Formulario de Cálculo de Varias Variables 1
February 5, 2023 | Author: Anonymous | Category: N/A
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ÊYNJ AN BFNMBFJR NSJBQJR BÊ@B^@E AN TJYFJR TJYFJI@NR AEBNMQN? FML. JMLN@ DNYMJMAE REQE R. ^MFAJA M·:? TNBQEYNR V R^WNYDFBFNR. D^MBFEMNR TNBQEYFJ@NR T NBQEYFJ@NR QNKJ
ANRBYFWBFÒM
DÒYK^@JR
MEQJR Beeranmjajs bf`ëmarfbjs? :.- (r, (r, γ) ns umj rnprnsnmtjbfòm pe`jr an `j preynbbfòm an W nm n` p`jme SV. Beeranmjajs bjrtnsfjmjs bjrtnsfjmjs?? @j pesfbfòm an um pumte nm n` nspjbfe Rfstnkjs 9.- _ ns `j afstjmbfj afrflfaj an (r, ( r, γ) j W S, trfafknmsfemj`ns Anpnmafnmae an `es n`nknmtes usjaes qunaj antnrkfmjaj nm beeranmjajs bjrtnsfjmjs per `j jisbfsj S, Beeranmjajs nsdærfbjs anBeeranmjajs beeranmjajs. bjrtnsfjmjs, bf`ëmarfbjs y nsdærfbjs
pjrj andfmfr um nm n` nspjbfe, sn afspemn anpumte`js afdnrnmtns bjrjbtnrfzjbfemns an burvjs y supnrdfbfns.
`j eranmjaj Vbf`ëmarfbjs? y `j j`turj (betj) _. W nm n` nspjbfe sn rnprnsnmtj Beeranmjajs ^m pumte per knafe an umj tnrmj eranmjaj (r, (r, γ, z) z) Beeranmjajs nsdærfbjs? ^m pumte W nm n` nspjbfe sn rnprnsnmtj per knafe an umj tnrmj eranmjaj (ϊ, γ, τ)
:.- ϊ ns `j afstjmbfj nmtrn W y n` erflnm, ϊ ≮ 7 9.- γ ns n` kfske êmlu`e utf`fzjae nm beeranmjajss bf`ëmarfbjs pjrj beeranmjaj pjrj r r ≮ 7 2.- τ ns n` êmlu`e nmtrn n` non z pesftfve y
⃜
n` snlknmte an rnbtj An `j trfjaj rnbtjmlu`jrns j `j trfjaj bf`ëmarfbjs y vfbnvnrsj? , , Tnbterns ^mftjrfes Qrjmsderkjbfòm an `j trfjaj rnbtjmlu`jr { } j `j trfjaj bf`ëmarfbj y nsdærfbj
, ,
Nstjs trjmsderkjbfemns mes pnrkftnm bemvnrtfr `j trfjaj rnbtjmlu`jr nmtrn `es afstfmtes tfpes an beeranmjajs trfafknmsfemj`ns trfafknmsfemj` ns nxfstnmtns, nxfstnmtns, pjrj ``nljr j djbf`ftjr afdnrnmtns bê`bu`es.
∜ ∜
Nm `js beeranmjajs bf`ëmarfbjs `j supnrdfbfn (bemstjmtn) rnprnsnmtj um bf`fmare f`fkftjae bem non an sfkntrëj z y rjafe fluj` j` vj`er an
Bekpemnmtns Tnbterfj`ns
, 7 ≡ τ ≡ χ
.
@j supnrdfbfn rnprnsnmtj umj nsdnrj bnmtrjaj nm n` erflnm y bem n` rjafe fluj` j . @j supnrdfbfn ns `j kfskj qun nm beeranmjajs bf`ëmarfbjs, um p`jme vnrtfbj` fmb`fmjae j um êmlu`e an
∙ ⁁ ∜ ∜ , Tnbterns ^mftjrfes
,
Bekpemnmtns Tnbterfj`ns
∜ ∜ ∜ ∜
An `j trfjaj rnbtjmlu`jrns j `j trfjaj nsdærfbj y vfbnvnrsj? , , Tnbterns ^mftjrfes
Bekpemnmtns Tnbterfj`ns
, Tnbterns ^mftjrfes
Bekpemnmtns Tnbterfj`ns
Dumbfemns Tnbterfj`ns. Epnrjbfemns burvjs nm n` y nspjbfe
Burvjturj an `ëmnjs
Rupnrdfbfns bf`ëmarfbjs
Ns umj dumbfòm vnbterfj`, jqun``j aeman `js dumbfemns bekpemnmtns d, l y c sem dumbfemns qun sn nmbunmtrjm rnspnbte j` pjrêkntre t.
rt r‹t≧
Rf B ns umj burvj sujvn bjrjbtnrfzjaj per `j trjynbterfj peankes andfmfr peankes nm teaes sus pumtes um vnbter tjmlnmtn bem `j kfskj erfnmtjbfòm an `j burvj, yj qun 7 buj`qufnrj qun snj t. Burvjturj? Rnj B umj burvj sujvn. @j burvjturj an B kfan kfan buêm rêpfajknmtn bjkifj `j burvj an afrnbbfòm nm um pumte. @j knafaj an bjkife an` vnbter umftjrfe tjmlnmtn bem rnspnbte j `j `emlftua an jrbe sn `j anmekfmj bem ‐H ‛ (Hjpppj).
W`jme?
̎?
Nspjbfe?
⃜⃜ ℍ
Rn andfmn beke? Q ns n` vnbter umftjrfe tjmlnmtn. Rn punan trjaubfr beke?
s | | Ü | | | | | ++ Dòrku`js an `j burvjturj?
, dumbfemns nmY2
, bujmae tfnmn
Yjafe an Burvjturj?
Rnj B umj burvj nm um p`jme y snj @ umj @js supnrdfbfns bf`ëmarfbjs qun tfnmnm `j Burvj Lnmnrjtrfz rnbtj me pjrj`n`j j nsn p`jme. J` pnrtnmnbfnmtn j `es p`jmes beeranmjaes y Ynbtjs lnmnrjtrfbns bemoumte an teajs `js rnbtjs pjrj`n`js j Wjrj`n`js j `es nons beeranmjaes tfnmnm umjs an `js sflufnmtns @ qun bertjm j B sn sn `n ``jkj bf`fmare. J derkjs? B sn `n ``jkj `j burvj lnmnrjaerj an` bf`fmare y j `js rnbtjs pjrj`n`js sn `ns ``jkj Ynbtjs lnmnrjtrfbns.
,,,, 77 ,, 7
t? Wjrêkntre (vj`er fmanpnmafnmtn) r(t)? Tnbterns (vj`er anpnmafnmtn) Tnbter pesfbfòm (x, y, z) Ajmae vj`erns jriftrjrfes jriftrjrfes j t N` pumte fmfbfj` ns n` erflnm N` pumte dfmj` ns n` pumte an `j burvj N` rjafe an burvjturj nm um pumte nm `j burvj ns, j lrjmans rjsles, n` rjafe an` bërbu`e qun knoer sn joustj j `j burvj ns nsn pumte. @j burvjturj sn andfmn beke `j kjlmftua an `j anrfvjaj bem rnspnbte j `j `emlftua an jrbe an umj dumbfòm vnbterfj` tjmlnmtn umftjrfj. Rf `j burvj me ns umj `ëmnj rnbtj, `j anrfvjaj an kfan kfan `j tnmanmbfj an `j tjmlnmtn j bjkifjr sn afrnbbfòm.
‹‹
,, 7 ,, 7 ,, 7
, Ru burvj lnmnrjtrfz ns pnrtnmnbfnmtn j` p`jme SV, y sus rnbtjs
lnmnrjtrfbns, snm lnmnrjtrfbns j` non _. Ru pjrj`n`js burvj lnmnrjtrfz ns pnrtnmnbfnmtn j` p`jme S_, y sus rnbtjs lnmnrjtrfbnss snm pjrj`n`js j` non V. lnmnrjtrfbn , Ru burvj lnmnrjtrfz ns pnrtnmnbfnmtn j` p`jme V_, y sus rnbtjs lnmnrjtrfbnss snm pjrj`n`js j` non S lnmnrjtrfbn
Rupnrdfbfns bujarêtfbjs
^mj supnrdfbfn bujarêtfbj derkj j pjrtfr an umj nbujbfòm ansnsnlumae lrjae bem trns vjrfji`ns S, V, _
Nstjs dflurjs rnspemanm j `j sflufnmtn nxprnsfòm bujarêtfbj lnmnrj`?
Wjrj qun rnprnsnmtn umj supnrdfbfn bujarêtfbj, anin pesnnr `js sflufnmtns bjrjbtnrëstfbjs? Nstjr prnsnmtns `js 2 vjrfji`ns (x, y, z) Rnr umj nxprnsfòm pe`fmòkfbj an snlumae lrjae J` knmes aes an `js vjrfji`ns aninm nstjr n`nvjajs j` bujarjae
9 9 9 7 Aeman? J, I, B….O sem bemstjmtns
^MFAJA M·9? ANYFTJAJR WJYBFJ@NR QNKJ
ANRBYFWBFÒM
Dumbfemns an vjrfjs vjrfji`ns
, umj dumbfòm d an aes Rnj vjrfji`ns ns umj rnl`j qun jsflmj j bjaj pjrnoj eranmjaj nm A um ÿmfbe nm mÿknre rnj` anmtjae . N` bemoumte A ns n` aekfmfe an d y su rjmle ns n` bemoumte an vj`erns qun tekj d 8 ns anbfr,
@ëkftns y bemtfmufaja
A ⊉ ℛ9
DÒYK^@JR
, ,, ,, |,, ∃ }
@ëkftns? Rnj d umj dumbfòm an aes vjrfji`ns buye aekfmfe A bemtfnmn, nmtrn etres, pumtes jriftrjrfjknmtn bnrbjmes j . Nmtembns, n` `ëkftn an , bujmae tfnman j tfnman ns ns @. Bemtfmufaja? Rn afbn qun umj dumbfòm d an aes vjrfji`ns ns bemtfmuj nm së? , d ns
dx,y j,ix,y j,i j,i ,↙, ,, , , `j,fki ℍ, ℍ, ℍ , ↙ ℍ, ,`↙fk ℍ, ℍ , bemtfmuj nm A sf d ns ns bemtfmuj nm teaes
`es pumtes ananA.aes vjrfji`ns, sus Rf d ns ns umj dumbfòm anrfvjajs pjrbfj`ns sem `js dumbfemns , andfmfajs per?
Anrfvjajs pjrbfj`ns
MEQJR Rf tnmnkes , umj dumbfòm an aes vjrfji`ns, Ru lrêdfbe sn andfmn beke n` bemoumte an pumtes (x, y, z) nm tj`ns qun . N` `uljr lnekætrfbe ns ``jkjae supnrdfbfn beke yj sn `e cjiëjm jmtfbfpjae.
2 , ℛ , ,, ↙ ,↙, 9 9 9 , ↙, ≧ ,, ↙ ,`f↙k, ,, ` f k ,, , ↙ , ? ? ⊉ ℛ ↙ ℛ ̎ ̎ ̎ ↙`fk ̎ ↙`fk ̎ ̎ ̎ , , , ,, ,, ,, ,, ,, 9 9
,,Tr, ??ℛℛ↙, Ücχℛ ↙ 9ℛℍ ?ℛÜℛ↙ℛ r,c↙χ9ℍ
Rf bujmae `e `jrle an umj trjynbterfj bujmae `j trjynbterfj aeman nmtembns me nxfstn
@ëkftns?
Rnjm
Bemtfmufaja?
. Rnj
j j y nm , um?
um pumte së y
^. Anbfkes qun d ns bemtfmuj nm
se`e sf?
Rf
, sn nsbrfin?
Wjrj bj`bu`jr umj bemstjmtn rnspnbte j x. Wjrj bj`bu`jr umj bemstjmtn rnspnbte j y.
, bemsfaærnsn j y beke y anrfvn j bem bem , bemsfaærnsn j y beke y anrfvn j bem bem
Fmtnrprntjbfòm lnekætrfbj? Fmtnrprntjbfemns an anrfvjajs pjrbfj`ns
@j pnmafnmtn k an `j tjmlnmtn ns j `j lrêdfbj nm n` pumte `j anrfvjaj an d nm n` pumte j, , sfnmae κ n` êmlu`e derkjae per `j rnbtj tjmlnmtn bem n` non ES.
,
κ
`fk∄↙↙`fk ∄, ∄,ℍ ∄ , , , ∄ ∄∄ , `fk∄↙ , 9 9 99 9999 9 9 99 9999 9 9 ∄ ∄ y V ,∄ ∄ ∄,∄∄ ∄, ∄, ∄ ∄ ∄ , yV ,, ,,
Aeman k ns `j pnmafnmtn an `j rnbtj tjmlnmtn, y befmbfan bem?
Rf
Anrfvjajs pjrbfj`ns an eranm supnrfer
Rf d ns umj dumbfòm an aes vjrfji`ns, nmtembns sus anrfvjajs pjrbfj`ns sem tjkifæm dumbfemns an aes vjrfji`ns, an keae qun sn bemsfanrjm sus anrfvjajs pjrbfj`ns , , y , qun sn ``jkjm snlumajs anrfvjajs pjrbfj`ns an d.
N` fmbrnknmte an umj vjrfji`n x ns n` juknmte e afskfmubfòm qun nxpnrfknmtj, ansan um vj`er j etre an su bjkpe an vjrfjbfòm. Fmbrnknmtes y afdnrnmbfj`ns
, y V ∄,∄ ∄ y V ∄
Rf y sem fmbrnknmtes an S , nmtembns `es afdnrnmbfj`ns an `js vjrfji`ns fmanpnmafnmtns S sem y y `j afdnrnmbfj` tetj` an `j vjrfji`n anpnmafnmtn z ns?
@js anrfvjajs pjrbfj`ns mes fmafbjrêm `j pnmafnmtn an umj rnbtj bembrntj tjmlnmtn j `j supnrdfbfn. @es vj`erns an
, , ,
anmetjm `js pnmafnmtns an `j supnrdfbfn nm `js afrnbbfemns an S y an V, an kjmnrj rnspnbtfvj @j anrfvjaj an eranm supnrfer bekprnman `js anrfvjajs j pjrtfr an `j snlumaj anrfvjaj j kês, y qun sn ndnbtÿj anrfvjmae tjmtjs vnbns beke sn fmafqun, nste tekjmae nm bunmtj rnspnbte j quæ vjrfji`n vjkes j anrfvjr, tekjmae j `js etrjs beke bemstjmtns
Fmbrnknmtes? Rnjm , fmbrnknmtes an S , nmtembns n` fmbrnknmte an z vfnmn ajae per? Afdnrnmbfj`ns? Rf y y sem fmbrnknmtes an S , nmtembns `es afdnrnmbfj`nss an `js vjrfji`ns fmanpnmafnmtns S n V ssem? afdnrnmbfj`n em? N` afdnrnmbfj` tetj` an `j vjrfji`n anpnmafnmtn z ns?
Bujmae sn bemsfanrj umj bjmtfaja vjrfji`n qun brnbn per lrjaes fmdfmftjknmtn pnqunúes y sn ansnj bemebnr n` vj`er an `es kfskes, `e qun sn cjbn ns antnrkfmjr n` vj`er an nsj bjmtfaja pjrj um fmstjmtn buj`qufnrj y n` vj`er an nsj kfskj bjmtfaja pjrj n` fmstjmtn fmknafjtjknmtn sflufnmtn8 nmtembns, `j afdnrnmbfj an nstes vj`erns ns n` fmbrnknmte (e n` anbrnknmte) qun rnbfin nsj bjmtfaja? j nste ns j `e qun sn ``jkj afdnrnmbfj` an nsj bjmtfaja.
Rnj Ynl`j an `j bjanmj
@j anrfvjbfòm per `j rnl`j an `j bjanmj sn jp`fbj bujmae iusbjkes anrfvjr umj bekpesfbfòm an dumbfemns. Rf tnmnkes umj dumbfòm bekpunstj an `j derkj? , puns su anrfvjaj ns?
, aeman d ns umj
dumbfòm afdnrnmbfji`n an S l .y Rf n y , sfnmae c dumbfemns an t, nmtembns w ns umj dumbfòm anrfvji`n an t.
,ℍ y V ,, 7
,, ,, , ,, ≧ 7 7 y V ,, , , , , , , ,,,,,, ,,,,,, , ,,≧7 , , , ↙`fk,, ,, ,, , ,, ,, ,, , ∌∌,, ,, ,⃜∌ ,, ∖,∖⃜ ∌∌,, Rf `j nbujbfòm andfmn j V fkp`ëbftjknmtn beke dumbfòm an S, nmtembns?
Rf `j nbujbfòm
andfmn j V fkp`ëbftjknmtn beke dumbfòm an S?
Anrfvjbfòm fkp`ëbftj
Rf `j nbujbfòm andfmn j _ fkp`ëbftjknmtn beke dumbfòm an S , fkp`ëbftjknmtn nmtembns?
Rf `j nbujbfòm
andfmn j _ fkp`ëbftjknmtn beke dumbfòm an S n V?
N` vnbter lrjafnmtn kjrbjrê `j afrnbbfòm an kêxfkj vjrfjbfòm an `j dumbfòm nm buj`qufnr pumte.
Anrfvjaj afrnbbfemj`? Rnj d umj dumbfòm
an aes vjrfji`ns S n V, snj um vnbter umftjrfe. Nmtembns `j anrfvjaj afrnbbfemj` an d nm `j afrnbbfòm an u sn anmetj per . Tnbter lrjafnmtn? Rf , nmtembns n` lrjafnmtn an d, qun sn anmetj knafjmtn .
@j anrfvjaj fkp`ëbftj an umj dumbfòm fkp`ëbftj sn eitfnmn anrfvjmae `j dumbfòm, anspuæs an anspnojr `j vjrfji`n y, qun ns `j qun sn bemsfanrj vjrfji`n anpnmafnmtn, bemsfanrjmae qun ns dumbfòm an x. ^mj dumbfòm fkp`ëbftj ns jqun``j qun `j vjrfji`n anpnmafnmtn me nstê anspnojaj.
Anrfvjaj afrnbbfemj`
Anrfvjajs afrnbbfemj`ns y n` vnbter lrjafnmtn
Nm afrnbbfòm an` vnbter umftjrfe?
Rf
Tnbter Lrjafnmtn
@j anrfvjaj ns n` n`preaubte nsbj`jr an` afrnbbfemj` lrjafnmtn per vnbter umftjrfe qun antnrkfmj `j afrnbbfòm.
W`jmes tjmlnmtns y rnbtjs merkj`ns
ℛ , , ∖ , , ,,7 ∬ ,,,,7 ̎ ⃜∌ , :,7,, 9 :,7,̎ , , , , , , ℛ ,,,,,, ≮ ≮ ,,, ,,, ,, ≡ , , , ,, ,,,,,≮≮≡ , ,, , , , ? ⊉ℛ ↙ℛ ∃ , y V ℛ ℛℛ Bujmae sn fmtnrprntj an kjmnrj lnekætrfbj `js anrfvjajs pjrbfj`ns, sn andfmn qun um vnbter afrnbtrfz an `j rnbtj tjmlnmtn pjrj`n`j j` p`jme _S, nm um pumte an `j supnrdfbfn 8 y um nstê ajae per vnbter afrnbtrfz an `j rnbtj tjmlnmtn pjrj`n`j j` p`jme _V nstê ajae per?
^mj dumbfòm an aes vjrfji`ns tfnmn um kêxfke rn`jtfve nm sf
nstê rnbfin bnrbj an . N`bujmae mÿknre n` mekirn an vj`er kêxfke rn`jtfve. Rf bujmae nstê bnrbj an , nmtembns ns um ns vj`er këmfke rn`jtfvj.
Tj`erns kêxfkes y këmfkes
Rf
W`jme tjmlnmtn j R nm ns umj supnrdfbfn s upnrdfbfn afdnrnmbfj afdnrnmbfji`n i`n nm
Nbujbfòm an `j rnbtj merkj`
@j rnbtj merkj` e derkj sfkætrfbj snrëj?
Rnj d umj umj dumbfòm andfmfaj nm umj rnlfòm qun bemtfnmn ( )
d @j dumbfòm tfnmn um këmfke rn`jtfve nm
sf
Wjrj teae nm um afsbe nm afs be jifnrte qun bemtfnmn @j dumbfòm d tfnmn um kêxfke rn`jtfve nm sf sf Wjrj teae nm um afsbe jifnrte qun bemtfnmn nm Tj`er nxtrnke Rnj , snj y snj ) um pumte Qnernkj an` Tj`er Nxtrnke? pnrtnmnbfnmtnn j `j dumbfòm, sn afbn qun? Rnj d umj dumbfòm bemtfmuj an aes pnrtnmnbfnmt vjrfji`ns S andfmfaj nm umj rnlfòm W ns um kêxfke jise`ute an d, sf pjrj teae x afstfmte an jbetjaj bnrrjaj nm n` p`jme SV? Nxfstn per `e knmes um pumte nm nm n` qun d tekj um vj`er këmfke. Nxfstn per `e knmes um pumte nm nm n` qun d tekj um vj`er kêxfke.
Rn ``jkj p`jme tjmlnmtn j umj supnrdfbfn sup nrdfbfn nm um pumte W an a n `j kfskj, j` p`jme qun bemtfnmn teajs `js tjmlnmtns j `js burvjs trjzjajs seirn `j supnrdfbfn per n` pumte W. Rn ``jkj rnbtj merkj` j umj supnrdfbfn j `j rnbtj qun pjsj per um pumte W y ns pnrpnmafbu`jr j` p`jme tjmlnmtn.
pnrtnmnbfnmtn pnrtnmnbfnmt n j` suibemoumte J, su fkjlnm ns knmer e fluj` qun `j an W ns um këmfke jise`ute an d, sf pjrj teae x afstfmte an pnrtnmnbfnmtnn j` suibemoumte J, su fkjlnm ns kjyer e fluj` qun pnrtnmnbfnmt `j an
@es kêxfkes y këmfkes an umj dumbfòm an aes vjrfji`ns kfanm j`tftuans kêxfkjs y këmfkjs seirnan `j `j supnrdfbfn qun bemstftuyn `j lrêdfbj dumbfòm (sem beke `js betjs an` pumte kês n`nvjae an umj be`fmj e an` pumte kês predumae an umj cemaemjaj). @es nxtrnkes an umj dumbfòm, sem `es vj`erns kês lrjmans (kêxfkes) e kês pnqunúes (këmfkes), qun tekj umj dumbfòm nm um pumte sftujae yj snj anmtre an umj rnlfòm nm pjrtfbu`jr an `j burvj (nxtrnke `ebj`) `ebj`) e nm n` aekfmfe an `j dumbfòm nm su tetj`faja (nxtrnke l`eij` e jise`ute).
^mfaja 2? Fmtnlrj`ns kÿ`tfp`ns QNKJ
ANRBYFWBFÒM
DÒYK^@JR
MEQJR
Rnj d umj umj dumbfòm an aes vjrfji`ns
↙`f,,k√Ü;,,√}; ,, /∄≡∄≡ ∬ ≡≡ ↙↙ .
Fmtnlrj`ns Aei`ns
Nvj`ujbfòm an fmtnlrj`ns aei`ns
J` sn `n
anmekfmj `j fmtnlrj` aei`n an d nm Y. Jankês, sf nxfstn nstn `ëkftn anbfkes qun d ns ns fmtnlrji`n nm Y. Rnj d umj dumbfòm an aes vjrfji`ns andfmfaj nm `j rnlfòm p`jmj
Ynj`fzjr umj fmtnlrj` aei`n bemsfstn nm rnj`fzjr aes fmtnlrj`ns sfku`têmnjs, umj nm prfknr `uljr nm dumbfòm an x (e an y), bemsfanrjmae qun `j y (e `j x), ns umj bemstjmtn8 y nm snlumae `uljr sn prebnan j fmtnlrjr n` rnsu`tjae nm dumbfòm an` tærkfme dj`tjmtn, (nm nstn bjse yj me nxfstfrê mfmlÿm tærkfme an` buj` sn cj fmtnlrjae nm prfknrj fmstjmbfj).
,, ,, ,, ,, ,, ,, ,, ,, Fmtnlrj`ns aei`ns
Rf d ns bemtfmuj nm Y nmtembns?
,, Ü ,, } , /≡≡ ∬ ≡≡ 9,, 9 ,, ≮≮ , 7 ℍ9 ℍ ℍ9 ℍ ,, ∠ ,, 9 9 9 , , Rnj Y umj rnlfòm an tfpe F bekprnmafaj nmtrn `js lrêdfbjs an y , sem bemtfmujs , aeman nm . Rnj Y umj rnlfòm tfpe FF bekprnmafaj nmtrn `js lrêdfbjs an y , aeman sem bemtfmujs nm
N` vj`er qun sn eitfnmn j` rnse`vnr `j fmtnlrj` ns um mÿknre rnj` qun sn utf`fzj u tf`fzj pjrj antnrkfmjr n` vj`er an` ve`uknm an` sò`fae sftujae an derkj vnrtfbj` seirn n` rnbtêmlu`e Y an` p`jme ESV ijoe `j supnrdfbfn , sfnkprn y bujmae . Bjin rnbj`bjr qun, tjkifæm peankes rnj`fzjr fmtnlrj`ns aei`ns
Ynlfòm an tfpe F? Rf d ns ns bemtfmuj nm Y, nmtembns?
Ynlfòm an tfpe F? Rf d ns ns bemtfmuj nm Y, nmtembns?
Rf sn ansnj nmbemtrjr `j fmtnlrj` aei`n
Fmtnlrj`ns aei`ns nm beeranmjajs pe`jrns
an , aeman Y ns umj rnlfòm pe`jr, y j` nmbemtrjr `es `ëkftns an fmtnlrjbfòm bem rnspnbte j , nstes j` rnj`fzjr `j fmtnlrj` vun`vnm bekp`noj su antnrkfmjbfòm, nmtembns, bem `j dfmj`faja an sfkp`fdfbjr nstn
Fmtnlrj` aei`n nm derkj pe`jr , ), ,
seirn rnbfmtes qun me snjm rnbtêmlu`es Wjrj nmbemtrjr umj fmtnlrj` nm beeranmjajss pe`jrns sn anin? beeranmjaj Nxprnsjr `j rnlfòm nm n` sfstnkj pe`jr, y antnrkfmjr `es `ëkftns an fmtnlrjbfòm. Rustftufr nm `j dumbfòm fmtnlrjmae `js beeranmjajss pe`jrns per su nqufvj`nmtn beeranmjaj nm beeranmjajs pe`jrns
prei`nkj sn nxprnsj `j rnlfòm beke umj rnlfòm pe`jr y sn antnrkfmjm `es `ëkftns an fmtnlrjbfòm nm nstn sfstnkj. Êrnj? Bemsfanrjmae umj rnlfòm p`jmj jbetjaj per ,y , nstj rnlfòm Y nstê bekpunstj per fmdfmftes rnbtêmlu`es (vnrtfbj`ns) nm ax. Jmê`eljknmtn, Jmê`eljknmtn, nm umj rnlfòm p`jmj jbetjaj per b , y per , nstj rnlfòm Y nstê bekpunstj per fmdfmftes rnbtêmlu`es (cerfzemtj`ns) (cerfzemtj`n s) nm ay. Te`uknm? Rnj d y l aes dumbfemns an aes vjrfji`ns andfmfajs y bemtfmujs nm `j rnlfòm p`jmj Y tj`ns qun . Rnj T n` ve`uknm an` sò`fae Jp`fbjbfòm an `js jbetjae supnrferknmtn per `j lrêdfbj fmtnlrj`ns aei`ns? Êrnj, ve`uknm, an `j dumbfòm l y jbetjae fmtnrferknmtn per `j lrêdfbj an `j dumbfòm d. kjsj, ntb. Kjsj? Rf sn nsbeln um pumte jriftrjrfe qun pnrtnmnzbj j Y, nmtembns `j kjsj an nstn sui-rnbtêmlu`e sn andfmn beke?
9 ≡≡ ≡≡ ≡ ℍ9 ≡≡ ℍ≡ , ≡ ≡ , . Rf snan juknmtj n` mÿknre an suifmtnrvj`es, keae qun `j merkj an `j pjrtfbfpjbfòm an W tfnmaj j bnre, sn tfnmn?
, ∄ `fk↙ ; ; , ∄
Ynnkp`jzjr n` afdnrnmbfj` an êrnj per su nqufvj`nmtn nm beeranmjajs pe`jrns. Nvj`ujr `j fmtnlrj` rnsu`tjmtn.
Êrnj Bujmae `js dumbfemns nmtrn `js qun sn andfmn n` êrnj vfnmnm ajajs nm dumbfòm an x, n` rnbtêmlu`e rnprnsnmtjtfve an` êrnj snrê vnrtfbj` y per tjmte n` eranm an fmtnlrjbfòm snrê . Wer etre `jae, sf `js dumbfemns vfnmnm ajajs nm dumbfòm an «y¸, n` rnbtêmlu`e rnprnsnmtjtfve rnprnsnmtjtfv e snrê cerfzemtj` y n` eranm an fmtnlrjbfòm snrê .
Êrnj bem ijrrfae vnrtfbj`?
ayax axay
Êrnj bem ijrrfae cerfzemtj`?
Te`uknm an um sò`fae nm n` nspjbfe?
,, ,
Kjsj an umj dflurj p`jmj?
, )∄ , `fk↙ ( ; ; ,
Te`uknm @e qun mes pnrkftn `j fmtnlrj` aei`n ns qun j` prebnanr j fmtnlrjr `es ve`ÿknmns fmdfmftnsfkj`ns an um sò`fae sn ``nlj j eitnmnr n` ve`uknm an teae n` sò`fae (nste ns `e qun cjbn `j fmtnlrj` nxtnrfer). Kjsj Kjsj @j fmtnlrj` aei`n sn usj pjrj antnrkfmjr `j kjsj an umj dflurj p`jmj me cekelæmnj, an êrnj Y, ns anbfr pjrj rnlfemns aeman `j anmsfaja vjrëj nm bjaj pumte.
Bnmtre an kjsj? @js beeranmjajs an` bnmtre an lrjvnaja e bnmtre an kjsj an umj dflurj p`jmj Y sn eitfnmnm j
̎ , 2 ℛ ≡≡,≡≡,≡≡ ≡≡,≡≡,≡≡} ,, Ü ,, Ü ,, ,,,/} ,,,, ̎, , ̎∄∄∄∄ 9,, ,, 9,,,, ,,,, trjvæs an
Rnj d umj dumbfòm an trns vjrfji`ns andfmfajs nm umj rnlfòm an ,
Fmtnlrj`ns trfp`ns
^mj dumbfòm an trns vjrfji`ns andfmfaj nm Z, pjrj qun nstj andfmfaj pjrtfbfòm snrëj an derkj?
Jp`fbjbfòm an fmtnlrj`ns trfp`ns
Ynlfòm tfpe F Rn afbn qun umj rnlfòm sò`faj N ns an tfpe : sf nstê nmtrn `js lrêdfbjs an aes dumbfemns bemtfmujs an , ns anbfr? Aeman A ns `j preynbbfòm an N seirn seirn n` p`jme . Qnmfnmae nm bunmtj qun n` `ëkftn supnrfer n fmdnrfer an `j nbujbfòm sem? , rnspnbtfvjknmtn. Wer `e qun sn kjmtfnmnm dfojs y sem bemsfanrjajs beke bemstjmtns, kfnmtrjs qun sn fmtnlrj rnspnbte j _.
Bnmtre an kjsj
Bnmtre an Kjsj @js fmtnlrj`ns aei`ns mes pnrkftnm bj`bu`jr tjmte `j kjsj an umj p`jbj p`jmj beke an `es keknmtes nstêtfbes an` sò`fae.
̎ : ,, ,: : ,, `fk ; ↙∝ ↙∝ ; ; ̎, , ̎∄∄∄∄ ∫↙∝,,, ,,, ,, ,
Wjrj teae Z, cjy qun bemsfanrjr umj bjmtfaja an pjrtfbfpjbfemns, pjrtfbfpjbfem ns, ns anbfr?
Wrepfnajans Qeaj dumbfòm bemtfmuj ns fmtnlrji`n Wesnn `fmnj`faja, kemetemëj, jaftfvfaja
Y ns j`lumj rnlfòm nm n` nspjbfe
trfafknmsfemj`. ns j`lumj dumbfòm bem vj`erns
nsbj`jrns qun tfnmn beke nmtrjaj pumtes nm n` nspjbfe trfafknmsfemj trfafknmsfemj`.`. ns umj umfaja an ve`uknm kuy pnqunúj.
Fmtnlrj`ns Qrfp`ns
Rn usj umj fmtnlrj` trfafknmsfemj` bjaj Ynlfòm an tfpe F?
vnz qun sn tnmlj qun ‐anspnajzjr‛ umj
, ∫ ,,, ,, ,,, ∫ ,,, , ,,,
rnlfòm trfafknmsfemj` nm fmdfmftes pnajzes, jsebfjr bjaj pnajze bem um vj`er y `unle sukjr teae. Nste ns ÿtf` pjrj nmbemtrjr n` ve`uknm an rnlfemns trfafknmsfemj`ns j` sukjr teaes `es ‐kfmf‛ ve`ÿknmns.
Ynlfòm an tfpe FF?
Ynlfòm tfpe FF Rem jqun``js supnrdfbfns aeman A ns `j preynbbfòm an N seirn seirn n` p`jme . @j supnrdfbfn pestnrfer y `j dremtj` sem Ynlfòm tfpe FFF Rem jqun``js aeman A ns `j preynbbfòm an N seirn n` p`jme , ns `j supnrdfbfn fzqufnraj y ns `j supnrdfbfn anrnbcj.
,, 9,, yz z ,9,
Ynlfòm an tfpe FFF?
∫ ,,, ,, ,,,
Bujmae fmtnlrjs seirn umj rnlfòm trfafknmsfemj` Y, ns ÿtf` fkjlfmjr qun `j bertjs nm fmdfmftes pnajzes fmdfmftjknmtn pnqunúes, bjaj ume bem ve`uknm . Bujmae sn trjijoj bem beeranmjajs bjrtnsfjmjs, nstes pnqunúes pnajzes sem bemsfanrjaes beke i`equns rnbtjmlu`jrns. rnbtjmlu`jrn s. Nm beeranmjajs nsdærfbjs, ns ÿtf` pnmsjr nm nstes pnajzes pnqunúes beke i`equns `flnrjknmtn
Ynberankes
Fmtnlrj`ns trfp`ns nm beeranmjajs bf`ëmarfbjs
∟ 9jm9 ,,, {jrbt
ℛ2 ?≡≡ ∫ ,,, ≡≡ ,, ≡≡ 9,,ℍ,,?≡≡ ,,, ,, ℍ9,,}}, ,x,y,,z
Rf umj rnlfòm R an tfnmn um non an sfkntrëj, `js fmtnlrj`ns trfp`ns sem kês dêbf`ns an nvj`ujr sf sn cjbn um bjkife an beeranmjajs an bjrtnsfjmjs j bf`ëmarfbjs. Rnj
`j
an
Nmtembns?
fmtnlrjbfòm.
Bemsfanrjkes d umj dumbfòm bemtfmuj nm R an trns vjrfji`ns , an keae qun sem `js beeranmjajs bf`ëmarfbjs an` pumte
burvnjaes qun nsdnrj. Bem n` dfm an qun"jirjzjm" bjaj bjrj umj an` pnajze rnprnsnmtn um vj`er bemstjmtn pjrj umj an `js beeranmjajs nsdærfbjs.
Bjkife an Tjrfji`n nm umj fmtnlrj` trfp`n
, ∫ ∫, ∫,,,,, ,,,, ,, N` ojbeifjme
Bjkife an vjrfji`ns nm fmtnlrj`ns kÿ`tfp`ns
sn eitfnmn beke? sn
ℛ2 ↙ ℛ ⊉ ℛ2 ,, ,,, ∃ ‹,‹,,⊉,,ℛ,‹2,∃,,‹ ⊉ℛ 2,,, ,, X [ [ ∫, ∫ ,,,, ∇,,, ∃ ‹ ,
Rnj d ? umj dumbfòm bemtfmuj, andfmfaj nm `j rnlfòm . Rnj Q umj dumbfòm fmynbtfvj qun trjmsderkj `es nm pumtes , knafjmtn `j nxprnsfòm . Rupemlj qun Q ns an b`jsn y qun `j anrfvjaj ns umj kjtrfz fmvnrsfi`n
Bjkife j beeranmjajs bf`ëmarfbjs
Bjkife j beeranmjajs nsdærfbjs
∫ ,,∫, ,,, 9
Nstn tnernkj mes pnrkftn, bem n` dfm an nvj`ujr umj fmtnlrj` andfmfaj an umj dumbfòm rnj` an vjrfji`n rnj` nm um fmtnrvj`e bnrrjae , bjkifjr `j vjrfji`n an fmtnlrjbfòm bem `j dfmj`faja an rnse`vnr afbcj fmtnlrj` an umj kjmnrj kês snmbf``j y nm su kjyerëj bem um prebnse kês rnaubfae.
,,
^MFAJA M·1 BÊ@B^@E TNBQEYFJ@ QNKJ
Bjkpes Tnbterfj`ns
Lrjafnmtn, afvnrlnmbfj retjbfemj` y
ℛ9 ℛ9
ANRBYFWBFÒM Rnj A um suibemoumte nm (rnlfòm p`jmj) um bjkpe vnbterfj` seirn , ns umj dumbfòm D qun qun jsflmj j bjaj pumte an A um vnbter an 9 afknmsfemns . Rnj N um um suibemoumte an um bjkpe vnbterfj` seirn ns umj dumbfòm D qun qun jsflmj j bjaj pumte nm N , um
DÒYK^@JR Bjkpe Tnbterfj`
,, ,, ,, ,,, ⃜∌ ,, ,, ,,
MEQJR ^m bjkpe vnbterfj` ns um afjlrjkj qun kunstrj `j kjlmftua y `j afrnbbfòm an `es vnbterns (vn`ebfajans, dunrzjs) nm afdnrnmtns `uljrns an` nspjbfe. ^m bjkpe vnbterfj` ns umj dumbfòm qun jsflmj j bjaj pumte um vnbter?
x,y Dx,y 2 ? ⊄ ↙ ℛ ℛ2 ,,, , 9, 2 ∃ , 9, 2, . , , , , , , , , , , , , , , , , , x, y , z x, y , z ∌∌,, ,, ,, ∌ , ∌∌,,, ,,, ,,, ,,, , ,,, ,, ∌∖,, ⌦ , ⌯ ∖ ⌦,, ⌯ ,, ∌∖,, ,,, ⃜ ⌦ , , , , , , ∌∖ , , ⌯ ∖ ⌦,,,⌯ ,,, ∌∖,,,
vnbter an aes trfafknmsfemj` . Bjkpe Lrjafnmtn? Rf d ns umj dumbfòm nsbj`jr an aes vjrfji`ns, su lrjafnmtn, (e lrja d ), ), sn andfmn beke `js anrfvjajs pjrbfj`ns an `j dumbfòm. An` kfske keae, sf d ns umj dumbfòm nsbj`jr an trns vjrfji`ns, su lrjafnmtn sn andfmn beke `js anrfvjajs pjrbfj`ns an `j dumbfòm.
Bjkpe lrjafnmtn pjrj dumbfemns an aes vjrfji`ns?
N` lrjafnmtn ns umj dumbfòm an vj`er vnbterfj`, j afdnrnmbfj an umj anrfvjaj, qun ns umj dumbfòm an vj`er nsbj`jr. J` fluj` qun `j anrfvjaj, n` lrjafnmtn rnprnsnmtj `j pnmafnmtn an `j `ëmnj tjmlnmtn j `j lrêdfbj an umj dumbfòm.
Bjkpe lrjafnmtn pjrj dumbfemns an trns vjrfji`ns?
@j afvnrlnmbfj kfan `j afdnrnmbfj nmtrn n` d`uoe sj`fnmtn y n` d`uoe nmtrjmtn an um bjkpe vnbterfj` seirn `j supnrdfbfn qun reanj j um ve`uknm an bemtre`, per tjmte, sf n` bjkpe tfnmn ‐dunmtns‛ `j afvnrlnmbfj snrê pesftfvj, y sf tfnmn ‐sukfanres‛, `j afvnrlnmbfj snrê mnljtfvj.
Afvnrlnmbfj an um bjkpe vnbterfj` nm n` p`jme
Afvnrlnmbfj? Ajae n` bjkpe vnbterfj` nm n` p`jme , nmtembns `j afvnrlnmbfj nm n` p`jme ns? Ajae n` bjkpe vnbterfj` nm n` nspjbfe , nmtembns su afvnrlnmbfj nm n` nspjbfe ns?
Afvnrlnmbfj an um bjkpe vnbterfj` nm n` p`jme
Yetjbfemj` an um bjkpe vnbterfj` Yetjbfemj`? Ajae n` bjkpe vnbterfj` , nmtembns su retjbfemj` snrê?
,,,,,, ,⃜ ,, ∌ Ü
∌ Ü,,, X ⃜[ ⃜ ⃜ ∖ ‹ ?? , , ,⃜,‹ , , ‹ , ∖ ,, ,, ∖ , ∖ ‹ ∖⃜
N` retjbfemj` ns um epnrjaer qun kfan `j retjbfòm nm n` kevfkfnmte an um d`ufae ansbrfte per um bjkpe vnbterfj` an trns afknmsfemns.
Derkj Afdnrnmbfj`? Nstjs fmtnlrj`ns an `ëmnj `js peankes fmtnrprntjr beke n` trjijoe qun tfnmn qun rnj`fzjr um bjkpe j` ansp`jzjr umj pjrtëbu`j seirn `j burvj B , sf anmetjkes j` trjijoe beke \ , nmtembns? Fmtnlrj`ns an `ëmnj
∨ ∖⃜ ,,, ,,, ,,∨≡,,≡∖⃜ ( ( ) ) , ,
. Bemnxfòm bem fmtnlrj`ns an `ëmnj? Rnj D tj` qun, y . Rf ,
nmtembns pjrj nvj`ujr andfmfkes?
,
Fmtnlrj` an `ëmnj. Derkj afdnrnmbfj`?
Rupemljkes qun
y qun y
), nmtembns ), Ynnkp`jzjmae
Bemnxfòm bem fmtnlrj`ns an `ëmnj?
@js fmtnlrj`ns an `ëmnj mes sem ÿtf`ns nm dësfbj j` keknmte an bj`bu`jr n` trjijoe qun rnj`fzj umj dunrzj seirn um eionte nm kevfkfnmte. Rf sn ``nlj j pjrjkntrfzjr `js burvj an tj` derkj qun tn kunvjs nm `j afrnbbfòm epunstj bemderkn t brnbn, n` vj`er an `j fmtnlrj` an `ëmnj sn ku`tfp`fbj per ∙: .
∨ ∖ ⃜ ∌∌,, ., 9 ⃜ ∖⃜ ∖⃜ ∖⃜ , ∖⃜ 7 Fmanpnmanmbfj an `j trjynbterfj
Bemsfaærnsn `j fmtnlrj` an `ëmnj
, seirn `j trjynbterfj B an um bjkpe
Fmanpnmanmbfj an `j trjynbterfj
vnbterfj` qun sn eitfnmn an` lrjafnmtn an um bjkpe nsbj`jr, Rn afbn qun ns um bjkpe bemsnrvjtfve. Nm tj` bjse?
Rf ns um bjkpe bemsnrvjtfve seirn umj burvj bnrrjaj
Qnernkj an Lrnnm pjrj rnlfòm Y an tfpe F?
≡≡ 9,,≡≡ ? ,, ≡≡ , , 9 ), 9 ,, (, ) ,
Qnernkj an Lrnnm
Rupemlj qun B ns umj burvj bnrrjaj sfkp`n sujvn per pjrtns bem umj erfnmtjbfòm pesftfvj qun `fkftj umj rnlfòm sfkp`nknmtn bemnxj Y. Rf sem bemtfmujs bemtfmujs seirn Y,
,, ,, ,,
nmtembns?
Qnernkj an Lrnnm pjrj rnlfòm Y an tfpe F?
≡≡ ? ℍ ≡≡ℍ9,,≡≡ ℍ , ℍ , 9 , ,, ℍ9, ℍ ℍ
Rf tnmnkes umj burvj B bnrrjaj bnrrjaj y nm n``j sn cjbn aes bertns, andfmfnmae aes pumtes. Rf jprnbfjs n` lrêdfbe rnsu`tjmtn nmtembns sn eitfnmnm aes trjynbterfjs buyj fmtnlrjbfòm an teaj `j burvj ns `j sukj an `js fmtnlrj`ns an `js trjynbterfjs. Wer etre `jae, sf tnmnkes um bjkpe bemsnrvjtfve nmtembns `js fmtnlrj`ns an `js trjynbterfjs sem fluj`ns y `j fmtnlrj` an `ëmnj aj bnre. Nstn tnernkj rn`jbfemj umj fmtnlrj` an `ëmnj j `e `jrle an umj burvj bnrrjaj sfkp`n nm n` p`jme bem umj fmtnlrj` aei`n nm `j rnlfòm nmbnrrjaj per B. N` tnernkj an Lrnnm se`e ns jp`fbji`n j burvjs qun nstêm erfnmtjajs nm snmtfae bemtrjrfe j `js kjmnbf``js an` rn`eo. Rf nstês fmtnlrjmae nm afrnbbfòm an `js kjmnbf``js an` rn`eo j`rnanaer an `j burvj y qufnrns jp`fbjr n` tnernkj an Lrnnm, tfnmns qun fmvnrtfr n` sflme an tu rnsu`tjae nm j`lÿm keknmte
Fmtnlrj`ns an supnrdfbfn nm bjkpes nsbj`jrns? Rnj R supnrdfbfn ajaj per fmynbtfvj y snj tj` qun ns
?⊉ ?∠ 2↙9,,↙∖2∖ Ü ((∠,, ∖) 9 2 ? 2?⊉ ↙ 2 ↙ ∠ ((,, ) ∖ Ü ∠ ∖ bemtfmuj.
Fmtnlrj`ns an supnrdfbfn an bjkpes vnbterfj`ns? Rnj R supnrdfbfn ajaj fmynbtfvj y snj per bemtfmuj seirn R. Nmtembns?
Qnernkj an `j Afvnrlnmbfj
Rnj T umj rnlfòm sò`faj jbetjaj per umj supnrdfbfn bnrrjaj R erfnmtjaj per um vnbter umftjrfe merkj` qun jpumtj cjbfj n` nxtnrfer an T . Rf D ns um bjkpe vnbterfj` bem anrfvjajs pjrbfj`ns bemtfmujs nm T, nmtembns?
∖ ∫∌∖ ∫∌∖
∖ ∖ ⃜m 9 9 9 ∟++1
Rf sn bemebn um vnbter umftjrfe merkj` j `j supnrdfbfn m
Nmtembns?
Fmtnlrj`ns an Rupnrdfbfn
Fmtnlrj` an supnrdfbfn
Aeman
punan snr an aes derkjs? Derkj fkp`ëbftj
||| 1 ∟ 9 999 :
Derkj nxp`ëbftj
⃜ |⃜⃜|
Rn trjtj an antnrkfmjr n` ndnbte an dumbfemns vnbterfj`n vnbterfj`nss jtrjvnsjmae umj supnrdfbfn R, pjrj nste sn nkp`njrê fmtnlrj`ns an supnrdfbfn. Nstj fmtnlrj` ns ``jkjaj ‐Fmtnlrj` an d`uoe‛. @j fanj an `j fmtnlrj` an supnrdfbfn ns `j kfskj qun `j an `js fmtnlrj`ns aei`ns, nxbnpte qun nm vnz an "sukjr" pumtes nm umj rnlfòm p`jmj an aes afknmsfemns, sukjrês pumtes nm umj supnrdfbfn nm n` nspjbfe, qun ns petnmbfj`knmtn burvj.
Tnbter merkj`fzja merkj`fzjae e
Qnernkj an `j afvnrlnmbfj
∥ ∖̉ ∫ ∫∌∖ ∫∌∖
∌∖ ,, ,, ,,
@j afvnrlnmbfj kfan n` d`uoe cjbfj jdunrj nm pumtes fmafvfauj`ns, kfnmtrjs qun n` d`uoe kfan `j bjmtfaja an d`ufae qun sj`n an teaj umj rnlfòm. Jsë qun j` sukjr teaes `es ‐bjbcftes ‛ an afvnrlnmbfj sn eitfnmn n` kfske vj`er qun n` d`uoe. J afdnrnmbfj an` tnernkj an Rtehns, n` tnernkj an `j afvnrlnmbfj se`e sn jp`fbj j supnrdfbfns bnrrjajs, `e qun sflmfdfbj supnrdfbfns sfm dremtnrjs.
Rnj R umj supnrdfbfn erfnmtjaj sujvn per pjrtns jbetjaj per umj burvj B bnrrjaj sfkp`n sujvn per pjrtns. Rnj?
,,, ,,, Qnernkj an Rtehns
um bjkpe vnbterfj` pjrj n` buj` W, Z, Y sem bemtfmujs y tfnmnm prfknrjs anrfvjajs pjrbfj`ns bemtfmujs nm umj rnlfòm jifnrtj an` nspjbfe trfafknmsfemj` qun bemtfnmn j R. Rf B sn rnberrn nm `j afrnbbfòm pesftfvj, nmtembns?
Aeman m ns umj merkj` umftjrfj j R nm `j afrnbbfòm an `j erfnmtjbfòm an R.
∠∭ ∖ ∖∭ ∖
Qnernkj an Rtehns
∖ ⃜ ∖ ∖ ∌Ü ∌Ü ,,, X [ ⃜ ,9,9:⌯: ⃜|∌|∌∌| ∟⌦, Ynberankes
Tnbter Merkj` ‐m‛ nstê ajae per
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