
spin-waited golaakaara harmonics

pratyeka vidhulalo, ganita saastramlo oka amsam, spin-waited golaakaara harmonics anevi praamaanika golaakaara harmonics yokka saadhaaraneekaranalu mariyu-saadhaarana golaakaara harmonics laaga-golampai vidhulu. saadhaarana golaakaara saamarasyaala maadirigaa kakunda, spin-waited samarasyalu scalar kshetraala kante gaze kshetraaluh ganitasaastraparamgaa, avi sanklishta rekha kattalo viluvalanu teesukuntaayi. spin-waited harmonics saadhaarana golaakaara harmonics maadirigaane degree dwara nirvahinchabadataayi, conny adanapu spin baruvunu kaligi untaayi, idhi adanapu samaruupatanu pratibimbistumdi. harmonics yokka pratyeka aadharaanni laplace golaakaara harmonics nundi pondavacchu, mariyu saadhaaranamgaa praamaanika laplace golaakaara harmonics nundi telisina saadhaarana paaraamitulu ekkada mariyu ekkada suuchimchabadataayi. eee pratyeka praatipadikana, spin-waited golaakaara harmonics vaastava vidhulugaa kanipistaayi, endukante dhruva aksham yokka empika gaze aspashtatanu parishkaristundi. spin-waited golaakaara harmonicsnu spin rising mariyu loying operatorlanu upayoginchadam dwara praamaanika golaakaara harmonics nundi pondavacchu. mukhyamgaa, spin baruvu yokka spin-waited golaakaara harmonics kevalam praamaanika golaakaara harmonics
lorenz samuuham yokka praatinidhya siddhaantaaniki sambandhinchi spin-waited golaakaara harmonics yokka khaaliilu modata gurtinchabaddaayi. avi taruvaata mariyu swatantramgaa gurutvakarshana vikiranaanni vivarinchadaaniki mariyu mallii diarock monopoles adhyayanamlo "monopole harmonics" anie pilavabade vaatini tirigi kanugoni, vartimpajeyabaddaayi.
spin-waited functionlu.
eee golaanni trimiteeya euclidean pradaesamloe ponduparachabadinatlu pariganinchandi. golampai oka binduvu vadda, tangent vectorla yokka saanukuula aadhaarita orthonormal aadharam oka jatha vectors vantidi
modati jatha sameekaranaalu mariyu tangent anie perkonnappudu, rendava jatha mariyu unit vectors anie, upanta sameekaranam mariyu orthogonal anie mariyu tudi sameekaranam kudi chethi aadharam anie chebutundi.
spin-weit function anedi oka point inputgaa mariyu tangent vectorla saanukuula aadhaarita orthonormal praatipadikagaa angeekarinche function, antey
prati bhramana koonaaniki.
taruvaata, anni spin-weit functionla sekarananu soochinchandi. sanklishta scaling kinda eee krindi sajaateeyata chattanni santruptipariche vidhulu anie ivi khachitamgaa ardham chesukobadatayi.
idhi ardha-puurnaamkam ayithe ardhame.
nairuupyamgaa, sanklishta projective linepy serr twist yokka antiholomorphic vector bundle antarleenamgaa unna smooth vector bandilku isomorphic. taruvaati bandillooni oka vibhagam anedi santruptikaramaina function
atuvantidi ichinatlayithe, hermician roopam yokka tagina saktitoe guninchadam dwara manam spin-weit functionnu utpatti cheyavachu.
pratyaekamgaa, idhi spin-weit function. oka saadhaarana sajatiya functionku spin-waited function yokka anubandham oka isomorphism.
operator.
spin baruvu kattalu differential operator (eehech) thoo amarchabadi untaayi. eee operator tappanisarigaa dalbielt operator, tagina gurtimpulu chesina tarvaata,
andukani,
spin-weit yokka vidhini nirvachistundi.
spin-waited harmonics.
sampradaaya golaakaara harmonics anevi golampai laplace-beltramy operator yokka aigen functionlu, spin-weit harmonics anevi spin-weit functionla kattalapai panichese laplace-beltramy operatorku igenfections.
vidhulugaa praatinidhyam.
uttara dhruvamgaa panicheyadaaniki golampai oka binduvunu enchukunna tarvaata spin-waited harmonicsnu golampai vidhulugaa suuchimchavachchu. nirvachanam prakaram, "spin weit" thoo kuudina function pole chuttoo tirugutuu maarutundi.
praamaanika golaakaara akshaamsaalalo pani chestu, oka functionpy panichese nirdishta operator nuu ilaa nirvachinchavacchuh
idhi manaku maroka function mariyu istundi. (operator samarthavantamgaa golamlo covarient derivative operator.)
kotta function yokka mukhyamaina lakshanam emitante, spin baruvu unte, spin baruvu untundi. eee vidhamgaa, operator oka function yokka spin baruvunu 1 penchutundi. adevidhamgaa, oka function yokka spin baruvunu 1 tagginche operatornu manam nirvachinchavacchuh
spin-waited golaakaara harmonics appudu saadhaarana golaakaara harmonics paramgaa ilaa nirvachinchabadataayih
appudu functionlu spin baruvutho maarutunna lakshanaanni kaligi untaayi.
itara mukhyamaina lakshanaalu eee krindi vidhamgaa unnaih
orthogonality mariyu paripuurnata.
harmonics mottam golamlo artogonalga untayih
mariyu paripuurnata sambandhaanni santrupti parachandi
lekkinchadam.
eee samarasyalanu aneka paddhatula dwara spashtamgaa lekkinchavacchu. rising ledhaa downing operatorlanu padepade vartimpajeyadam valla spashtamaina punaraavruta sambandham erpadutundi. pratyaksha ganana kosam suutraalu dwara pondabaddaayi. vaari suutraalu kandan-shortley dasha kosam paatha empikanu upayogistaayani gamaninchandi. udaaharanaku, krinda enchukunna sampradaayam ganitasastramto ekibhavistundi.
goldberg marinta upayogakaramgaa untundi, modalainavi.suutraalu eee krindi vidhamgaa unnaih
ekapaksha spin-waited golaakaara harmonicsnu lekkinchadaaniki eee suutraanni upayoginche ganitasastrapu notebook ikkada chudavachu.
ikkada dasha samavesamtoh
modati konni spin-waited golaakaara harmonics.
modati konni orthonormalised spin-waited golaakaara harmonics kosam vislaeshanhaatmaka vyakteekaranaluh
wigner bhramana maatrikalaku sambandhinchinadi.
eee sambandham-maatrala kosam punaraavruta sambandhaalanu upayoginchi spin harmonicsnu lekkinchadaaniki anumatistundi.
triple integral.
3-chihnam paramgaa ivvabadina sandarbhamlo triple integral

