S ‹ J ¯ ‹ m n J J N J q ´ S N T H KAH N H^ M × N boVXW Pp f V Pp id × f VXW‰VXW V VXy z Õ }>[ VXy°P π i ՛Vj[Q k1PR[~PRp²VXWEy [ H b T‚H kVXky‘inV JAHq i JBVXN k H˜^ T i H ՛Vj[Q KAH:N J²M Vjynynp }>q [  f(p) M N=M ^ xN f 0 0 p M a b0 0 M id f M −−− N ⇔ ×f M M −−− − M × N „ p K W m Vjk JAzN T }y }N ô1ri PRN W J k m PW PN r H:J™J i KEMBKlPN W K yŽk m PW PRN r1kP[ k1V M VXpXp }>q [ =ה£a:m ôri PN W J k m PW PRN r J VX‰ P N J V a:m (E, M, π) ‰ PRN inbcPy S P‰ H W π : E M P–r PN p K y JDVjN iniLV JHq i JBVXN k H-^ T i H‹ ՛Vj[Q KlH>N = û a:T m VXQ KXPN W K yLM PRk˜JnP²H ôJBr1N i PN KW J = k m PW PRN r H:JÃM Vj[ JnH iLk m PW PRN r1kP[ÙkV M VXpXp }q [ S ‰PŽb K k1YVXk ôriL}bV r‚Vjk Vj[ i Vji y „ p f : M

M V a:m boV J ynp VXN y‘V+Pp (E, M, π) k m PW PRN r1kP[ S ‰1P π ◦ f = id =†„ boH V J K ynp NVXN J ynVx[ ‰1PWEbcPp N JP N J Sec‹¯M ES ‹æM VXW:՛VXW Hq WuÕ }>q [ HB= H ù PŽK V a:N mJN pKEJ ∈ M‹¯M [ Hq iLk m Hq VXpXqV JBVxNS r‚VXk HM [ a:Hm ‹ iLY J k P N PRJ iLVXH kYN ynpXK Vji V x VXW P ôriL}b[ inY k P P y PW՛VXW WuÕ }[ P[B[ i™V boV ynp VXy W [ PW yLPkoPRp y (x) n} q a;a:M†VXN k m Vj[ zN iLk1PR[×WZV =´”£a:m f boV J ynp VXN y k‹ PRYÚiLVjk1Y } ^ ՛V J‘‹ Õ V N J´T VXY KZa Pp (x , y , y , ., y ) [ H]H iLY K k P N J P n} q a;a:M†VXN k m Vj[ÙynV azN J y V N a¬V N M VXW;՛VXW ‹ WZVjboVXpXp }>q [ = x f(x) (x, f(x)) x M α i α i Példák. α α i α i1.ik ?:=²„ p (R × R, R, π) k m PW PRN rIboV J ynp VXN y‘VƒkVXbb PN y S b K k J V a:m f : R R M PW H N y n} q a;a:M†VXN k m= Ó „ π : R R kVXb b PN y S b K k J PpƒVXWZy H ^ [ H b T‚H kVXky‘iLV JAHq i JBVXN k H ^ T i H ՛Vj[Q KlH>N = Ô

8B=²„ (TM, M, π) k m PW PN r boV J ynp VXN ynV K Pp M M Vj[ JLH inboVXp H;^ K¯= Ó π : TM M S =Ô v p š VxiLb VXN ynpXV J VXy4b H N Y H k M VXpXV J ‰V JH ^ r‚V T i H ՛Vj[Q KAH N P-bcP a PyLPRrrœiLVXkY }™^ ՛V J‘‹ Vx[ J Vji VxN r H ^ W p Pp)PW`PRQXy H k m Pr1rÖiLVXkY }×^ ՛V J‘‹ Vx[ J Vji VjN r‚V = “ VXN WZY PN }W J Vj[ K k J ‰V JRH ^ P Jk E  πk−1 y (xi , yµ , ., yµi1ik−1 , yµi1 ik )  πk−1 y (xi , yµ , ., yµi1ik−1 ) Bm K W MOPN k M PW H>N S ‰ H;a:m ‰1Pb K kYVXk x TH k J VjynV JBVXN k÷P ξ ∈ J E P π T i H ՛Vj[Q KlH N bcP a Õ N PRN rP=´k ” M PRk S PplPRp π (ξ)HBH = K0 S NPRJ>[]N [ H Ji¬P ξK [ J HBH JiLHY^ K k PN N a:J>PmN KS ξ =K J(x a:, m0, ., 0, y PWZPR[ }1PR[ pXVx[#Pp y [ iLY k P P[ Vj[ k ‰V [ } b k V Jk−1 E  k k−1 µ i1.ik ) i k−1 µ i1 .ik . × T} E |T × {z JlzN T }y }ÖN ynp K boboV J i K [B}1y J VXkp H iLboVXp H ^ [ H b T‚H kVXkynV K  ∂ Œ . ξ ∈ Vji

π − ε(ξ) = y dx ⊗ . ⊗ dx ⊗ ∂y ûBp PRN b H W PN ynyLPRW KZa Pp H WZ‰1P J1H>N S ‰ H;a:m PoYVjô1k zN Q KAH N [ H iniLVx[ JlS PplPp~Pp ξ‹ ‰VXp zN W m b H N Y H k iLVjk1YVXW J–J VXkp H iLboVXp H ^ kVXb n} q a;a P˜[ H]H iLY K k P N J PRiLVjk1YynpjVji MUPN W`PRynp JPN y P N J1H N W =´š Vj‰ P N J Œ Vji π − S T ⊗ E V a:mÚK p H b H inô1pXb }y S£VXN yŽP k times µ i1 .ik k ε i1 i1 µ k ∗ k−1 ε y H i°V a:m V a plPR[ J y H i H plP JlS PRplPpŽPƒy H inrPkoV a:m b p VjN ynVj[ H W m PRk H [ S ‰ H;a:m V a:mDK [IWZVj[ V N T VXp VXN yŽ[ V N T V [VXp H ^ WZVj[ V N T VXp VjN y-bcP a Õ P N M PW = πk−1 P N y£} J>PN k×ynpXVxiLV T W H+^ J VxiLVj[ VjN yOWZVj[ V N T V ‹ T‚H k JLH yLPk#boV a V a:m VXp K [IP i PN [ H:q M V J‘‹ 0 −−− Sk T ∗ ⊗ E −−− Jk E −−− Jk−1 E −−− 0, 3.2 Lineáris PDE formális integrálhatósága • V a:m VXk E VXN y F } :a m P k1Pp H k M VXWZV J‘JLK k m PW PN r

S°VjN y²WZV a:m VXk P : Sec(E) − Sec(F), f − Pf V a:m k‹ PYinVXkY }^ Y KEg VjinVXkQ KlPN W H:T Vji P N JLH i =7” [][ H i S PoVXkk mBK r‚VXk K ynboVxi J Vj[ f‹ kVj[P T‚H k J rPk k‹ PY iLVXkY KZa PYVji KhMUPN W J ՑP K¯S PR[B[ H iboV a J }Y՛}>[fN b H kYPRk K P p ∈ M N JBN N J Ó PplPpÚP Pf‹ kVj[fP p‹ rVXk P 0‹ PY iLVXkY }^ ՛V J‘‹ Õ V N J Ôx= „ W J PW PRN k H ynPk@ Pf(p) Vji Vj[ V PoVjk1k mDK rVXk K ynboVji J Vj[ f‹ kVj[ÚP p ∈ M T‚H k J rPk k + l‹ VXYÚiLVXkY KZa P˜YVji KEMUPN W J ÕØP K¯S PR[]‹ [ H iÙS boÓ V acJ }Y;՛}[‹ b H kY1P‹ k K P K Pf‹ l‹ VXYwiLVj^ k1Y KZJ‘a ‹ PÚN J YÔ =†Vji „ KEMUPN W N Ja:ÕØm P K k1PRT‚[ H:JnPJ p VjN i JBVxN [ V N J P PpXPpÙP Pf k1Vx[ P p r‚VXW l VXY inVXkY } ÕØV Õ V p z [P p r‚VXk p0 (P) p0 (P) WZVx[ V N T Vjp VXN ynVj[V J P P‹ ‰VXp Pyny‘p H Q KXPN W J b H inô1pXb }ynk1PR[ S°K WEWZV JnM VIP P‹ ‰VXp Py‘ynp H Q KlPN W J bM H inKô1pXb~JnM }y l‹

VXY K ‹ [ T i H W HHq k a´q PN W SÚJ Õ PN N k1PR[IkV H:M T VjpXp }>qN JL[ H =” pX‹ VjkwS-WZVjK [ V N TJnM VXp VXN y‘Vj[øbc‹ P a Õ P N J R ^ ‹ PW WZWZV V R WZVXWO՛VXW W|Õ }>[ VXyÚP P Vji P i k PY WZWEV V k + l VXY iLVXkY } Jk E −−− F, k+l Jk+l E −−− Jl F, k ob V a;H WENY JPRN y JLPN K k1PR[ k1V J M VXpXp }>q [ = T Ó R H ⊂JJN J E VXN y R ynpjVXboW VXW V VXpjVXk VjiLVx[Ö[P QXy W`P P  == == k k  π y k+3 ⊂ Jk+l E k+l =dԘ„ [ :Hq M V J [VXp H ^ Y K P a iŠP ==  πk+3 y Rk+2 −−− Jk+2 E −−−   πk+2 π k+2 y y Rk+1 −−− Jk+1 E −−−   πk+1 π y y k+1  π y 3 J2 F  π y 2 J1 F  π y 1 Rk −−− Jk E −−− F ® ý Æ ÀÄj¹Š¼À Í ç À]ÁEÂùnÄj¹Š¼ŊÁ ÆA¾ Ç Éxö ¹LÄ Æ ¾ ¿ É ÄA¿ ÊæÉ ÄLÒ Æl¾ Ç ÁZÐ Æ ¼oÁE¼¿›¹Lº;Ä ÆA¾ ÇZÈÆ ¿ É ¾ ¼ Æ ½ ¼¹ÎR¹ ÑXÑ4Í Ë ½ Ï ÈÆ~Æ Ç ¹ ½ƒ¹

¾ ö ¹ Ñ ¹Š¾ Ðx¹ ½ ÒoÁZ¼À¹Lº»Á ½ ¹–Ð Ñ´Í Ë Ä þ ¹ ½ ¿ ü ¾ ÎBå ù P P `H iLb PN W K yLPk K k J V a i PN WE‰P J1H>N S PR[][ H i/P π ynp }>q i¯Õ›Vj[ JLKEMDKhJPN yLPOb K P JnJ V a:m™J V J ynp H ^ WEV a VXy JBN M H m S P K kVj[ M V J;}q WZV J V JcV N TT VXk P ξ S PpXPp+P ξ ∈ R VjynV VXk Pk W Pk ξ ∈R ‹ ^ a;H N KEJ J JH ^ V a:m k + 1‹ VXY iLVXkY } ^ boV a;H WZY PN yny P N =°” p ξ k PY iLVjk1Y } boV WZY PyO[ VjiæÕØVjynp ‰V }H J1H N rN r K N S P Jπ =E=Ey‘=£p ” }>q i¯Õ›Vj[ JLKhMDKEJ>PRN yLN PÖN b K P JnJ ‹ [ KEJ VjiæÕØVja:ynmop J M†‰NV a:JJH ^ V a:m k + ^ 2‹ Ó Vj `YH inVXkN Y K }f^ Ô boV aa‹‹ WZY Py‘y P y r pXpXVXW’Pp™VjWuÕ PRi PynynPW P kVj[cV V VjWZVXkciLVXkY } iLb PRW y boV H WZY PN y P N J [:P T ՛}>[ = 8 Definı́ció Pk πk+l k k k k+1 k+1 k k k 3.3 Differenciáloperátor szimbóluma • V a:m VXk P : Sec E Sec F V a:m k‹ PRYcinVXkY }c^ Y KEg VjiLVXkQ KXPRN W

H:T Vji P N JLH i = ù P)QXyLP[cP k‹ PYƒiLVXkY } ^ J P a;H [ H k MBK pXy a´PN W|ÕØ}>[7PU‰1P J>PRN y P N JlS PR[][ H i’[P T ՛}[ƒPRp H:T Vji P N JLH i@y‘p K b˜r H N WE}1b P N Jl= V‘Õ Hq W VXN y VjN iLVƒ‰PRynpXk PN Wu՛}>[:P σ (P) ynp K b~r H N WZ}b H:Jl=Ú VX‰ P N J   0 σ0 (P) Jk E ξ = (x, 0.0, yµi1ik ) −−− F = J0 (F) . σ0 (P)ξ = p0 (P)ξ IF K k J P K k J Pp J boV a b } J P J‘J }>[ P (x, 0.0, y ) JlzN T }y }wN VjWZVXboVj[ J VjinV×boV a V ‹ a:m VXp K [oPp S T ⊗ E‹¯M VXW SzN a:m P+y‘p K b˜r H N WE}1b K k J Vji T iLV J>PN WZ‰1P J1H N b K k J S T ⊗ E F WZVx[ V N T Vjp VXN yX µ i1.ik k ∗ σ0 (P) Sk T ∗ ⊗ E −−−  ε y Jk E p0 (P) F  ε y −−− F k ∗ ù PRy H k W H N Pk#WZVX‰V J P T i H W H k a4PRN W J PR[ÚynV a>zN J y V N a/V N M VjW¬r‚V M VXpXV J k K P P ynp K b~r H N WZ}b PN 1k PR[ T i H W H k ´a P N W J Õ P N J  σl (P) „ a N J ‹¯M S σ (P) bcP Õ P g

PW P 0 Sk+l T ∗ ⊗ E −−− Sl T ∗ ⊗ F   ε ε y y JF a N J ‹ c b P Õ P E W X V > W Ø Õ VjW Hq WuÕ }>q [ = σ (P) g pl (P) −−− Jk+l E 0 l l l ¸¬¹Lº»¹Š¼ ¹Lº»G¹Š¼À¹ Ñ ¹¿‘¿œä ÆA¾ Ñ ÁZÐ ÆÖÆ ý ¼¹ ½ å –¹ÎR¹ Ñ ¹¿ þ‚Í Ë ½ Æ ¹L¾ Äl¿ ¹ ¾ ½ ¹ ½ Äj¹ Æ ¿›¹nÄj¹ ½ ¹¿nå ® Ñ ä ÆA¾ Ñ ÁZÐX¿U¹Lº» ½ Î ÆA¾ Ñ Á ý Äj¹Lº Í Ç Æ ¾ ÄLÁZÐ)ä ÆA¾ Ñ ÁZÐn¼ Æ ½ ¼¹ÎR¹ ÑXÑ´Í Ë ½ Ï È‚Æ À]ÁZÒ g (P) = À]ÁZÒ g (P) + XÀ]ÁZÒ g (P) . ® ÀBÁhÂùLÄj¹L¼‚ÅLÁ ÆA¾ ÇâÉxö ¹nÄ Æ ¾ ¿ É Ä’Ð Ñ ÁEÒ ä ɐ¾ Ç Í Ò Æ ¾ ¿BÁE¼Î ÉÇ Í ¿ ü ¾ ΐ¼ Æ ½ ¼¹ÎR¹ ÑXÑ4Í Ë ½ Ï ÈÆOÇ ¹¾ ¿›¹ Ñ Á ½ ÈÉ ÑXÑ Æ ¾ ½ Î ÆA¾ Ñ Á ý Äj¹Lº Í Ç Æ ¾ ÄLÁZЃä ÆA¾ Ñ ÁZÐXå 9 Definı́ció (Kvázi-reguláris bázis, involutivitás) dc . E = {e1 , ., en} Tp M j=  . 1, .,

n (gk )p,e1 .ej = A ∈ gk (P)p | ie1 A = = iej A = 0 E n−1 k+1 x k x k x,e1 .ej j=1 3.4 Kovariáns deriválás „ p R JBVjN inr‚VXk+WZV a:m VXk X = X ∂ VXN y Y = Y ∂ [ V N JÃM Vj[ JLH inboVXp H>^ =/„ p Y M Vj[ JLH iLboVXp H ^ K JLK [ HM PRi KXPRN k1y–YVji KEMOPN W J Õ PN k1PR[×k1V M VXpXp }>q [ ∇ Y M Vj[ JLH inboVXp H:^ JlS P‰ H W X y‘pXVji k Ó t;Ô . ∇ Y = X(Y )∂ . š Vj‰ P N J PRp X ynpjVji K k J YVji KhMUPN WZk K [VXWEW/Pp Y [ H b T‚H kVXky n} q a;a:M†VXN k m V KhJl= „ ∇ PYY KhJAzN M b K kY[ V N J£MUPN W JnH p H N Õ PRN rPk S W K k1V PRN i K y X‹ r‚VXk VXN y J VjW ‹ Tulajdonságok: ՛VXy zN JLK P ∇ (fY) = (Xf)Y + f∇ Y ynplPr PRN W mBJl= VjboQXyLPR[ M Vj[ JLH iLboVXp H ^ [V JlS ‰PRk1Vjb n} q a;a:M†VjN k m Vj[V JXS/J VXkp H iLboVjp H ^ [V J)K y)WEVX‰V J YV ‹ iJ KhMUPN WZH k K¯S =´„ p)K VXW M b K kH Y KZa T‚}H a:m PRkPRK p:°‰1P TH V a:TmÚH J VXkp H K i S PR[B[ H i ∇ K TJnK K y²WZV a:KEMUm

VXN k J V a:= m Vjk1p i P kVj[cPœ[ b kVXkynV P T [ b k1Vjk1y‘V kVj[ X ynpXVxi k YVji PW ÕØP ù PøV a:m M Pp R ‹ kVj[ V a:m i VjN ynpXy H [Py P N a P ScVXN y KhJnJ PR[PRiL}k>[—r‚V M VjpXV J k K V a:m K W m VXk#YVji KEMOPN W PN y JXS PR[][ H i M PkV a:m [ K y T i H rW VXN bcPB°kVXb r K p JLH y S ‰ H;a:m P VXk JnK VXW M Pa:Wh[:m PWEbcPp JPN y P N M J PWUH [P T‚H:N Jna>J S ∇ YH r‚VXkkV T M PH k PRN p VjN i S K k JH;H:^ a:JBm VjN inrKVXk = ù P H T VXK Y KZa PRp HBH M‹ V Pry‘p iŠPR[ y [:PRy P PR[B[ i PpÚP i r1WÓ VXbcP ‰ k kQXyœ[Pk k [D}yœ[ i Y K k P N J PiLVXkYynpXVji S P K inV MH kP J [ H p J P J‘M P P t;Ô V a:m VXkWZV JnJ VXWUr‚V M VXpXV J ‰V J k VXN k>[—P [ HAM H Pi N KlPN K kK y°YVjHBi H KEMUPN K W J P N JlJ =4” p VjN i J#} N a:m M VXJ pjV J JLÕ H }>q [×H N r‚VŽP ∇ nq Ya;Vja:i M†KhMUN PN W m PRN y JlS JX‰ S£H;N a:m boM V a PRYJ ՛}>q [ P™W [ PW y y/[ iLY k P PRiLVXkYynpXVjin‰1Vjp PRi p Γ

(x) } Vjk Vj[V VXy´r‚V VXpjV Õ }>[ n i j i j X j X X j X  X n X k ij P ∇J æ‹ J TP ∇ S ∂ H;=.a:m Γ ∂ `H iLb~}1W P N M PRW =/„ VXk JLK1J }1WZPÕØY H kyny P N a;H [œboV a [ H:q M V J V J W VjN y V N M VXW Pp [P ÕØ}>[ ‰
Ó <9Ô . ∇ Y = X(Y ) + Γ (x)X Y ∂ . F V a ՛V a:m VXpXp }>q [ S ‰ H;a:m Pp R [:Pk H k K [B}1y‚[ H kkVxž KlH N Õ PRN ‰ H p J Pi JLH p H N Γ (x) n} q a;a:M†VjN k m Vj[ Pp H k H ynPkÞk]}WZW PRN [ = F V a ՛V a:m VXpjp }>q [ JnHMOPRN rr P N S ‰ H;a:m PoVXkk mBK r‚VXk Y VjN i JBVxN [VGP ‹ N S H ‹ ‹ K H T‚H kVXkynV K+V N TT Vjk§Pp Y [ H b ‹ p ∈ M r‚VXkøp VjiL}y PR[B[ i+P ∇ Y k1PR[ p r‚VjW [ b T‚H kVXkynV K kVj[ X ynpXVji K k JLK YVji KEMOPN W J ÕØP =£” p+kVXboQXyLP[ M Vj[ JLH inboVXp H ^ [BinV S ‰1PkVXb J VXk ‹ p H inboVXp H ^ [BinV K y KZa PRp = „ [ HAM PRi KlPN kyßY KEg VxiLVXkQ KlPN W PN yßP J VXkp H i H [ F J Vji VjN k V a:m VXWZy H ^ iLVXkY } ^

Megjegyzés  YJ KEg VjiLVXq kS Q KXPN H;W a:H:mT Vji P N JLH i S PRb K ‰VXp = J PRi JnH p H N p (P) : J F T ⊗ F WZVx[ V N T Vjp VXN y‘iLV VXWu՛VXy }W ‰ p (P) ◦ ε = id ∂i j k ij k k X k ij i j k n k ij X 1 1 1 3.5 The Cartan-Kähler theorem š ‰VŽk H:JLKZH k H; @K k MH WE} JnKEMDKEJmS: ‰ K Q ‰  VŽy‘‰PRWZWy J }1Y mK k J ‰VŽkVxž J y‘VXQ JLKZH k S PRWZW HA y }yéâì JLDH ë QŠ‰VXé QL[ J ‰1V `H ì iLbcPW K k J V a iLPRr é :K Wé KEJíøm×ë K k B} KEéJ V)P˜yé K b T WZV  ë P m  é ’éæð ì é é é í ò ñ ð ñ ð Ó k H kVXVXY ð JLH Q ‰ð VXQL[ J ð‰1P J ò ñ J ‰V)ð bcP ð T y π π PRinV H k JLòÙHÌÔxð = R YX C6kD 9X %lR ?> % ?> % A e/ A D ) ` % 7 9X 696 k L 6 % B ?> % A )*B ?> %gfihdj 6 %;:8) =< m6 k+l ¸¬¹¿ ä>¹ ÆcÇ ÁZ¼¹ Æ Ä ö/Æ ÄA¿âÁ ÆAÇ ÀBÁhÂùLÄj¹L¼¿æÁ ÆlÇUÉxö ¹LÄ Æ ¿ É Älå Í öö£É Ðx¹c¿ ÈÆ ¿

ÎR¹XÅA¿ É Äœä Í ¼À Ç ¹ É ¼ Áõå ¹å ÁZЖÁZÐßÄj¹Lº Í ÇâÆ ÄAå ­ Ê ÆXä ã ¿ ¹-Л»>Ò ä ÁEвÁZ¼Î ÁZÐ É ¼¿æÁõ¿ ÎRÉ ¹XÏ Ï ¿ È ¹Š¼ ã P ÁZÐ È ÊæÉ ÄLÒ ÆAÇZÇ »cɐÁZÇ ¼¿›¹Lº;Ä Æ Éä ÇÇ Í¹Aå 14 Tétel. (Cartan-Kähler) P Rk P πk : Rk+1 − Rk gk+1 (P) ÁZÐ Æ First compatibility conditions for a PDO ù VxiLV  V -K WZW¬Vxž T W`P K k‰ H JLH ô1k1Y K k J V a iŠPRr K W KhJ›m Q H kY KEJLKZH ky H i)‰ H JLH QŠ‰VXQL[ J ‰VUyn}>i¯Õ›VXQ JLKEMDKhJ›mÙH; π = r1y J iL}Q JLKEH k1y JnH7J ‰1V K k J V a iŠPr K W KEJmS PRWZy H QlPWZWEVXY ð ë ò é‘ë;ì S PRi K ynV)P JßJ ‰ K y²y J P a V = • V J P : Sec(E) − Sec(F) rVIP kJ ‰ H iLYVji W K kVlPRi “ e ˜= V։P M V J ‰V `H WZW H-K k a Y K P a iŠP# on ) k pn 0 q2 0  τ / / T∗ ⊗ F /K O  ε ∇ ε  p (P)  1 /  / / Rk+1 Jk+1 E J1 F O  πk π0 πk  

/   po (P) / Jk E /F Rk  σk+1 Sk+1 T ∗ ⊗ E /0 Ó <?AÔ   0 0  1‰ VxiLV K YVXk H:J Vjy J ‰VIQ H [VjiLkVXW H; -J ‰VÖb H i T ‰ K y‘b σ  K := Tb ⊗ F PkY K KEJ m K H JLKZH k H k F =U„ QXW`Pyny K QlPW‚iLVXy‘}1W J™K k ‰ H bσ H W H;a;K QXPW ∇ yŽPRkPinr iŠPRi W kVlPRi²Q kk1VjQ PW a VjriŠP a;KEM VXy J ‰V `H WZW H-K k a 15 Állı́tás ª È ¹LÄj¹ƒ¹ ÁZÐX¿æÐ Æ Ò É Ä ö‚È ÁZÐnÒ ϕ : R − K Ð Í Å È ¿ ÈÆ ¿ß¿ È ¹-Ðx¹ Í ¹Š¼ÅX¹ ∗  k+1  G k π ϕ k Rk+1 −−− Rk −−− K k+1 ÁZЙ¹ Æ Å¿nå ­ ¼ ö/Æ Äl¿æÁâÅ Í ÇâÆ Äxϒ¿ È ¹ŽÒ É Ä ö@È ÁZÐ‘Ò π ÁZÐ É ¼¿ É Á Ê´Æ ¼À É ¼ Ç »ÖÁ Ê ϕ = 0 å • V J }y°Q H k1y J iL}Q J£J ‰V-bcP T ϕ =°‡UH ky K YVji z ∈ R SJ PR[V z ∈ J E y‘}1QŠ‰ J ‰1P J H T J = Vƒ‰1P M V π z = z PkY Q b } V p (P)z sr  k k q2 k 1 1 1 k+1 1 û K kQX=˜V ‡UJ ‰H VœynV

B}VXkJ QXV m#TJ ⊗ F K JJ F F K y™VxžPRQ JlS‚J ‰ J K y T J i HAM VXy J ‰1P J p (P)zÓ ∈K b ε kynV B}VXk W ‰VjinV Vxž y y A ∈ T ⊗ F yn}Q ‰ ‰1P εA = p (P)z A y }k K B• }VXJ W m Y1TV J VjJ iLb K kVXYcr‚VXQXP}ynV ε K y K k;՛VXQ JLKhM V Ôx= V }y } ϕ(z) = τA. Vb~}1y J)T i HAM V J ‰1P J ϕ(z) Y H VXy+k H:J Y1V T VXkY H k J ‰V)Q ‰ H;K QjV H; z =ߕ V J z PRWZy H rVÙPk#VXWEVXboVXk J²H; R yn}Q ‰ J ‰1P J π z = z = ™ Q H }>iLy‘V  V²ô1kY J ‰1P J π (z − z ) = 0 S y H z − z ∈ b ε =£• V J A r‚V-PkoVXWZVjboVXk J H; T ⊗ F yn}QŠ‰ J ‰P J εA = p (P)z = Vob }y J Q ‰VXQL[ J ‰1P J τA = τA SÛK¯= V = Œ M A − A ∈ Vji τ = b σ . V)‰P V π0 p1 (P)z1 = po (P)πk z1 = po (P)z = 0. / ∗ /  1 1 ∗ / 1 1 1 2 0 1 1 k ∗ 0  0 1 0 1 0 t2 1 2 0 1 0  1 Mn 0 k 1 k+1 0 1 k+1 ε(A − A 0 ) = p1 (P)z10 − p1 (P)z1 = p1 (P)(z10 − z1 ), ‰VXkQXV b σ ) } J ε K y H k JLH>S y

H A − A ∈ b σ =°š ‰ K y T i AH M VXy J ‰1P J ϕ K y  VXWZW ‹ YVjô1k1VjY = HA WZV J }y²QŠ‰VXQL[ J ‰P J K H JnH ϕ = 0 ⇐⇒ π y k . VƒÕ›}y J kVXVXY JLHcT i HAM V J ‰1P J+Œ Vji ϕ = b π . V ‰1P M V ϕ(z) = τA S -KEJ ‰ A yn}QŠ‰ J ‰1P J εA = p (P)z PkY z yn}Q ‰ J ‰P J π z = z = H J J τA = 0 ⇐⇒ A ∈ b σ ⇐⇒ ∃ B ∈ S T ⊗ E yn}QŠ‰ ‰1P A = σ B. • V J }1y–Q H ky K YVji εB  Vƒ‰1P M V a 
ε(A − A 0 ) ∈ p1 (P)ε(Sk+1 T ∗ ⊗ E) = ε σk+1 (Sk+1 T ∗ ⊗ E) = ε(  0  k+1 k+1 k 2  1 1  2 k 1  k 1 k+1 ∗ k+1 k+1 u PkY y H e Vjô1kV p1 (P)εB = εσk+1 B = εA = p1 (P)z1 = V)‰1P M V p1 (P)(z1 − εB) = 0 z := z1 − εB  ‰ K Q ‰ T i HM Vjy J ‰1P J v2 J ‰1P J²K y z1 − εB ∈ Rk+1 . πk (z) = z − πk εB = z yn}QŠ‰ J ‰1P J K¯= V =1Œ Vji ϕ = b π = 2 š ‰B}y J ‰Vƒyn}>i¯Õ›VXQ JLKhMDKEJmÚH; π QlPk r‚VƒQŠ‰1VjQŠ[VjY r m ‘y ‰ HAŽK k

a˜J ‰1P J ϕ = 0 = V -K WZWk HA V ž T W`P K kډ HÛJ ‰ K yßQlPRk r‚V)QXPRini K VjY H } Jl=  ϕ(z) = 0 ⇐⇒ ∃ z ∈ Rk+1 πk (z) = z, k q2 k wmxzy5{}|z~ 
€9‚y‚{ƒ|Z~„{}|U€m5€?O|9†‡9;ˆ {ƒ†M‡Zy‰Šˆ5{E‹Œ€‹~„ŽTQˆ‰‹
| y‰Oˆ‹€9€m‘’|;“ 3.6 Példa: lineáris konnexió • V a:m VXk ∇ Pp M y H [:PRy P N a;H k V a:m W K kV PN i K yO[ H kkVxž KAH>N =£š Vx[ K k J ‰V JH ^ ∇ V a:m ∇ : X(M) −−− Sec(T ∗ ⊗ T ) VXWEy H ^ iLVXkY } ^ Y Khg VjiLVxiLVXkQ KlPN W H:T Vji P N JnH iLk1PR[ ÓS P‰ H W ∇Y : X(M) X(M) WZVj[ V N T VXp VXN yƒP `H N M KlN Jl=4„ <9Ô V a:m VXkWZV J PW`P T Õ PN k ∇ Y W H [ PN W K yLPkÚP X 7− ∇ Y iLb }W P PW‚YVjô1k PW 7− Y X ∇Y X  ∂Y k  ∂ . k j ∇X Y = X i + Γ (x)Y , ij ∂xi ∂xk [ V N T WZV JnJ VjWPY‰1P J1H N boV a>S´VXN y zN a:m P ∇ Y KEg VjiLVXkQ KXPN W H:T Vji P N JLH iL‰ H p J PRi JLH p H N b H i‘ôpjb }y p0

(∇) : −−− T ∗ ⊗ T J1 T
x, Yik +Γijk (x)Y j . (x, Y j , Yik ) 7− „ p H:T Vji P N JnH iUynp K b˜r H N WZ}b PRplPp7P K YVXk LJ K [B}1yßWZVx[ V N T Vjp VXN yX Szimbólum, involutivitás σ1 (∇) : T ∗ ⊗ T − T ∗ ⊗ T kVXbb PRN y S b K k J PRp σ1 (∇) σ1 (∇) (x, Aki ) −−−   εy T ∗ ⊗ T −−− T ∗ ⊗ T   εy p1 (∇) p1 (∇) Ó „ [ K ynp PRN b zN JPN ynk PN W J Vj[ K k J }k>[™V a:m A = A dx ⊗e J VXkp H iLboVjp H:^ JlS Pp ε WZVj[ V N T VXp VXN ynynVXW V a:mH W m Pk k‹ PYoiLVXkY }˜^ ՛V J؋ [ VjN k JUK k J Vji T iLV JPN Wu՛}>[ S P K kVj[ Pp™PRW`PQXy H k m PRrroiLVXkY } ^ J P a ÕØP K PpXPpb H y J P 9R‹ PY÷iLVXkY } ^ YVji KEMOPN W J PR[Bk1PR[ boV a; VXWZVjW H ^ [ HBH iLY K k P N J>PN [ k]}1WEW PRN [ S bcP՛YÚboV a k VXN pXp }>q [ S ‰ H;a:m b KEJ QXy K k PRN W M VXWEV+P~Y Khg VjiLVjk1Q KlPN W H:T Vji P N JLH i S PplPRp P p (∇) = „ ynp K b˜r H N WZ}b T i H W

H k a´PN W J Õ PN k1PR[VXynV JBVXN k#‰1Py H kW H N PkIPY H N Y K [ S ‰ H;a:m J1 T −−− T ∗ ⊗ T k j j (x, 0, Aki) −−− (x, Aki ) k 0 σ2 (∇) p2 (∇) XV N yÖVXkkVj[ boV a; VXWZVXW H ^ Vjk P T i H W H k a4PRN W J ynp K ˜b r H N WZ}bcP§P NT N T ⊗ T ⊗ T WZVx[ V Vjp VXyŽP ∗ ∗ −−− p1 (∇) (x, 0, 0, Akiji) −−− (x, 0, Akji) −−− J1 (T ∗ ⊗ T ) J2 T σ2 (∇) (x, Akji )   εy S2 T ∗ ⊗ T −−− T ∗ ⊗ T ∗ ⊗ T   εy σ2 (∇) : S2 T ∗ ⊗ T − σ2 (∇)(B)(Z, X, Y) = B(Z, X, Y) K k>[BW }N p KAH>N =´” pXVXk#ynp K b~r H N WZ}b H [ bcP a Õ PRN iŠP+P Œ Œ g (∇) = Vji σ (∇) = {0}, g (∇) = Vji σ (∇) = {0}, 1 1 2 2 PY H N Y K [ S+VXN y zN a:m P p N ∈ M TH k J rPk MOPN W`Pynp JLH:JnJ+J V J ynp H ^ WEV a VXy {e , ., e } r PRN p K y‘iŠP a:mcJ jV iLb VXN y‘pXV J VXy‘VXk g (P) = {0}. Y K b g (∇) = 0, Y K b g (∇) = 0, Y K b g (P) = 0 VXN y-P Y K b g (P) = Y K b g

(P) + XY K b g (P) VXW JDV N J VXW J VjWu՛VXy }q W S ‰ K ynpXVjk 0 = 0 =£š VX‰ P N J P ∇ ynp K b~r H N WZ}bcP K k MH WZ} JlzN M= 1 k n  x,e1 .ej 1 2 k x,e1 .ej n−1 k+1 x k x k x,e1 .ej j=1 Integrálhatósági feltételek meghatározása „ Ó <?AÔ Y K P a iŠPok1PR[×P˜[ H k>[]i V N J VXynV J rVXk P σ2 (∇) R2 −−−  π y 1 S2 T ∗ ⊗ T −−− T ∗ ⊗ (T ∗ ⊗ T ) −−−   ε ε y y p1 (∇) 2 J1 (T ∗ ⊗ T )  π y 0 −−− J2 T  π y 1 ‡ßH [Vxi σ (∇)−− 0 po (∇) Y K P a iLP VXWEVXW‚boV a>=£FIKEM VXW@Pœynp K b~r H N WZ}b VXN y T i H W H k a´PN W J ÕØP K k;ÕØVx[ JAzN MS VXpjVj[ÚbcP a ÕØP P J i KEMBKlPN W K yOp VjN iL}yLPRW JBVjN i S PRplPp g (∇) = 0 SOVXN y g (∇) = 0 =†” kkVj[ÖPW`P T Õ PN k Y K b ‡UH [Vji σ (∇) = Y K b (T ⊗ T ⊗T )−iŠPk>[ σ = n − n (n + 1) = n (n − 1) , 2 2 NVXy ‡ßH [Vxi σ (∇) K p H b H iLy Λ T

⊗ T ‹æM VXW S PplPRp Pp Pk JnK ynp K boboV J i K [B}1y M Vj[ JLH i VjN i JDVjN [ } ^ J Vjk1p H i H [ J VjiLV = ù P r‚V M VXpXV J Õ }q [ P τ : T ⊗ T ⊗ T − Λ T ⊗ T Pk JLK ynp K boboV J i K p PN W H N WZVj[ V N T VXp VjN y J P R1 −−− (T ∗ ⊗ T ) −−− J1 T 1 ∗ 2 2 ∗ 2 2 ∗ 2 ∗ 3 2 2 ∗ 2 ∗ τ(C)(X, Y) = C(X, Y) − C(Y, X). ` H iLb~}W P N M PW S PR[][ H iOk mDK W MOPN kb K k1YVXk ‰1P M V B ∈ S2 T ∗ ⊗ T ynp K boboV J i K [B}1y J VXkp H iOVXy‘V JBVXN k τ ◦ σ2 (B)(X, Y) = σ2 (B)(X, Y) − σ2 (B)(Y, X) = B(X, Y) − B(Y, X) = 0. FIKEM VjW τ k mDK W MUPN k M PRW H N PkÚynp }>q i¯Õ›Vj[ JlzN MS’zN a:m P σ τ S2 T ∗ ⊗ T −−−2 T ∗ ⊗ (T ∗ ⊗ T ) −−− Λ2 T ∗ ⊗ T −−− 0 y H i)JDN V J a plJlP=°[ Jl” =o” [B[ H i a:Jm VX‰ P N J P τ ynV azN J y V Na:a¬mÖV N M H VXW´m [ K y‘pM PN b JLzNH J ‰1P J ÕØ}>H>^ [ S PÚ[ H b mDK T P JLK r K W KEJPN y K VXW V VXW

‰‰VXpÙWZV Vjk Y ∈ X(M) V W Pk Vj[ iLboVXp PRboVXW [cP p ∈ M T‚H k J rPk§V a:m VXWZy H ^ inVXkY }w^ boV a;H WZY PN y S PplPp j (Y) VXWZV a V J+J VXy‘poP p (∇)j (Y) = 0 VXW JDV N J VXWEk1Vx[ S PRplPp (∇Y) = 0 = ûBp PN b zN J yn}>[[ K P ϕ(j Y) = τ ∇(∇X)
. ù P ‹æM Hq q H KAHÙN JLH iLp KAH N Õ P N JlS PRplPp7P T (X, Y) =. ∇ Y − ∇ X − [X, Y] T VXWD՛VXW WuÕ }>[oP ∇ [ k1kVxž `H iLb~}W P N M PWYVjô1k KlPN W JUJ VXkp H i JlS PR[][ H iŽP p TH k J rPk 1 p 0 1 1 x0 x0 X Y τ(∇∇Y)(X, Z) = (∇∇Y)(X, Z) − (∇∇Y)(Z, X) = ∇X (∇Y)(Z) − ∇Z (∇Y)(X) = ∇X ∇Z Y − ∇∇X Z Y − ∇Z ∇X Y + ∇∇Z X Y = R(X, Z)Y + ∇T (X,Z) Y = R(Y, Z)Y ‰ K ynpjVXk ∇ Y
= 0 S boVji J ∇X
= 0 = ” pXVj[ÛynpjVji K k J P [ H k1kVxž KAH N a1Hq inr }q WZV JBVXN kVj[ L} q a;a:M†VXN k m†VjN r‚VXk M PRk M P a:m k K kQXy K k ‹ J V a i PN WZ‰1P J1H N y P N a;K VXW JBV N J VXWZV)P ∇ K k J V a i PN WZ‰1P

J1H N y P N a´PN k1PR[ ?:= ‰1P R = 0 S PR[B[ H i k K kQXy K k J V a i PN WZ‰1P J1H N y P N a;K) VXW JDV N J VXWEVwP ∇Y = 0 Y KEg V ‹ inVXkH ^ Q KlPN WZV a:m ^ VXkWZV J a;kH Vj[ S N PR plPpwP ‡ PRJi H J^ PÓk T ‹›ŒœH Pq H ‰WZVja4i N JDV N J J1VXHÌNW Ô VjN i a:J m#VXWEb M†NVjN a:r‚J VXk b K k1YVXk ^ VjWZy iLVXkY }øboV WEY PRy VXWZVjboVXWZ‰V i W k PRWZ‰1P V V VXWZVXkiLVXkY } boV a;H WZY PN yny P N 8B= ‰1P R 6= 0 S PR[][ H i H W m PRk Y VXWZy H ^ iLVXkY }#^ boV a;H WZY PN yOr K p JnH yLPk kVXbVXboVXWZ‰V JH ^ VjW Ó K WZW =T i H W H k a4PRN WZ‰1P J1HÌN Ô V a:m M†V N a:J VXWZVjkøiLVXkY }w^ boV a;H WZY PN yny P N S PoVXW mDK [#kVXb J VXWu՛VXy zN JnK Pp R(XZ)Y = 0 VXW JDV N J VXW J b K k1YVXk X, Z ∈ T VjynV JBVXN k = ” rr‚Vjk PRa:pÙm VXynV J r‚J VXk } N a:m [qVXWZW¬VXW|Õ PNS iLkBN }k>N [ S ‰ H;a:m P q [:P T‚MDH:K JnJ™a´ N VXW JDJ V N J KVXW J r‚J VXWZV M VXK ynynH:p^ M‚}>q [N J PJnpJ V VXkWZV iLVjk1YynpjVji

}k>[Br‚V VXy };ՍiŠP#[VXpXY;Õ }[ pXy PW`P P k>[P P [ r z V V a:a:mm VXkWZV JJ iLVjk1JlYS ynpjVjiniLVXW S PplPpœV a:m¬} q JnJŽMBK pXy a´PN W|ÕØ}>[ P ∇Y = 0 VXN y R(., )Y = 0 V VXkWZV Vj[V PplPp+P T (X,Z) p p u  ∇X Y = 0, R(X, Z)Y = 0, ∀X ∈ T ∀X, Z ∈ T V a:m VXkWZV J iLVjk1YynpjVji Jl=´„ p–VXWZy H ^ n b zN a P7b PN y H Y K [ n(n − 1) Y1PRiLPRr T PRiLQ KlPN W K y Y KEg VjiLVXkQ KXPN WZV a:m VXkWZV J V J PY#PRp K y‘boVjiLV J WEVXk Y [ H b ‚T H kVXkynV K iLV = 1 2