
multivector

bahurekhiya beejganit mein, ack bahubhaajak, jise kabhi-kabhi clifford sankhya kahaa jaataa haye, ack sadish sthaan key baahari beejganit kaa ack tatva haye. yeh beejganit shrenibaddh, sahayogi aur vaikalpik haye, aur isamen saral-vektaron (jinhen apghatit-vector yaa-blade key roop mein bhee jaanaa jaataa haye) key raikhik sanyojan hotey hain.
jahan sootra _ 2 hain.
aeye-vector ack aisaa raikhik sanyojan haye joe degree kaa "sajaatiya" haye (sabhi shabd ack hii key liye-blade hain). lekhakon key aadhar parr, ack "bahu-vaahak" yaa tou ack-vaahak yaa baahari beejganit kaa koee bhee tatva hoe sakata haye (sambhaavit roop seey bhinn moolyon key saath-blade kaa koee bhee raikhik sanyojan).
vibhedak jyaamiti mein, ack-vector sparshrekha sadish sthaan key baahari beejganit mein ack sadish haye; arthaat, yeh kuchh purnank key liye sparshrekha sadish key baahari utpaad key raikhik sanyojan koo laker praapt ack pratisamitiya tanyata haye. ack vibhedak-roop sparshrekha sthaan key dohare key baahari beejganit mein ack-vaahak haye, joe sparshrekha sthaan key baahari beejganit kaa bhee dohra haye.
aur,-vektaron koo aksar kramshah "scalar", "vactor", "bivector" aur "trivector" kahaa jaataa haye; way kramshah dohare seey 0-roop, 1-roop, 2-roop aur 3-roop hain.
baahari utpaad.
bahukriyaon key nirmaan key liye upayog kiya jaane vaalaa baahari utpaad (jise ways utpaad bhee kahaa jaataa haye) bahurekhiya (pratyek input mein raikhik), sahayogi aur vaikalpik hotaa haye. iska matlab haye kii vector sthaan "v" mein vactor u, v aur w key liye aur scalar "α", "β" key liye, baahari utpaad mein guna hainh
key" vactor key baahari utpaad yaa aise utpaadon key yog (ack ekal "key" key liye) koo grade "key" multivector, yaa "key"-vector kahaa jaataa haye. ack bahubhaajak kaa adhikatam grade vector sthaan "v" kaa ayam haye.
vaikalpik guna key saath donon mein seey kissy bhee input mein raikhikta kaa arth haye dusare input mein raikhikta. baahari utpaad kii bahurekhiyata ack bahubhaajak koo "v" key aadhar vactors key baahari utpaadon key raikhik sanyojan key roop mein vyakt karane kii anumati detee haye. v" key "k" aadhar sadish kaa baahari utpaad "k"-sadish key sthaan key liye pratyek aadhar tatva key nirmaan kaa maanak tarika haye, jiska ayam () ack "n"-ayami sadish sthaan key baahari beejganit mein haye.
kshetrafal aur matra.
key" key baahari utpaad seey praapt "key"-vector mein "aine"-ayami sthaan mein alag-alag vactor hotey hain joe vactors dwara faile "key"-samanantar-aayatan key anumaanit-matra koo paribhaashit karate hain. inn ghatakon key vargon key yog kaa vargamool "key"-samanantar lotop key aayatan koo paribhaashit kartaa haye.
nimnalikhit udaharanon seey pataa chalata haye kii dou aayamon mein ack dwibhaajak ack samanantar chaturbhuj key kshetra koo maapata haye, aur teen aayamon mein ack dwibhaajak kaa parimaan bhee ack samanantar chaturbhuj key kshetra koo maapata haye. isi tarah, teen aayamon mein ack teen-vaahak ack samanantar pipe kii matra koo maapata haye.
yeh jaanchana aasaan haye kii chaar aayamon mein teen-vector kaa parimaan inn vactors dwara faile samanantar pipe kii matra koo maapata haye.
aar2 mein bahu-vaahak.
bahu-sadishon key gunon koo dou ayami sadish sthaan parr vichaar karke dekhaa jaa sakata haye. maan leejiye kii aadhar vactor e1 aur e2 hain, isaliye u aur v dwara diye gaye hain
aur multivector, jise bivector bhee kahaa jaataa haye, kii ganana kii jaati haye
urdhvaadhar pattiyan matricks key nirdhaarak koo darshaati hain, joe vactor u aur v dwara faile samanantar chaturbhuj kaa kshetra haye. iska parimaan iss samanantar chaturbhuj kaa kshetrafal haye. dhyaan dein kii kyonki "v" kaa ayam dou haye, isaliye aadhar dwibhaajak λ "v" mein ekmatra bahubhaajak haye.
ack bahubhaajak key parimaan aur vactor dwara faile kshetra yaa aayatan key beach sambandh sabhi aayamon mein ack mahatvapurna visheshta haye. isake alawa, ack bahubhaajak kaa raikhik karyatmak sanskaran joe iss aayatan kii ganana kartaa haye, ack vibhedak roop key roop mein jaanaa jaataa haye.
aar3 mein bahu-vaahak.
tri-ayami sadish sthaan parr vichaar karke bahu-sadishon kii adhik visheshtaon koo dekhaa jaa sakata haye. iss mamle mein, aadhar vactor koo e1, e2 aur e3 honey dein, isaliye u, v aur w dwara diye gaye hain.
aur dwibhaajak kii ganana kii jaati haye
iss bivector key ghatak cross utpaad key ghatakon key samaan hain. iss dwibhaajak kaa parimaan isake ghatakon key vargon key yog kaa vargamool haye.
yeh darshaata haye kii dwibhaajak kaa parimaan vactor yuu aur vee dwara faile samanantar chaturbhuj kaa kshetra haye kyonki yeh tri-ayami sthaan "vee" mein sthit haye. dwibhaajak key ghatak teen nirdeshank vimaanon mein seey pratyek parr samanantar chaturbhuj key anumaanit kshetra hain.
dhyaan dein kii kyonki "v" kaa ayam teen haye, isaliye λ "v" mein ack aadhar teen-vector haye. teen-vector kii ganana karein
sootra _ 11
yeh darshaata haye kii teen-vector kaa parimaan teen vactors u, v aur w dwara faile samanantar pipe kaa aayatan haye.
uchch-ayami sthaanon mein, ghatak teen-vector tri-sthaanon key nirdeshank parr ack samanantar pipe kii matra key anuman hain, aur tri-vector kaa parimaan samanantar kii matra haye kyonki yeh uchch-ayami sthaan mein baithata haye.
ghaas key aadami nirdeshank.
iss khand mein, humm ack prakshepya sthaan "pee" aine "parr bahu-vaahak parr vichaar karate hain, joe rekhaaon, vimaanon aur atee-vimaanon key liye nirdeshank kaa ack suvidhajanak sett pradaan kartaa haye, jinamen binduon key sajaatiya nirdeshank key samaan guna hotey hain, jinhen ghaas key nirdeshank kahaa jaataa haye.
ack vaastavik prakshepya sthaan "p" "n" mein binduon koo sadish sthaan r "n" + 1 key mool key madhyam seey rekhaaon key roop mein paribhaashit kiya jaataa haye. udaaharan key liye, prakshepya tala "p" 2 r3 key mool key madhyam seey rekhaaon kaa samooh haye. iss prakaar, r "n" + 1 parr paribhaashit bahu-sadishon koo "p" n "parr bahu-sadishon key roop mein dekhaa jaa sakata haye.
pee" aine "parr ack bahubhaajak koo dekhane kaa ack suvidhajanak tarika haye kii ise" pee "aine" key ack sanlagn ghatak mein jancha jaae, joe kii ack chayanit hyperplane key saath aar "aine" + 1 key mool key madhyam seey rekhaaon kaa pratichhedan haye, jaise kii. aar3 key mool key madhyam seey rekhaaen prakshepya tala key ack sanlagn sanskaran koo paribhaashit karane key liye tala koo kaatati hain jisamen kewal una binduon kaa abhaav hotaa haye jinake liye anant parr bindu kahaa jaataa haye.
pee" 2 parr bahu-kaarak.
prakshepya tala key affine ghatak mein binduon mein nirdeshank hotey hain. dou binduon kaa ack raikhik sanyojan aur r3 mein ack tala koo paribhaashit kartaa haye joe p aur q mein shamil honey vaali rekha mein e koo kaatata haye. multivector dwara diye gaye r3 mein ack samanantar chaturbhuj koo paribhaashit kartaa haye
dhyaan dein kii p key liye pratisthaapan iss bahubhaajak koo ack sthiraank seey gunaa kartaa haye. isaliye, key ghatak r3 key mool key madhyam seey tala key liye sajaatiya nirdeshank hain.
pee aur kyu key madhyam seey rekha parr binduon kaa sett samtal key saath paribhaashit tala kaa pratichhedan haye. yeye bindu santusht karate hain, arthaat,
joe ack rekha key sameekaran koo saral banaata haye
yeh sameekaran α aur β key vaastavik moolyon key liye binduon seey santusht haye.
rekha koo paribhaashit karane vaale teen ghatakon koo rekha kaa ghaas kaa nirdeshank kahaa jaataa haye. kyonki teen sajaatiya nirdeshank ack bindu aur ack rekha donon koo paribhaashit karate hain, binduon kii jyaamiti koo prakshepya tala mein rekhaaon kii jyaamiti key liye dohri kahaa jaataa haye. ise dwait kaa siddhant kahaa jaataa haye.
pee" 3 parr bahu-kaarak.
tri-ayami prakshepya sthaan, "pee" 3 mein aar4 kii utpatti key madhyam seey sabhi rekhaaen hoti hain. tri-ayami atee-tala, binduon dwara paribhaashit prakshepya sthaan kaa sambaddh ghatak hoe. multivector dwara diye gaye r4 mein ack samanantar koo paribhaashit kartaa haye
dhyaan dein kii p key liye pratisthaapan iss bahubhaajak koo ack sthiraank seey gunaa kartaa haye. isaliye, key ghatak r4 key mool key madhyam seey 3-sthaan key liye sajaatiya nirdeshank hain.
affine ghatak mein ack tala aich key pratichhedan mein 3-sthaan key saath paribhaashit binduon kaa samooh haye. yeye bindu santusht karate hain, arthaat,
joe ack samtal key sameekaran koo saral banaata haye
yeh sameekaran "α", "β" aur "γ" key vaastavik moolyon key liye binduon seey santusht haye.
samtal koo paribhaashit karane vaale chaar ghatakon koo samtal kaa ghaas kaa nirdeshank kahaa jaataa haye. kyonki chaar sajaatiya nirdeshank prakshepya sthaan mein ack bindu aur ack tala donon koo paribhaashit karate hain, binduon kii jyaamiti samtalon kii jyaamiti key liye dohri hoti haye.
dou binduon key judaav key roop mein ack rekhaah prakshepya sthaan mein dou binduon p aur q key madhyam seey rekha "λ" koo r4 mein tala key saath sanlagn sthaan key pratichhedan key roop mein dekhaa jaa sakata haye. bahubhaajak rekha key liye sajaatiya nirdeshank pradaan kartaa haye.
inhen rekha key plucker nirdeshank key roop mein jaanaa jaataa haye, halanki way ghaas key nirdeshank kaa ack udaaharan bhee hain.
dou vimaanon key pratichhedan key roop mein ack rekhaah prakshepya sthaan mein ack rekha "μ" koo una binduon key samooh key roop mein bhee paribhaashit kiya jaa sakata haye joe dou vimaanon "π" aur "ρ" key pratichhedan koo grade teen bahubhaajak dwara paribhaashit karate hain, isaliye bindu x raikhik samikaranon key samadhan hain.
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taba
tou, rekha "μ" key plucker nirdeshank dwara diye jaate hain
kyonki ack rekha key chhah sajaatiya nirdeshank dou binduon key judaav yaa dou vimaanon key pratichhedan seey praapt kiye jaa sakate hain, isaliye rekha koo prakshepya sthaan mein swa-dohri kahaa jaataa haye.
clifford utpaad.
doubleu. key. clifford ney vector sthaan parr paribhaashit aantarik utpaad key saath bahu-vaahak koo sanyukt kiya, taaki atee-jatil sankhyaon key liye ack samanya nirmaan praapt kiya jaa sakey jisamen samanya jatil sankhyaaen aur hamilton key chaturthak shamil hain.
dou vactor yuu aur vee key beach clifford utpaad baahari utpaad kii tarah dwirangi aur sahayogi haye, aur isamen atirikt guna haye kii multivector yuvi koo clifford key sambandh dwara aantarik utpaad seey jodaa jaataa haye,
clifford kaa sambandh vactor key liye anticommuting guna koo barkarar rakhta haye joe lambavat hain. ise aar "aine" mein parasparik orthogonal ikai vactors seey dekhaa jaa sakata haih clifford kaa sambandh upaj
joe darshaata haye kii aadhar vactor parasparik roop seey virodhi hain,
baahari utpaad key vipareet, ack vector kaa clifford utpaad apane aap mein shoonya naheen haye. ise dekhane key liye, utpaad kii ganana karein
joe upaj detaa haye
clifford key utpaad kaa upayog karke nirmit bahu-sadishon kaa samooh ack sahayogi beejganit detaa haye jise clifford beejganit key roop mein jaanaa jaataa haye. vibhinn gunon vaale aantarik utpaadon kaa upayog vibhinn clifford algebra key nirmaan key liye kiya jaa sakata haye.
jyaamitiya beejganit.
key-blade" shabd kaa upayog "clifford beejganit seey jyaamitiya cullon" (1984) mein kiya gayaa thaa.
bahu-kaarak bhautik vigyaan key ganitiya nirmaan mein ack kendriya bhoomika nibhaate hain jise jyaamitiya beejganit key roop mein jaanaa jaataa haye. david hesten key anusaar,
2003 mein ack bahubhaajak key liye "blade" shabd jise [ack scalar aur] vactor key ack sett key baahari utpaad key roop mein likhaa jaa sakata haye, kaa upayog see dwara kiya gayaa thaa. dorran aur aeye. lessenby. yahaan, "kissy bhee bahubhaajak koo blade key yog key roop mein vyakt kiya jaa sakata haye", iss kathan dwara, scalar koo nihit roop seey 0-blade key roop mein paribhaashit kiya gayaa haye.
jyaamitiya beejganit mein, ack bahubhaajak koo vibhinn-shreni "k"-blade key yog key roop mein paribhaashit kiya jaataa haye, jaise kii ack scalar, ack vector aur ack 2-vector kaa yog. kewal "key"-shreni ghatakon key yog koo "key"-vector yaa "sajaatiya" bahu-vaahak kahaa jaataa haye.
ack sthaan mein uchchatam shreni tatva koo "chhadm kaalar" kahaa jaataa haye.
yadi koee diyaa gayaa tatva ack grade "key" kaa sajaatiya haye, tou yeh ack "key"-vector haye, lekin jaruri naheen kii ack "key"-blade hoe. iss tarah kaa tatva ack "key"-blade haye jaba ise "key" vactor key baahari utpaad key roop mein vyakt kiya jaa sakata haye. 4-ayami sadish sthaan dwara utpann ack jyaamitiya beejganit ack udaaharan key saath bindu koo darshaata haih xy-tala seey liye gaye kissy bhee dou blade kaa yog aur zw-tala seey liye gaye dusare key saath ack 2-vector kaa nirmaan karega joe 2-blade naheen haye. 2 yaa 3 ayam key sadish sthaan dwara utpann ack jyaamitiya beejganit mein, 2-blade key sabhi yog koo ekal 2-blade key roop mein likhaa jaa sakata haye.
udaaharan.
aayatan roop kii upasthiti mein (jaise kii ack aantarik utpaad aur ack abhivinyas diyaa gayaa haye), chhadm-kaarak aur chhadm-varnakon koo vactor aur scalar key saath pahachaana jaa sakata haye, joe vector cullon mein niyamit haye, lekin aayatan roop key binaa yeh ack manmana vikalp banaae binaa naheen kiya jaa sakata haye.
bhautik sthaan key beejganit mein (euclidean 3-space kaa jyaamitiya beejganit, (3 + 1)-spacetime key model key roop mein upayog kiya jaataa haye), ack scalar aur ack vector kaa yog ack paravector kahaa jaataa haye, aur spacetime mein ack bindu kaa pratinidhitva kartaa haye (vector space, scalar samay).
bivectors.
ack bivector ack sparshrekha sthaan key pratisamitiya tensor utpaad kaa ack tatva haye.
jyaamitiya beejganit mein, ack dwibhaajak dou vactors key ways utpaad key parinamswarup ack grade 2 tatva (ack 2-vector) haye, aur isaliye yeh jyaamitiya roop seey ack "unmukh kshetra" haye, usee tarah ack "vector" ack unmukh rekha khand haye. yadi a aur b dou vactor hain, tou bivector key paas haye
dwibhaajak chhadm-vaahak seey judey hotey hain, aur jyaamitiya beejganit mein ghoornan kaa pratinidhitva karane key liye upayog kiye jaate hain.
jaisa kii dwibhaajak ack sadish sthaan key tatva hain λ2 "v" (jahan "v" ack parimit-ayami sadish sthaan haye), iss sadish sthaan parr ack aantarik utpaad koo nimnanusar paribhaashit karna samajhadaari haye. pahale, kissy bhee tatva koo aadhar key roop mein likhein
jahan einstein yog parampara kaa upayog kiya jaa rhaa haye.
abb iss baat parr jore dekar ack maanchitra koo paribhaashit karein kii
jahan sootra _ 31 sankhyaon kaa ack samooh haye.
anuprayog.
bivector bhautiki mein kaee mahatvapurna bhoomikaaen nibhaate hain, udaaharan key liye, vidyut chumbakiya kshetron key vargeekaran mein.