
prachaalak (ganit)

ganit mein, ack prachaalak aam taur parr ack maanchitran yaa kaary haye joe ack sthaan key tatvon parr kaary kartaa haye taaki dusare sthaan key tatvon kaa utpaadan kiya jaa sakey (sambhavatah aur kabhi-kabhi ack hii sthaan honey kii aavashyakta hoti haye). prachaalak" kii koee samanya paribhasha naheen haye, lekin iss shabd kaa upayog aksar "kaary" key sthaan parr kiya jaataa haye jaba domen karyon yaa anya sanrachit vastuon kaa ack samooh hotaa haye. saath hii, ack prachaalak key kshetra koo spasht roop seey chihnit karna aksar mushkil hotaa haye (udaaharan key liye ack abhinn prachaalak key mamle mein), aur ise vistaarit kiya jaa sakata haye taaki sambandhit vastuon parr kaary kiya jaa sakey (ack prachaalak joe karyon parr kaary kartaa haye vaha vibhedak samikaranon parr bhee kaary curr sakata haye jinake samadhan aise kaary hain joe sameekaran koo santusht karate hain). anya udaharanon key liye prachaalak (bhautiki) dekhein.
sabse buniyaadi prachaalak raikhik maanchitra hain, joe sadish rikt sthaan parr kaary karate hain. raikhik prachaalak raikhik maanchitron koo sandarbhit karate hain jinka kshetra aur seema ack hii sthaan haye, udaaharan key liye sootra _ 1 seey sootra _ 1.
aise prachaalak aksar gunon koo sanrakshit karate hain, jaise kii nirantarta. udaaharan key liye, vibhedan aur anishchit ekikaran raikhik prachaalak hain; joe prachaalak unase banaae jaate hain unhein vibhedak prachaalak, abhinn prachaalak yaa samakalan-vibhedak prachaalak kahaa jaataa haye.
prachaalak kaa upayog ganitiya sanchaalan key prateek koo darshaane key liye bhee kiya jaataa haye. yeh computer prograaming mein "operator" key arth seey sambandhit haye, operator (computer prograaming) dekhein.
raikhik prachaalak.
sabse aam prakaar key prachaalak "raikhik prachaalak" hain. maan leejiye kii "u" aur "v" ack kshetra "k" parr vector rikt sthaan hain. ack maanchitran "a": "u" "v" raikhik haye yadi sabhi x key liye sootra _ 3, "u" mein y aur sabhi "α" key liye, "k" mein "β".
iska matlab haye kii ack raikhik prachaalak sadish sthaan sanchaalan koo sanrakshit kartaa haye, iss arth mein kii isase koee fark naheen padata kii aap jodd aur scalar gunan key sanchaalan seey pahale yaa baad mein raikhik prachaalak koo lagu karate hain yaa naheen. adhik takniki shabdon mein, raikhik prachaalak vector rikt sthaan key beach rupankan hain.
parimit-ayami mamle mein raikhik prachaalak koo nimnalikhit tareeke seey matricks dwara darshaaya jaa sakata haye. sootra 4 koo ack kshetra honey dein, aur sootra 5 aur sootra 6 koo sootra 4 parr parimit-ayami sadish sthaan honey dein. aaie humm sootra 5 mein ack aadhar sootra 8 aur sootra 6 mein sootra 10 kaa chayan karein. phir sootra 12 koo sootra 5 mein ack manmana sadish honey dein (einstein parampara koo maanate hue), aur sootra 14 ack raikhik prachaalak hoe. phir formula _ 15 phir formula _ 16 nishchit aadharon mein prachaalak sootra _ 17 kaa matricks haye. sootra _ 18 sootra _ 19 key chayan parr nirbhar naheen kartaa haye, aur sootra _ 20 yadi sootra _ 21 haye. iss prakaar nishchit aadharon mein n-by-m matricks sootra _ 5 seey sootra _ 6 taka raikhik operators key liye dwi-abhikshepit patrachar mein hain.
seemit-ayami sadish rikt sthaan key beach prachaalak seey seedhe sambandhit mahatvapurna avadhaaranaaen shreni, nirdhaarak, vyutkram prachaalak aur igenspace kii hain.
raikhik prachaalak bhee anant-ayami mamle mein ack badee bhoomika nibhaate hain. shreni aur nirdhaarak kii avadhaaranaon koo anant-ayami matricks taka naheen badhaaya jaa sakata haye. yahee kaaran haye kii anant-ayami mamle mein raikhik prachaalak (aur samanya roop seey prachaalak) kaa adhyayan karate samay bahut alag takneekon kaa upayog kiya jaataa haye. anant-ayami mamle mein raikhik prachaalak key adhyayan koo karyatmak vishleshan key roop mein jaanaa jaataa haye (isaliye kahaa jaataa haye kyonki karyon key vibhinn varg anant-ayami sadish rikt sthaan key dilchasp udaaharan banaate hain).
vaastavik sankhyaon key anukramon kaa sthaan, yaa kissy bhee sadish sthaan mein sadishon key adhik aam anukram, swayam ack anant-ayami sadish sthaan banaate hain. sabse mahatvapurna mamle vaastavik yaa jatil sankhyaon key anukram hain, aur yeye sthaan, raikhik upasthanon key saath, anukram sthaan key roop mein jaane jaate hain. inn rikt sthaan parr prachaalak anukram parivartan key roop mein jaane jaate hain.
maanak prachaalak maanak key sambandh mein banach sthaan parr bandhe raikhik prachaalak ack banach beejganit banaate hain. banach algebra kaa siddhant spectra kii ack bahut hii samanya avadharana vikasit kartaa haye joe eigenspace key siddhant koo sunder roop seey samanya banaata haye.
baadhya prachaalak.
maan leejiye kii "u" aur "v" ack hii krambaddh kshetra parr dou sadish sthaan hain (udaaharan key liye, sootra _ 24), aur way maandandon seey lais hain. taba "u" seey "v" taka key ack raikhik prachaalak koo baadhya kahaa jaataa haye yadi "c"> 0 maujood haye tou "u" mein sabhi x key liye aisaa sootra _ 25 haye.
bounded operator ack vector space banaate hain. iss sadish sthaan parr humm ack maanak pesh curr sakate hain joe "u" aur "v" key maandandon key saath sangat haih sootra _ 26in operators key mamle mein "u" seey yeh dikhaaya jaa sakata haye kii iss sampatti key saath formula _ 27 koo ack abhinn maanak beejganit kahaa jaataa haye. iss tarah key algebra key liye varnakramiya siddhant koo samanya banaana sambhav haye. see *-algebra, joe kuchh atirikt sanrachana key saath banach algebra hain, quantum yantriki mein ack mahatvapurna bhoomika nibhaate hain.
udaaharan.
jyaamiti.
jyaamiti mein, vector rikt sthaan parr atirikt sanrachanaon kaa kabhi-kabhi adhyayan kiya jaataa haye. aise vector rikt sthaan koo dwi-uddeshyapurna roop seey maanchitrit karane vaale prachaalak inn adhyayanon mein bahut upayogi hain, way swabhavik roop seey sanrachana dwara samooh banaate hain.
udaaharan key liye, ack sadish sthaan kii sanrachana koo sanrakshit karane vaale dwi-abhikshepak prachaalak sateek roop seey vyutkramaniya raikhik prachaalak hain. way sanrachana key tahat samanya raikhik samooh banaate hain. way prachaalak key jodd key tahat ack vector sthaan naheen banaate hain, ee.jee. aaee. dee. aur-aaee. dee. donon vyutkramaniya (dwi-abhikshep) hain, lekin unakaa yog, 0, naheen haye.
aise sthaan parr euclidean metrick koo sanrakshit karane vaale prachaalak samamit samooh banaate hain, aur joe mool koo thheek karate hain way ack upsamooh banaate hain jise orthogonal samooh key roop mein jaanaa jaataa haye. orthogonal samooh mein sanchaalak joe vector tuple key abhivinyas koo bhee sanrakshit karate hain, vishesh orthogonal samooh, yaa ghoornan key samooh kaa nirmaan karate hain.
sambhaavyata siddhant.
prachaalak sambhavana siddhant mein bhee shamil hain, jaise kii apeksha, bhinnata aur sahprasaran. vaastav mein, pratyek sahprasaran mool roop seey ack bindu utpaad haye; pratyek prasaran apane aap mein ack sadish kaa ack bindu utpaad haye, aur iss prakaar ack dwighaat maanak haye; pratyek maanak vichalan ack maanak haye (dwighaat maanak kaa vargamool); iss bindu utpaad key liye sambandhit cosine nashpati sahsambandh gunank haye; apekshit moolya mool roop seey ack abhinn prachaalak haye (antariksh mein bhaarit aakaron koo maapane key liye upayog kiya jaataa haye).
cullon.
karyatmak vishleshan key drishtikon seey, cullon dou raikhik prachaalak kaa adhyayan haih vibhedak prachaalak sootra _ 28, aur voltera prachaalak sootra _ 29.
forier shrinkhala aur forier parivartan.
forier parivartan vyaavhaarik ganit, vishesh roop seey bhautiki aur sanket prasanskaran mein upayogi haye. yeh ack aur abhinn prachaalak haye; yeh mukhya roop seey upayogi haye kyonki yeh ack (asthaayi) domen parr ack function koo dusare (aavritti) domen parr ack function mein parivartit kartaa haye, ack tarah seey prabhaavi roop seey vyutkramaniya haye. koee jaankaari naheen khou jaati haye, kyonki ack vyutkram parivartan prachaalak haye. aavart falan key saral mamle mein, yeh parinam iss pramey parr adharit haye kii kissy bhee nirantar aavart falan koo saain tarangon aur cosine tarangon kii ack shrinkhala key yog key roop mein darshaaya jaa sakata haih sootra _ 30-tupal ("a" 0 ", a" 1 ", b" 1 ", a" 2 ", b" 2 ",)...) vaastav mein ack anant-ayami sadish sthaan "l" kaa ack tatva haye, aur iss prakaar fourier shrinkhala ack raikhik prachaalak haye.
samanya falan sootra _ 31 key saath kaam karate samay, parivartan ack abhinn roop letaa haih sootra _ 32
lapless parivartan.
lapless transform" ack anya abhinn prachaalak haye aur vibhedak samikaranon koo hull karane kii prakriya koo saral banaane mein shamil haye.
f" = "f" ("s") diye jaane parr, ise sootra _ 33 dwara paribhaashit kiya jaataa haih
scalar aur vector kshetron parr maulik prachaalak.
vector cullon key liye teen prachaalak mahatvapurna hainh
vector cullon prachaalak key bhautik vigyaan, engineering aur tensor space key vistaar key roop mein, grade, dive aur curl prachaalak bhee aksar tensor cullon key saath-saath vector cullon key saath judey hotey hain.

