
multivector

bahularekhiya beejaganitamlo, konnisaarlu clifford sankhya anie piluvabade bahula-karakamu, vector space yokka bahya beejaganitamlo oka moolakam. eee beejaganitam vargeekarinchabadindi, anubandhamgaa mariyu pratyamnaayamgaa untundi mariyu sarala-vahakalu (decomposable-vahakalu ledhaa-bladelu anie kooda pilustaaru) yokka sarala kalayikalanu kaligi untundi.
suutram _ 2 ekkada undhi.
a-vector anedi degree yokka "sajatiya" gaa unde oka sarala kalayika (anni padaalu oke vidhamgaa-blades). rachayitalanu batti, "multivector" anedi-vector ledhaa bahya beejaganitamloni edainaa moolakam kaavacchu (-blades yokka edainaa sarala kalayika, vaati yokka vibhina viluvalatho).
avakalana jyaamitiloo, a-vector anedi tangent vector space yokka bahya beejaganitamlo oka vector; antey, idhi konni puurnaamkaalaku, tangent vectorla yokka bahya utpatti yokka sarala kalayikalanu teesukovadam dwara pondina antisemetric tensor. avakalana-roopam anedi sparsha sthalam yokka dwandwa yokka bahya beejaganitamlo oka-vector, idhi sparsha sthalam yokka bahya beejaganitaaniki kooda dvandvamgaa untundi.
kosam mariyu,-vektarlanu tarachugaa varusaga "scalars", "vectors", "bivectors" mariyu "trivectors" anie pilustaaru; avi varusaga dwandwa nundi 0-roopaalu, 1-roopaalu, 2-roopaalu mariyu 3-roopaalu.
bahya utpatti.
multivectorlanu nirminchadaaniki upayoginche bahya utpatti (cheelika utpatti anie kooda pilustaaru) bahularekha (prati inputlo saralamgaa), anubandha mariyu pratyamnaayamgaa untundi. dheeni ardham vector space "v" loe vektarlu u, v mariyu w kosam mariyu scalars "α", "β" kosam, bahya utpatti lakshanaalanu kaligi untundih
k" vectorla yokka bahya utpatti ledhaa atuvanti utpattula mottaanni (oke "k" kosam) grade "k" multivector ledhaa "k"-vector anie pilustaaru. multivector yokka garishta grade vector space "v" yokka parimaanam.
pratyaamnaaya lakshanamtho kalisi edainaa inputlo saralata anedi itara inputlo saralatanu suuchistumdi. bahya utpatti yokka bahularekheeyata anedi "v" yokka aadhara vahakalu yokka bahya utpattula yokka sarala kalayikaga vyaktaparachadaaniki bahula vahakanni anumatistundi. v" yokka "k" basis vectors yokka bahya utpatti anedi "k"-vectors yokka space kosam prati basis elementnu nirminche praamaanika maargam, idhi "n"-dimensional vector space yokka bahya beejaganitamlo dimension () kaligi untundi.
praantam mariyu parimaanam.
n"-dimensional space loe "k" pratyeka vectorla yokka bahya utpatti nundi pondina "k"-vectorlo vectorla dwara vistarinchi unna "k"-samaantara lotop yokka anchanaa-vaalyuumlanu nirvachinche bhagalu untaayi. eee bhagala yokka vargaala mottam yokka varga muulam "k"-samaantara lotop yokka parimaanaanni nirvachistundi.
rendu kolatalalo unna bivector samaantara rekha yokka vaisaalyaanni kolustundani, moodu kolatalalo unna bivector yokka parimaanam kooda samaantara rekha yokka vaisaalyaanni kolustundani eee krindi udaaharanalu chuupistunnaayi. adevidhamgaa, moodu kolatalalo moodu-vector samaantara paipu yokka parimaanaanni kolustundi.
naalugu kolatalalo moodu-vector parimaanam eee vectorla dwara vistarinchi unna samaantara pipe yokka parimaanaanni kolustundo ledho tanikhi cheyadam sulabham.
r2loe multivectors.
dvi-miteeya vector sthalaanni parigananaloki teesukovadam dwara bahula vahakalu yokka lakshanaalanu chudavachu. basis vektarlu e1 mariyu e2 gaa undanivvandi, kabatti u mariyu v luu eee vidhamgaa ivvabadataayi
mariyu bivector anie kooda piluvabade multivector, ilaa lekkinchabadutundi
niluvu baarlu maatrika yokka nirnayadhikaaraanni suuchistaayi, idhi vektarlu u mariyu v dwara vistarinchi unna samaantara rekha yokka vaisaalyam. parimaanam eee samaantara chaturbhujam yokka vaisaalyam. v" kii rendu parimaanaalu unnanduna λ "v" loe basis bivector matrame multivector anie gamaninchandi.
multivector yokka parimaanam mariyu vectors dwara vistarinchi unna praantam ledhaa valume madhya sambandham anni kolatalalo oka mukhyamaina lakshanam. inca, eee vaalyuumnu lekkinche multivector yokka leanier functional versionnu differential roopam antaaru.
aarr3loe multivectors.
trimiteeya vector sthalaanni parigananaloki teesukovadam dwara multivectorla yokka marinni lakshanaalanu chudavachu. eee sandarbhamlo, basis vektarlu e1, e2 mariyu e3 gaa undanivvandi, kabatti u, v mariyu w luu eee vidhamgaa ivvabadataayi
mariyu bivector lekkinchabadutundi
eee bivector yokka bhagalu cross product yokka bhaagaalato samanamga untaayi. eee bivector yokka parimaanam daani bhagala yokka vargaala mottam yokka varga muulam.
idhi dvibhaajaka parimaanam anedi tri-miteeya sthalam "v" loe unnanduna u mariyu v vahakalu vistarinchi unna samaantara chaturbhuja praantam anie choopistundi. bivector yokka bhagalu moodu samanvaya samatalaalo pratidaanipai samaantaragrandhi yokka anchanaa praantaalu.
v" kii moodu parimaanaalu unnanduna, λ "v" loe oka aadharam moodu-vector undani gamaninchandi. moodu-vektarlanu lekkinchandi
suutram _ 11
idhi moodu-vector yokka parimaanam anedi u, v mariyu w anae moodu vectorla dwara vistarinchi unna samaantara pipe yokka parimaanam anie choopistundi.
adhika-dimensional pradaesaalaloo, bhaagam tri-vahakalu anevi tri-pradaesaala samanvayampai samaantara paipu yokka parimaanam yokka anchanaalu, mariyu tri-vaahaka parimaanam anedi adhika-dimensional pradaesamloe koorchunnanduna samaantara paipu yokka parimaanam.
grossman coordinatelu.
eee vibhaagamlo, memu projective space "p" n "paie multivectorlanu pariganistaamu, ivi grossman coordinatelu anie piluvabade paayintla sajatiya coordinatela maadirigaane lakshanaalanu kaligi unna linelu, plainlu mariyu hyperplanelaku anukuulamaina coordinatelanu andistaayi.
nijamaina projective space "p" n "looni paayintlu vector space r" n "+ 1 yokka muulam dwara vachhe rekhalagaa nirvachinchabadataayi. udaaharanaku, projective plain" p "2 anedi r3 yokka muulam dwara vachhe rekhala samiti. anduvalla, r" n "+ 1 paie nirvachinchina multivectorlanu" p "n" paie multivectorlugaa chudavachu.
p" "n" paie multivectornu chuudataaniki anukuulamaina maargam emitante, daanini "p" "n" yokka anubandha bhaagamlo pariseelinchadam, idhi r "n" + 1 yokka muulam dwara enchukunna hyperplanetho rekhala khandana, udaaharanaku ". r3 yokka muulam gundaa unna rekhalu, anantam vadda unna binduvulu anie piluvabade paayintlu matrame laeni projective plain yokka affine versionnu nirvachinchadaaniki plaennu kalustaayi.
p" 2 paie bahula vahakalu.
projective plain yokka affine bhaagamlooni paayintlaku coordinatelu untaayi. rendu paayintla sarala kalayika mariyu p mariyu q kalise rekhalo enii kalusukunna r3loe oka plain nii nirvachistundi. multivector r3 loe samaantara chaturbhujaanni nirvachistundi
p kii pratyaamnaayam eee multivectornu sthiraamkamtho gunistundani gamaninchandi. anduvalla, r3 yokka muulam dwara samatalaaniki sajatiya akshaamsaalu.
p mariyu q dwara rekhapai unna paayintla samiti anedi samatalamtho nirvachinchabadina samatalam yokka khandana. eee paayintlu santrupti chendutaayi, antey,
idhi oka rekha yokka sameekaranaanni saralikrutam chestundi
eee sameekaranam α mariyu β yokka vaastava viluvalaku paayintla dwara santrupti chendutundi.
λ" anae rekhanu nirvachinche moodu bhaagaalanu rekha yokka gaddi coordinatelu antaaru. moodu sajatiya akshaamsaalu oka binduvu mariyu oka rekha rendintinee nirvachistaayi kabatti, paayintla jyaamiti projective plaenlooni linela jyaamitiki dvandvamgaa untundi. dheenini dwaita suutram antaaru.
p" 3 paie bahula vahakalu.
trimiteeya prakshepaka sthalam, "p" 3 anedi r4 yokka muulam dwara anni rekhalanu kaligi untundi. trimiteeya hyper plain, paayintla dwara nirvachinchabadina prakshepaka sthalam yokka anubandha bhaagam avanivvandi. multivector r4 loe samaantara paipnu nirvachistundi
p kii pratyaamnaayam eee multivectornu sthiraamkamtho gunistundani gamaninchandi. anduvalla, r4 yokka muulam dwara 3-space kosam bhagalu sajatiya akshaamsaalu.
affine bhaagamlo oka plain anedi h yokka koodalilo 3-space dwara nirvachinchabadina paayintla samiti. eee paayintlu santrupti chendutaayi, antey,
idhi oka plain yokka sameekaranaanni saralikrutam chestundi
eee sameekaranam "α", "β" mariyu "γ" yokka nijamaina viluvala kosam paayintla dwara santrupti chendutundi.
aaa naalugu bhagalu "λ" samatalaanni nirvachistaayi, veetini samatalamloni gaddi coordinatelu antaaru. naalugu sajatiya akshaamsaalu projective space loe oka binduvu mariyu oka plain rendintinee nirvachistaayi kabatti, paayintla jyaamiti plains jyaamitiki dvandvamgaa untundi.
rendu paayintla kalayikaga oka rekhah projective space loe p mariyu q anae rendu paayintla dwara "λ" anae rekhanu r4 looni plain thoo affine space yokka khandanaga chudavachu. multivector line kii sajatiya coordinatelanu andistundi.
veetini rekha yokka plucker coordinatelu anie pilustaaru, ayinappatikee avi gaddi coordinatelaku kooda oka udaaharana.
rendu samatalaala kuudaligaa oka rekhah projective space loe oka rekha "μ" nuu grade moodu multivectors dwara nirvachinchabadina rendu samatalaala "π" mariyu "ρ" yokka kuudalini yerparuchukune x binduvula samitigaa kooda nirvachinchavacchu, kabatti x binduvulu sarala sameekaranaalaku parishkaralu.
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appudu
kabatti, "μ" rekha yokka plucker coordinatelu ilaa ivvabaddaayi
oka rekha yokka aaru sajatiya akshaamsaalanu rendu binduvula kalayika ledhaa rendu samatalaala koodali nundi pondavacchu kabatti, rekhanu projective space loe selph dual anie antaaru.
clifford utpatti.
doubleu. keey. clifford vector sthalamlo nirvachinchina antargata utpattitoe multivectorlanu militam chesi, saadhaarana sanklishta sankhyalu mariyu hamilton yokka quaternianlanu kaligi unna hyper complexes sankhyala kosam saadhaarana nirmaanaanni pondadaaniki.
rendu vektarlu u mariyu v madhya clifford utpatti bahya utpatti vale dwiparshwa mariyu anubandhamgaa untundi, mariyu clifford yokka sambandham dwara multivector uv lopali utpattitoe jatacheyabade adanapu lakshanaanni kaligi untundi,
clifford yokka sambandham lambamgaa unde vahakalu kosam anticommuting lakshanaanni kaligi undhi. dheenini r "n" looni paraspara orthogonal unit vectors nundi chudavachuhu clifford yokka relation digubadi
idhi aadhara vahakalu parasparam vyatiraekata kaligi unnayani choopistundi,
bahya utpattiki viruddhamgaa, vector yokka clifford utpatti sunna kaadu. deenni chuudataaniki, utpattini lekkinchandi
edhi digubadi istundi
clifford yokka utpattini upayoginchi nirminchina bahula vahakalu clifford beejaganitam anie piluvabade anubandha beejaganitamnu istundi. vividha lakshanaalatoo kuudina antargata utpattulanu vividha clifford algibralanu nirminchadaaniki upayoginchavacchu.
rekhaganita beejaganitam.
keey-blade" anae padaanni "clifford beejaganitam nundi rekhaganita ganana" (1984) loe upayoginchaaru.
jyaamiteeya beejaganitam anie piluvabade bhouthika shaastram yokka ganita suutreekaranaloo bahula karakalu pradhana paatra poeshistaayi. david hesteness prakaram,
2003loe [oka scalar mariyu] vectorla samiti yokka bahya utpattigaa vrayagala multivectorku "blade" anae padaanni c upayoginchindi. dorran mariyu a. lassenby. ikkada, "edainaa multivectornu bladela mottamgaa vyakteekarinchavachhu" anae prakatana dwara, scalarlu 0-bladelugaa parokshamgaa nirvachinchabadataayi.
rekhaganita beejaganitamlo, oka bahula-vahakanni scalar, vector mariyu 2-vector yokka sankalanam vanti vividha-grade "k"-bladela mottamgaa nirvachistaaru. k"-grade bhagala mottaanni matrame "k"-vector ledhaa "sajatiya" multivector antaaru.
oka pradaesamloe atyadhika grade muulakaanni "sudoskelar" antaaru.
icchina moolakam grade "k" kii sajaateeyamgaa unte, adhi "k"-vector, conny tappanisarigaa "k"-blade kaadu. atuvanti moolakam "k"-blade, dheenini "k" vectorla yokka bahya utpattigaa vyakteekarinchavachhu. 4 dimensional vector space dwara utpatti cheyabadina rekhaganita beejaganitam oka udaaharanhatho binduvunu vivaristundihi xy-plain nundi teesukunna edainaa rendu bladela mottam mariyu zw-plain nundi teesukunna marokati 2-blade kaani 2-vektarnu yerparustundi. 2 ledhaa 3 parimaanam gala vector space dwara utpatti cheyabadina rekhaganita beejaganitamlo, 2-bladela mottam mottaalanu oke 2-bladelugaa vraayavacchu.
udaaharanalu.
valume roopam samakshamlo (lopali utpatti mariyu dhorani ivvadam vantivi), sudovectorlu mariyu sudoskelarlanu vektarlu mariyu skelarlatho gurtinchavacchu, idhi vector calculuslo nityakrutyam, conny valume roopam lekunda idhi ekapakshamgaa empika cheyakunda cheyalemu.
bhouthika sthalam yokka beejaganitamlo (euclidean 3-space yokka rekhaganita beejaganitam, (3 + 1)-antariksha samayam yokka namoonaga upayoginchabadutundi), oka scalar mariyu vector yokka mottaanni paravector anie pilustaaru mariyu antariksha samayamlo oka binduvunu suuchistumdi (vector space, scalar samayam).
bivectors.
bivector anedi oka sparsha sthalam yokka pratisamiteeya tensor utpatti yokka oka moolakam.
rekhaganita beejaganitamlo kooda, bivector anedi rendu vectorla cheelika utpatti phalitamgaa vachhe grade 2 moolakam (2-vector), kabatti idhi rekhaaganitaparamgaa "oriented area", adhe vidhamgaa "vector" anedi oriented line segment. a mariyu b rendu vahakalu ayithe, bivector kaligi untundi
bivectorlu sudovectorlatho anusandhaaninchabadi untaayi mariyu rekhaganita beejaganitamlo bhramanaalanu suuchimchadaaniki upayoginchabadataayi.
bivectorlu vector space λ2 "v" yokka muulakaalu kabatti (ikkada "v" anedi parimita-dimensional vector space), eee vector space paie antargata utpattini eee krindi vidhamgaa nirvachinchadam ardhame. modata, edainaa muulakaanni praatipadika paramgaa ilaa vraayandi
ikkada inestein summation convention upayoginchabadutondi.
ippudu nokki cheppadam dwara myapnu nirvachinchandi
ikkada suutram _ 31 anedi sankhyala samiti.
applications.
bhouthika saastramlo bivectorlu aneka mukhyamaina paatralanu poeshistaayi, udaaharanaku, vidyudayaskaanta kshetraala vargeekaranalo.

