
3dee rotation samooh

yantriki aur jyaamiti mein, 3dee ghoornan samooh, jise aksar (3) key roop mein darshaaya jaataa haye, sanrachana key sanchaalan key tahat tri-ayami euclidean antariksh sootra _ 1 kii utpatti key baare mein sabhi ghurnanon kaa samooh haye.
paribhasha key anusaar, utpatti key baare mein ack ghoornan ack parivartan haye joe utpatti, euclidean doori (isaliye yeh ack samamitita haye), aur abhivinyas (i.ee., sthaan kii "hastata"). dou ghurnanon kii rachana key parinamswarup ack aur ghoornan hotaa haye, pratyek ghoornan mein ack advitiya vyutkram ghoornan hotaa haye, aur pehchaan maanchitra ack ghoornan kii paribhasha koo santusht kartaa haye. uparokt gunon key kaaran (samagr ghoornan key sahayogi guna key saath), sabhi ghurnanon kaa samooh sanrachana key tahat ack samooh haye.
pratyek gair-tuchchh ghoornan isake ghoornan kii dhuri (mool key madhyam seey ack rekha) aur isake ghoornan konn seey nirdhaarit hotaa haye. ghoornan parivartaniya naheen hotey hain (udaaharan key liye, x-y tala mein "r" 90° aur usake baad y-z tala mein "s" 90° ghoomana "s" key baad "r" key samaan naheen hotaa haye), jisase 3d ghoornan samooh ack gair-abelian samooh bunn jaataa haye. isake alawa, ghoornan samooh mein ack praakritik sanrachana hoti haye jisake liye samooh sanchaalan sucharu roop seey avakalaniya hotey hain, isaliye yeh vaastav mein ack jhooth samooh haye. yeh compact haye aur iska ayam 3 haye.
ghoornan sootra _ 1 key raikhik parivartan hain aur isaliye sootra _ 1 key aadhar koo chune jaane key baad matricks dwara darshaaya jaa sakata haye. vishesh roop seey, yadi humm sootra _ 1 kaa ack orthonormal aadhar chunate hain, tou pratyek ghoornan koo ack orthogonal 3 × 3 matricks (i.ee., vaastavik pravishtiyon key saath ack 3 × 3 matricks, jise jaba isake pargaman seey gunaa kiya jaataa haye, tou nirdhaarak 1 key saath pehchaan matricks mein parinam hotaa haye. isaliye samooh koo (3) matricks gunan key tahat inn matricks key samooh key saath pahachaana jaa sakata haye. inn matricks koo "vishesh orthogonal matricks" key roop mein jaanaa jaataa haye, joe sanketan koo samajhaate hain tou (3).
samooh kaa upayog kissy vastu kii sambhaavit ghoornan samroopata key saath-saath antariksh mein kissy vastu key sambhaavit abhivinyas kaa varnan karane key liye kiya jaataa haye. isake pratinidhitva bhautiki mein mahatvapurna hain, jahan way purnank spin key praathamik kanon koo janm dete hain.
lambai aur konn.
kewal lambai koo sanrakshit karane key alawa, ghoornan vactors key beach key konon koo bhee sanrakshit karate hain. yeh iss tathya seey milata haye kii dou vactors u aur v key beach maanak bindu utpaad koo vishuddh roop seey lambai key sandarbh mein likhaa jaa sakata haih
sootra _ 5
iska anusaran yeh haye kii sootra _ 1 mein pratyek lambai-sanrakshit raikhik parivartan bindu utpaad koo sanrakshit kartaa haye, aur iss prakaar vactors key beach kaa konn. ghoornan koo aksar raikhik parivartanon key roop mein paribhaashit kiya jaataa haye joe aantarik utpaad koo sootra _ 1 parr sanrakshit karate hain, joe unhein lambai koo sanrakshit karane kii aavashyakta key barabar haye. iss adhik samanya drishtikon key upchaar key liye shaastriya samooh dekhein, jahaan ack vishesh mamle key roop mein dikhaai detaa haye.
orthogonal aur rotation matricks.
pratyek ghoornan sootra _ 1 key ack orthonormal aadhar koo dusare orthonormal aadhar parr mapp kartaa haye. parimit-ayami sadish rikt sthaan key kissy bhee raikhik parivartan kii tarah, ack ghoornan koo hamesha ack matricks dwara darshaaya jaa sakata haye. ack diyaa gayaa aavartan honey dein. sootra _ 1 key maanak aadhar key sambandh mein stambhon koo isake dwara diyaa jaataa haye. choonki maanak aadhar orthonormal haye, aur choonki konon aur lambai koo sanrakshit kartaa haye, isaliye stambh ack aur orthonormal aadhar banaate hain. iss orthonormality sthiti koo iss roop mein vyakt kiya jaa sakata haye
jahan pehchaan matricks key sthanantaran koo darshaata haye aur haye. jinn matricks key liye yeh guna dhaaran kartaa haye, unhein orthogonal matricks kahaa jaataa haye. sabhi orthogonal matricks key samooh koo darshaaya jaataa haye, aur isamen sabhi uchit aur anuchit ghoornan hotey hain.
lambai koo sanrakshit karane key alawa, uchit ghoornan koo abhivinyas koo bhee sanrakshit karna chaahiye. ack matricks iss aadhar parr abhivinyas koo sanrakshit yaa ulat dega kii matricks kaa nirdhaarak sakaratmak haye yaa nakaratmak. ack orthogonal matricks key liye, dhyaan dein kii iska tatparya haye, taaki. nirdhaarak key saath orthogonal matricks key upsamooh koo "vishesh orthogonal samooh" kahaa jaataa haye, jise darshaaya jaataa haye.
iss prakaar pratyek ghoornan koo ikai nirdhaarak key saath ack orthogonal matricks dwara vishisht roop seey darshaaya jaa sakata haye. isake alawa, choonki ghoornan kii sanrachana matricks gunan key anuroop haye, isaliye ghoornan samooh vishesh orthogonal samooh key liye samroop haye.
anuchit ghoornan nirdhaarak key saath orthogonal matricks key anuroop hotey hain, aur way ack samooh naheen banaate hain kyonki dou anuchit ghoornan kaa utpaad ack uchit ghoornan haye.
samooh sanrachana.
ghoornan samooh falan sanrachana (yaa raikhik parivartanon kaa utpaad) key tahat ack samooh haye. yeh samanya raikhik samooh kaa ack upsamooh haye jisamen vaastavik 3-sthaan sootra _ 1 key sabhi vyutkramaniya raikhik parivartan shamil hain.
isake alawa, ghoornan samooh gair-abelian haye. yaani, jiss crum mein ghoornan kiye jaate hain, usase fark padata haye. udaaharan key liye, ack chauthai sakaratmak "x"-aksh key chaaron or mudataa haye aur usake baad ack chauthai sakaratmak "y"-aksh key chaaron or mudataa haye, joe pahale "y" aur phir "x" key chaaron or ghumaane seey praapt ack alag ghoornan haye.
orthogonal samooh, jisamen sabhi uchit aur anuchit ghoornan hotey hain, pratibimbon dwara utpann hotaa haye. pratyek uchit ghoornan dou pratibimbon kii sanrachana haye, joe carton-diodone pramey kaa ack vishesh mamla haye.
seemit upsamuhon kaa poorna vargeekaran.
sootra _ 12 key seemit upsamooh poori tarah seey vargeekrit hain.
pratyek parimit upsamooh samtal samamitiyon key dou ganana yogya anant parivaaron mein seey ack key tatva key liye samroop haih chakreeya samooh sootra 13 yaa dihedral samooh sootra 14, yaa teen anya samoohon mein seey ack key liyeh tetrahedral samooh sootra 15, octahedral samooh sootra 16, yaa icosahedral samooh sootra 17.
ghoornan kii dhuri.
3 aayamon mein pratyek gair-tuchchh uchit ghoornan sootra _ 1 key ack advitiya 1-ayami raikhik upasthaan koo taya kartaa haye jise "ghoornan kaa aksh" kahaa jaataa haye (yeh euler kaa ghoornan pramey haye). iss tarah kaa pratyek ghoornan iss aksh key liye samtal orthogonal mein ack sadharan 2-ayami ghoornan key roop mein kaary kartaa haye. choonki pratyek dwi-ayami ghoornan koo ack konn "φ" dwara darshaaya jaa sakata haye, ack manmana tri-ayami ghoornan koo iss aksh key aspas ghoornan key konn key saath ghoornan key ack aksh dwara nirdisht kiya jaa sakata haye. (takniki roop seey, kissy koo aksh key liye ack abhivinyas nirdisht karane kii aavashyakta haye aur kyaa iss abhivinyas key sambandh mein ghoornan koo ghadee kii disha mein yaa ghadee kii vipareet disha mein liya jaataa haye).
udaaharan key liye, sakaratmak "z"-aksh key konn "φ" key baare mein ghadee kii vipareet disha mein ghoornan iss prakaar diyaa gayaa haye -
sootra _ 1 mein ack ikai sadish n aur ack konn "φ" diye jaane parr, maan leejiye kii "r" ("φ", n) n key madhyam seey aksh key baare mein ack ghadee kii vipareet disha mein ghoornan kaa pratinidhitva kartaa haye (n dwara nirdhaarit abhivinyas key saath). taba
inn gunon kaa upayog karke koee bhee yeh dikhaa sakata haye kii kissy bhee ghoornan koo 0 ≤ φ ≤ kii seema mein ack advitiya konn "φ" aur ack ikai sadish n dwara darshaaya jaa sakata haye jaise kii
agale khand mein, ghoornan key iss pratinidhitva kaa upayog (3) sthalaakritik roop seey tri-ayami vaastavik prakshepya sthaan key saath pehchaan karane key liye kiya jaataa haye.
topology.
jhooth samooh tou (3) vaastavik prakshepya antariksh sootra _ 21 key liye bhinn haye
trijya key sootra _ 1 mein thhos gend parr vichaar karein (yaani, sootra _ 1 key sabhi bindu joe mool seey doori key hain yaa usase cumm hain). uparokt koo dekhate hue, iss gend mein pratyek bindu key liye ack ghoornan hotaa haye, jisamen bindu aur mool key madhyam seey aksh hotaa haye, aur ghoornan konn mool seey bindu kii doori key barabar hotaa haye. pehchaan kaa ghoornan gend key kendra mein bindu key anuroop hotaa haye. 0 aur − key beach key konon key madhyam seey ghoornan ack hii aksh parr bindu aur mool seey doori key anuroop haye lekin mool key vipareet disha mein haye. ack shesh mudda yeh haye kii donon ghoornan-key madhyam seey samaan hain. isaliye humm gend kii satah parr antipodal binduon kii pehchaan karate hain (yaa "ack saath gond"). iss pehchaan key baad, humm ghoornan samooh key liye ack topological space homomorphic parr pahunchate hain.
vaastav mein, antipodal satah binduon key saath gend kii pehchaan ack chikani manifold haye, aur yeh manifold ghoornan samooh key liye bhinn haye. yeh vaastavik 3-ayami prakshepya antariksh sootra _ 24 key liye bhee bhinn haye isaliye baad vaalaa ghoornan samooh key liye ack topological model key roop mein bhee kaam curr sakata haye.
yeye pahachaanen darshaati hain kii isaliye (3) judaa huaa haye lekin kewal judaa huaa naheen haye. baad vaale key roop mein, antipodal satah binduon kii pehchaan key saath gend mein, "uttari dhruv" seey seedhe aantarik seey neeche dakshini dhruv taka jaane vaale maarg parr vichaar karein. yeh ack bund loop haye, kyonki uttari dhruv aur dakshini dhruv kii pehchaan kii gaee haye. iss loop koo ack bindu taka sikudaaya naheen jaa sakata haye, kyonki isase koee fark naheen padata kii aap loop koo kaise vikrit karate hain, shuruat aur anta bindu koo antipodal rehna hoga, anyatha loop "toot jaayega". ghoornan key sandarbh mein, yeh loop "z"-aksh key baare mein ghoornan key ack nirantar anukram kaa pratinidhitva kartaa haye joe pehchaan (gend key kendra) seey shuru hocker (udaaharan key liye), dakshini dhruv key madhyam seey, uttari dhruv parr koodata haye aur pehchaan ghoornan (i) parr phir seey samaapt hotaa haye.ee., ack konn "φ" key madhyam seey ghoornan kii ack shrinkhala jahaan "φ" 0 seey 2 taka chalata haye).
ashcharya kii baat haye kii yadi aap dou baar raaste seey gujarate hain, tou i.ee., uttari dhruv seey neeche dakshini dhruv taka dauden, uttari dhruv parr vaapas kooden (iss tathya kaa upayog karate hue kii uttari aur dakshini dhruvon kii pehchaan kii gaee haye), aur phir phir seey uttari dhruv seey neeche dakshini dhruv taka dauden, taaki "φ" 0 seey 4 taka chaley, aapako ack bund loop milata haye jise "ack bindu taka sikudaaya jaa sakata haih pahale path koo gend kii satah parr lagaatar leey jaaen, phir bhee uttari dhruv koo dou baar dakshini dhruv seey jodein. dusare maarg koo taba binaa maarg badale antipodal paksh mein pratibimbit kiya jaa sakata haye. abb hamaare paas gend kii satah parr ack sadharan bund loop haye, joe uttari dhruv koo ack badey vritt key saath khud seey jodata haye. iss vritt koo binaa kissy samasya key uttari dhruv taka sikudaaya jaa sakata haye. plate trick aur isi tarah kii tricks ise vyaavhaarik roop seey pradarshit karati hain.
ack hii tark samanya roop seey kiya jaa sakata haye, aur yeh darshaata haye kii (3) kaa maulik samooh crum 2 kaa chakreeya samooh haye (dou tatvon vaalaa ack maulik samooh). bhautiki anuprayogon mein, maulik samooh kii gair-tuchchata (ack seey adhik tatva) spinners key roop mein jaani jaane vaali vastuon key astitva kii anumati detee haye, aur spin-saankhyiki pramey key vikas mein ack mahatvapurna upkaran haye.
tou (3) kaa saarvbhaumik aavaran ack jhooth samooh haye jise spin (3) kahaa jaataa haye. samooh spin (3) vishesh ekaatmak samooh key liye samroop haye, yeh ikai 3-kshetra key liye bhee bhinnaroop haye aur ise vyutkramon key samooh (nirpeksh maan 1 vaale chaturthak) key roop mein samajha jaa sakata haye. chaturthak aur ghoornan key beach sambandh, aamtaur parr computer graphics mein upayog kiya jaataa haye, chaturthak aur sthaanik ghoornan mein samajhaaya jaataa haye. s" 3 seey laker (3) taka kaa naksha joe "s" 3 key prati-pratiroopan binduon kii pehchaan kartaa haye, vaha jhooth samoohon kaa ack surjective homomorphism haye, jisamen kernell {±1} haye. sthalgat roop seey, yeh maanchitra dou-seey-ack koo cover karane vaalaa maanchitra haye. (plate trick dekhein.)
tou (3) aur sue (2) key beach sambandh.
iss khand mein, humm dou-seey-ack aur sue (2) key surjective homomorphism key dou alag-alag nirmaan dete hain.
ikai maanak key chaturthansh kaa upayog karna.
samooh dwara diye gaye maanchitra key madhyam seey ikai maanak key chaturthak key liye samroop haye
sootra _ 25
sootra _ 26 taka seemit jahan sootra _ 27, sootra _ 28, sootra _ 29, aur sootra _ 30, sootra _ 31.
aaie abb sootra _ 33 key saath sootra _ 1 kii pehchaan karein. phir koee yeh satyaapit curr sakata haye kii yadi sootra _ 34 sootra _ 1 mein haye aur sootra _ 36 ack ikai chaturthansh haye, tou
sootra _ 37
isake alawa, maanchitra sootra 38 sootra 39 kaa ack ghoornan haye, isake alawa sootra 40 sootra 41 key samaan haye. iska matlab haye kii ikai maanak key chaturthansh seey 3 dee ghoornan samooh mein ack samroopata haye.
iss samroopata koo spasht roop seey kaam kiya jaa sakata haih ikai chaturthak, key saath
sootra _ 42
rotation matricks mein mapp kiya gayaa haye
sootra _ 43
yeh ack konn seey sadish key chaaron or ack ghoornan haye, jahan aur. ack baar aksh ghatakon key sanket taya hoe jaane key baad isake liye uchit sanket nihit hotaa haye. donon key baad seey spasht haye aur ack hii maanchitra parr haye.
mobius parivartanon kaa upayog karna.
iss khand kaa samanya sandarbh haye. kshetra parr bindu
uttari dhruv koo chhodkar, samtal parr paribhaashit binduon key saath ack-seey-ack dwibhajan mein rakhaa jaa sakata haye, chitra dekhein. maanchitra koo stereographic prakshepan kahaa jaataa haye.
nirdeshank koo chaalu honey dein. rekha seey gujarati haye aur ise iss tarah seey parametries kiya jaa sakata haye
yeh maang karate hue kii sootra _ 46 barabar haye, koee paataa haye
hamaare paas sootra _ 48 haye isaliye naksha
jahan, baad mein suvidha key liye, vimaan kii pehchaan jatil samtal sootra _ 50 seey kii jaati haye.
isake vipareet key liye, iss prakaar likhein
aur khojane kii maang karate hain aur iss prakaar
yadi ack ghoornan haye, tou yeh antahsthaapit sthaan sootra _ 39 parr apani maanak kaarravaai dwara binduon parr bindu legaa, iss kriya koo ack parivartan praapt karane key saath,
iss prakaar sootra _ 55 kaa ack parivartan haye joe sootra _ 1 key parivartan seey judaa huaa haye.
yeh pataa chalaa haye kii iss tarah seey darshaaya gayaa haye jise ack matricks key roop mein vyakt kiya jaa sakata haye (jahaan sanketan koo matricks key liye usee naam kaa upayog karane key liye punarnaveenikaran kiya jaataa haye jaisa kii sootra _ 55 key parivartan key liye yeh darshaata haye). iss matricks kii pehchaan karane key liye, pahale ack konn key madhyam seey ack ghoornan parr vichaar karein,
isaliye
joe, aashcharyajanak roop seey, jatil tala mein ack ghoornan haye. ack samaan tareeke seey, yadi ack konn key madhyam seey ack ghoornan haye, tou
joe thode beejganit key baad bunn jaataa haye
yeye dou ghoornan, sootra _ 62 iss prakaar key dwiaadhaari parivartanon key anuroop hain, arthaat, way mobius parivartanon key udaaharan hain.
ack samanya mobius parivartan dwara diyaa jaataa haye
ghoornan, sootra 64 sabhi utpann kartaa haye aur mobius parivartanon key sanrachana niyam bataate hain kii sootra 64 kii koee bhee sanrachana mobius parivartanon kii sambandhit sanrachana mein anuvaad karati haye. mobius parivartanon koo matricks dwara darshaaya jaa sakata haye
kyonki radd karane kaa ack samanya kaarak haye.
isi kaaran seey, matricks koo vishisht roop seey paribhaashit naheen kiya gayaa haye kyonki gunaa kaa nirdhaarak yaa mobius parivartan parr koee prabhaav naheen padata haye. mobius parivartanon kaa sanrachana niyam sambandhit matricks kaa paalan kartaa haye. nishkarsh yeh haye kii pratyek mobius parivartan dou matricks seey mell khaata haye.
iss patrachar kaa upayog karke koee bhee likh sakata haye
yeye matricks ekaatmak hain aur iss prakaar. euler konon key sandarbh mein samanya ghoornan key liye paaya jaataa haye
ack haye
 & 0\\
\cos\frac {\heta} {2} aur i\ sin\frac {\heta} {2}\\
e ^ {i\frac {\psi} {2}} & 0\\
\anta {pematrics}\\
&= \ pee. aim.
\cos\frac {\heta} {2} e ^ {i\frac {\fi + \psi} {2}} aur i\ sin\frac {\heta} {2} e ^ {i\frac {\fi-\psi} {2}}\\
\anta {pmatrix}.
isake vipareet key liye, ack samanya matricks parr vichaar karein
pratisthaapan karein
pratisthaapan key saath, () key daahine haath kii or (rhs) kaa roop dhaaran kartaa haye, joe usee key saath () key rhs key roop mein ack matricks key neeche mell khaata haye. jatil maapdandon key sandarbh mein,
ise satyaapit karane key liye, () key rhs parr matricks key tatvon koo pratisthaapit karein. kuchh herfer key baad, matricks () key rhs kaa roop dhaaran kartaa haye.
euler konon key sandarbh mein spasht roop seey yeh spasht haye kii maanchitra
abhee varnit ack chikna, aur surjective samooh samroopata haye. isaliye yeh saarvbhaumik aavaran samooh seey saarvbhaumik aavaran sthaan kaa ack spasht vivaran haye.
jhooth beejganit.
pratyek jhooth samooh key saath judaa huaa iska jhooth beejganit haye, joe jhooth samooh key samaan ayam kaa ack raikhik sthaan haye, joe jhooth koshthak naamak ack dwilingi vaikalpik utpaad key tahat bund hotaa haye. key jhooth beejganit koo sootra _ 72 dwara darshaaya jaataa haye aur isamen sabhi tirchha-samamit metrices hotey hain. ise orthogonality sthiti koo alag karke dekhaa jaa sakata haye. sootra _ 72 key dou tatvon kaa jhooth koshthak, pratyek matricks samooh key jhooth beejganit key liye, matricks commutator dwara diyaa gayaa haye, joe phir seey ack tirchha-samamit matricks haye. jhooth beejganit koshthak jhooth samooh utpaad key saar koo ack arth mein pakadata haye jise becker-campbelle-hosdorff sootra dwara sateek kiya gayaa haye.
sootra _ 72 key tatva ghoornan key "aseem janitra" hain, i.ee.way manifold key sparshrekha sthaan key tatva hain isaliye (3) pehchaan tatva parr. yadi sootra _ 75 ikai sadish sootra _ 76 dwara nirdisht aksh key baare mein konn φ key saath ghadee kii vipareet disha mein ghoornan koo darshaata haye tou
iska upayog yeh dikhaane key liye kiya jaa sakata haye kii jhooth beejganit sootra _ 72 (commutator key saath) jhooth beejganit sootra _ 1 (cross utpaad key saath) key liye samroop haye. iss samroopata key tahat, ack euler vector sootra _ 80 sootra _ 82 dwara paribhaashit raikhik maanchitra sootra _ 81 seey mell khaata haye.
adhik vistaar seey, aksar ack vector sthaan key roop mein sootra _ 72 key liye ack upayukt aadhar haye
inn aadhar tatvon key sanchaar sambandh hain -
joe cross utpaad key tahat sootra _ 1 key teen maanak ikai vactors key sambandhon seey sahmat hain.
jaisa kii oopar ghoshit kiya gayaa haye, koee bhee iss jhooth beejganit mein kissy bhee matricks kii pehchaan euler vector sootra _ 87 seey curr sakata haye.
iss pehchaan koo kabhi-kabhi topi-maanchitra kahaa jaataa haye. iss pehchaan key tahat, sootra _ 72 koshthak sootra _ 1 mein cross utpaad seey mell khaata haye,
vector sootra _ 92 key saath pehchaane gaye matricks mein yeh guna haye kii
jahan baain or hamaare paas sadharan matricks gunan haye. iska tatparya haye kii sootra _ 92 tirchha-samamit matricks key shoonya sthaan mein haye jisake saath ise pahachaana jaataa haye, kyonki sootra _ 95
jhooth algebra parr ack nott.
beejganit pratinidhitva key aadhar parr, samooh isaliye (3) compact aur rank 1 kaa saral haye, aur isaliye isamen ack swatantra casimir tatva haye, joe teen generators kaa ack dwighaat aparivartaniya kaary haye joe una sabhi key saath aataa haye. ghoornan samooh key liye hatyaa kaa roop kewal kronekar delta haye, aur isaliye yeh casimir aparivartaniya kewal generator key vargon kaa yog haye, beejganit kaa sootra _ 96
yaani, casimir aparivartaniya dwara diyaa jaataa haye
ekaatmak aparivartaniya pratinidhitvon key liye, iss aparivartaniya key igen maan vaastavik aur asatat hain, aur pratyek pratinidhitva koo chihnit karate hain, joe parimit ayami haye, ayami sootra _ 99 kaa. yaani, iss casimir operator key igen maan hain
jahan purnank yaa ardh-purnank haye, aur spin yaa koniya samveg key roop mein sandarbhit kiya jaataa haye.
tou, oopar pradarshit 3 × 3 generator l triple (spin 1) pratinidhitva parr kaary karate hain, jabaki neeche 2 × 2 generator, t, double (spin 1/2) pratinidhitva parr kaary karate hain. kronekar utpaadon koo baar-baar apane saath laker, koee bhee sabhi uchch aparivartaniya pratinidhitvon kaa nirmaan curr sakata haye. yaani, teen sthaanik aayamon mein uchch spin system key liye parinami generator, manmaane dhang seey badey key liye, inn spin operators aur seedhi operators kaa upayog karke ganana kii jaa sakati haye.
pratyek ekaatmak apavartaniya abhyavedan key liye ack samatulya haye. sub 
anant-ayami apuraniya pratinidhitva gair-ekaatmak honah chaahiye, kyonki samooh sankshipt haye.
quantum yantriki mein, casimir aparivartaniya "koniya-gati-varg" prachaalak haye; spin key purnank maan bosonic pratinidhitva kii visheshta hain, jabaki ardh-purnank moolya fermionic pratinidhitva hain. oopar upayog kiye gaye antihermitian matricks kaa upayog spin operators key roop mein kiya jaataa haye, jaba unhein gunaa kiya jaataa haye, tou way abb hermician hain (jaise pauli matricks). iss bhasha mein,
aur isaliye
inke liye spasht abhivyaktiyaan hain,
jahan manmana aur sootra _ 104 haye.
udaaharan key liye, spin 1 (sootra _ 105) key liye parinami spin matricks hain
haalaanki, dhyaan dein kii yeye cartasian aadhar parr uparokt l kii tulanaa mein ack samatulya, lekin alag aadhar, golakar aadhar mein kaise hain.
uchch spin key liye, jaise spin (sootra _ 107):
spin key liye (sootra _ 109),
sue (2) key saath samroopata.
jhooth algebra sootra _ 72 aur sootra _ 112 samroop hain. sootra _ 112 key liye ack aadhar diyaa gayaa haye
yeye polly matricks seey sambandhit hain
pauli matricks jhooth algebra key liye bhautikavidon key parampara kaa paalan karate hain. usa parampara mein, shoonya beejganit tatvon koo gunaa kiya jaataa haye, ghaatiya maanchitra (neeche) koo ghatank mein ack atirikt kaarak key saath paribhaashit kiya jaataa haye aur sanrachana sthiraank samaan rahate hain, lekin unaki "paribhasha" ack kaarak praapt karati haye. isi tarah, sanchaar sambandh ack kaarak praapt karate hain. sootra _ 116 key liye sanchaar sambandh hain
purnataya samamitirodhi prateek kahaan haye? sootra _ 72 aur sootra _ 112 key beach samroopata koo kaee tarikon seey sthaapit kiya jaa sakata haye. baad mein suvidha key liye, sootra _ 72 aur sootra _ 112 koo maanchitran dwara pahachaana jaataa haye.
aur raikhikta dwara vistaarit.
ghaatiya maanchitra.
ghaatiya maanchitra haye, kyonki yeh ack matricks jhooth samooh haye, jise maanak matricks ghaatiya shrinkhala kaa upayog karke paribhaashit kiya gayaa haye,
kissy bhee tirchha-samamit matricks key liye, hamesha ander hotaa haye. pramaan matricks ghatank key praathamik gunon kaa upayog kartaa haye
matricks aur awagaman key kaaran, ise aasaani seey tirchha-samamit matricks sthiti key saath saabit kiya jaa sakata haye. yeh yeh dikhaane key liye paryaapt naheen haye kii yeh sambandhit jhooth beejganit haye, aur ise alag seey saabit kiya jaayega.
pramaan kii kathinaaee kaa star iss baat parr nirbhar kartaa haye kii ack matricks samooh koo beejganit kaise paribhaashit kiya jaataa haye. jhooth beejganit koo matricks key samooh key roop mein paribhaashit kartaa haye
iss mamle mein yeh tuchchh haye. pehchaan parr lii gaee pehchaan key madhyam seey chikane vakr khandon key paribhasha vyutpann key liye upayog kiya jaataa haye, iss mamle mein yeh kathin hotaa haye.
ack nishchit key liye, ack bhooganitiya key saath ack-parameter upsamooh haye. kii yeh ack-parameter upsamooh detaa haye joe ghaatiya maanchitra key gunon seey seedhe anusaran kartaa haye.
ghaatiya maanchitra mool key ack pados aur pehchaan key ack pados key beach ack vibhedak aakriti pradaan kartaa haye. pramaan key liye, bund upsamooh pramey dekhein.
ghaatiya maanchitra surjective haye. yeh iss tathya seey anusarega kii pratyek ghoornan ack nishchit aksh (euler kaa ghoornan pramey) chhodata haye, aur roop key ack khand vikarn matricks key liye sanyugmit hotaa haye.
kii aur vaha
iss tathya key saath joe kii kii sahayak kaarravaai key tahat bund haye, jiska arth haye kii.
iss prakaar, ee.jee.lokpriya pehchaan kii jaanch karna aasaan haye
jaisa kii oopar dikhaaya gayaa haye, pratyek tatva ack sadish seey judaa hotaa haye, jahan ack ikai parimaan sadish hotaa haye. kyonki yeh shoonya sthaan mein haye, yadi koee abb kissy anya orthogonal matricks key madhyam seey ack nae aadhar parr ghoomata haye, tou aksh key roop mein, nae aadhar mein rotation matricks kaa antim collum aur pankti shoonya hogi.
iss prakaar, humm ghatank key sootra seey pahale seey jaanate hain jise nishchit chhod denaa chaahiye. iss prakaar key aadhar key liye ack seedha sootra pradaan karna ganitiya roop seey asambhav haye, kyonki iska astitva baalon vaali gend pramey kaa ullanghan karega; lekin pratyaksh ghatank sambhav haye, aur upaj detaa haye.
jahan sootra _ 130 aur sootra _ 131. ise konn dwara aksh key chaaron or ghoomane key liye ack matricks key roop mein pahachaana jaataa haih cf. rodrigue kaa ghoornan sootra.
laghuganak maanchitra.
diye gaye sootra koo 132 pratisamitiya bhaag koo darshaata haye aur sootra koo 133 maan lein, tou laghuganak iss prakaar diyaa jaataa haye -
yeh rodrigue key sootra key mishrit samroopata roop key nireekshan seey prakat hotaa haye,
jahan daahine haath kii or pahala aur antim pada samamit hain.
ack samaan yaadrichhik namunakaran.
sootra _ 136 ikai chaturthak key samooh dwara doguna cover kiya gayaa haye, joe 3-kshetra key liye samroop haye. choonki ikai chaturthak parr haar maap 4 aayamon mein kewal 3-kshetra maap haye, sootra _ 136 parr haar maap 3-kshetra maap kaa kewal aage badhaana haye.
natijatan, sootra _ 1 mein ack samaan yaadrichhik ghoornan utpann karna 3-kshetra parr ack samaan yaadrichhik bindu utpann karane key barabar haye. yeh nimnalikhit sootra dwara puuraa kiya jaa sakata haye _ 139
jahan sootra _ 140 sootra _ 141 key samaan roop seey yaadrichhik namoone hain.
becker-campbelle-hosdorf sootra.
maan leejiye kii beejganit diyaa gayaa haye. unake ghatank, aur, ghoornan matricks hain, jinhen gunaa kiya jaa sakata haye. choonki ghaatiya maanchitra ack adhikshep haye, kuchh logon key liye jhooth beejganit mein, aur koee bhee asthaayi roop seey likh sakata haye
kuchh abhivyakti key liye aur. taba jaba aur yatra karate hain, jatil ghatank key vyavhaar kii nakal karate hain.
samanya mamla adhik vistrit bch sootra dwara diyaa jaataa haye, joe nested jhooth koshthakon kaa ack shrinkhala vistaar haye. matricks key liye, jhooth koshthak commutator key samaan sanchaalan haye, joe gunan mein commutavity kii kamee kii nigrani kartaa haye. yeh samanya vistaar iss prakaar hotaa haye,
bch sootra mein anant vistaar ack sankshipt roop mein cumm hoe jaataa haye,
upayukt trikonamitiya falan gunank key liye.
dwara diyaa jaataa haye
kahaan
key liye
aantarik utpaad hilbert-schmitt aantarik utpaad haye aur maanak sambandhit maanak haye. topi-samroopata key tahat,
joe aur key liye karakon kii vyakhya kartaa haye. yeh konn key liye abhivyakti mein girata haye. 
iss samagr ghoornan generator koo iss roop mein likhana saarthak haye
iss baat parr jore dene key liye kii yeh ack "jhooth beejganit pehchaan" haye.
uparokt pehchaan sabhi wafadar pratinidhitvon key liye haye. ack jhooth beejganit samroopata kaa kernell ack aadarsh haye, lekin saral honey key kaaran, koee gair-tuchchh aadarsh naheen haye aur isaliye sabhi gair-tuchchh pratinidhitva wafadar hain. yeh vishesh roop seey doublet yaa spinor pratinidhitva mein hotaa haye. isi spasht sootra kaa paalan pauli matricks, see. aff. key madhyam seey saral tareeke seey kiya jaataa haye. sue (2) key liye 2 × 2 vyutpatti.
usee bch sootra kaa pauli sadish sanskaran kuchh hudd taka saral samooh sanrachana niyam haye.
kahaan
cosine kaa golakar niyam. (nott konn hain, uparokt naheen.)
yeh spasht roop seey oopar key samaan praarup kaa haye, 
key saath
taaki
shamil mithya beejganit mein janitron key samaan saamaanyikaran key liye, pauli matricks koo-matricks key sandarbh mein vyakt karein, taaki 
taba satyaapit karane key liye yeye uparokt samaan gunank hain, gunank key anupaat kii ganana karein, 
anta mein, pehchaan dee gaee. 
samanya mamle key liye, koee bhee ref kaa upayog curr sakata haye.
dou ghoornan rb aur ra kii sanrachana kaa chaturthak nirmaan bhee seedhe ghoornan aksh aur samagr ghoornan rc = rbra key konn koo praapt kartaa haye.
maan leejiye kii ack sthaanik ghoornan r seey judey chaturthansh kaa nirmaan isake ghoornan aksh s aur ghoornan konn "φ" iss aksh seey kiya jaataa haye. sambaddh chaturthansh dwara diyaa jaataa haye,
phir r key saath ghoornan rr kii sanrachana ghoornan aksh aur konn key saath ghoornan rc = rbra haye jise chaturthak key gunanphal dwara paribhaashit kiya gayaa haye.
vaha haye
iss utpaad koo praapt karane key liye iska vistaar karein
iss sameekaran key donon pakshon koo pehchaan seey vibhajit karein, joe ack gole parr cosine kaa niyam haye,
aur ganana karein
yeh dou ghurnanon kii akshon key sandarbh mein paribhaashit ack samagr ghoornan kii aksh key liye rodrigues kaa sootra haye. unhone 1840 mein yeh sootra praapt kiya (prishth 408 dekhein).
teen ghoornan aksh a, b aur c ack golakar trikon banaate hain aur iss trikon kii bhujaaon dwara banaae gaye talon key beach key dwi-paridhiya konon koo ghoornan konon dwara paribhaashit kiya jaataa haye.
ghoornan kii praapti.
hamane dekhaa haye kii ghoornan kaa pratinidhitva karane key kaee tareeke hainh
golakar samanjasya.
tri-ayami euclidean ghoornan key samooh kaa hilbert sthaan parr ack anant-ayami pratinidhitva haye.
jahan sootra _ 164 golakar samanjasya hain. isake tatva gole parr varg samakalan yogya jatil-moolya vaale kaary hain. iss sthaan parr aantarik utpaad dwara diyaa jaataa haye
yadi ikai kshetra parr paribhaashit ack manmana varg samakalan falan haye, tou ise iss prakaar vyakt kiya jaa sakata haye
jahan vistaar gunank diye jaate hain
lorentz samooh kriya key roop mein vyakt kii jaati haye aur key liye pratibandhit haye
yeh kriya ekaatmak haye, jiska arth haye kii
claybush-gorden apghatan kaa upayog karke uparokt seey praapt kiya jaa sakata haye, lekin unhein adhik aasaani seey ack visham-ayami-pratinidhitva (3-ayami ack bilkul haye) key ghatank key roop mein vyakt kiya jaataa haye. iss mamle mein sthaan badey kareene seey apghatit hocker aparivartaniya visham parimit-ayami pratinidhitvon key ack anant pratyaksh yog mein vighatit hoe jaataa haye
yeh anant-ayami ekaatmak pratinidhitvon kii visheshta haye. yadi ack alag karane yogya hilbert sthaan parr ack anant-ayami ekaatmak pratinidhitva haye, tou yeh seemit-ayami ekaatmak pratinidhitva key pratyaksh yog key roop mein vighatit hoe jaataa haye. iss prakaar iss tarah kaa pratinidhitva kabhi bhee aparivartaniya naheen hotaa haye. sabhi aparivartaniya parimit-ayami pratinidhitvon koo aantarik utpaad key uchit vikalp dwara ekaatmak banaaya jaa sakata haye.
jahan samakalan saamaanyikrit parr advitiya aparivartaniya samakalan haye, yahaan euler konn parametrization kaa upayog karke vyakt kiya gayaa haye. samakalan key ander aantarik utpaad koee bhee aantarik utpaad hotaa haye.
saamaanyikaran.
ghoornan samooh apane maanak euclidean sanrachana key saath "aine"-ayami euclidean sthaan, sootra 166 key liye kaafee swabhavik roop seey saamaanyikrit hotaa haye. aine" aayamon mein sabhi uchit aur anuchit ghoornan key samooh koo orthogonal samooh oh ("aine") kahaa jaataa haye, aur uchit ghoornan key upsamooh koo vishesh orthogonal samooh kahaa jaataa haye, joe ayam kaa ack jhooth samooh haye.
vishesh saapekshata mein, koee vyakti 4-ayami sadish sthaan mein kaam kartaa haye, jise 3-ayami euclidean sthaan key bajaay minkowski sthaan key roop mein jaanaa jaataa haye. euclidean sthaan key vipareet, minkovsky sthaan mein anishchit hastakshar key saath ack aantarik utpaad haye. haalaanki, koee bhee abhee bhee "saamaanyikrit ghoornan" koo paribhaashit curr sakata haye joe iss aantarik utpaad koo sanrakshit karate hain. iss tarah key saamaanyikrit ghoornan koo lorentz parivartan key roop mein jaanaa jaataa haye aur aise sabhi parivartanon key samooh koo lorentz samooh kahaa jaataa haye.
ghoornan samooh tou (3) koo ee + (3) key upsamooh key roop mein varnit kiya jaa sakata haye, euclidean sootra key pratyaksh samamitiyon kaa euclidean samooh yeh badaa samooh ack kathor pind kii sabhi gatiyon kaa samooh haih inmein seey pratyek ack manmana aksh aur ack anuvaad key baare mein ack ghoornan kaa sanyojan haye, yaa alag tarah seey kahein tou, ack tatva kaa sanyojan haye (3) aur ack manmana anuvaad.
samanya taur parr, kissy vastu kaa ghoornan samooh pratyaksh samamitiyon key samooh key bheetar samamiti samooh haye; dusare shabdon mein, poorna samamiti samooh aur pratyaksh samamitiyon key samooh kaa pratichhedan. kieral vastuon key liye yeh poorna samroopata samooh key samaan haye.