
dell

dell, ledhaa nabla, ganitamlo (mukhyamgaa vector calculuslo) vector differential operatorga upayoginche operator, saadhaaranamgaa nabla chihnam ρ dwara suuchimchabadutumdi. oka dimensional domainlo nirvachinchina functionku vartimpajesinappudu, idhi calculuslo nirvachinchina vidhamgaa function yokka praamaanika utpannaanni suuchistumdi. oka "feeled" (bahula-dimensional domainpai nirvachinchina function) kuu vartimpajesinappudu, adhi vartimpajese vidhananni batti moodu operationlalo deeninainaa suuchimchavachchuh scalar feeled yokka pravanata ledhaa (sthaanikamgaa) nitaruga vaalu (ledhaa konnisaarlu navier-stokes sameekaranaalalo vale vector feeled); vector feeled yokka divergence; ledhaa vector feeled yokka curl (rotation).
dell anedi aaa moodu operationlaku (pravanata, divergence mariyu curl) chaalaa soukaryavantamaina ganita sanketam, idhi aneka sameekaranaalanu vraayadaaniki mariyu gurthunchukovadaaniki sulabhataram chestundi. dell chihnanni (ledhaa nabla) trimiteeya vector operatorga adhikarikamga nirvachinchavacchu, dheeni moodu bhagalu sambandhita pakshika utpanna operatorlu. vector operatorga, idhi scalar mariyu vector feeldlapy moodu ververu margallo panicheyagaladu, idhi moodu ververu avakalana kaaryakalaapaalaku daariteestundihi modatidi, idhi scalar feeldlapy "formal" scalar gunakaaram dwara panicheyagaladu-gradient anie piluvabade vector feeldnu ivvadaaniki; rendavadi, idhi vector feeldlapy "formal" dott product dwara panicheyagaladu-divergence anie piluvabade scalar feeldnu ivvadaaniki; chivaraga, idhi vector feeldlapy "formal" cross product dwara panicheyagaladu-curl anie piluvabade vector feeldnu ivvadaaniki. eee "adhikaarika" utpattulu itara operatorlu ledhaa utpattulato tappanisarigaa prayaaninchavu. eee krinda vivarinchina eee moodu upayogaalu ilaa sangrahinchabaddayih
nirvachanam.
cartician coordinate system formula 4 loe coordinates formula 5 mariyu standard basis formula 6 thoo, dell anedi vector operator, dheeni formula 7 bhagalu partial derivative operators formula 8; antey,
ikkada kundaleekaranaalalo vyakteekarana oka roow vector. tri-miteeya cartician coordinate system formula _ 10 loe coordinates formula _ 11 mariyu praamaanika aadharam ledhaa akshaala suutram _ 12 yokka unit vektarlatho, dell ilaa vraayabadutundi
vector operatorga, dell sahajamgaa scalar gunakaaram dwara scalar kshetraalapai panichestundi mariyu sahajamgaa chukkala utpattulu mariyu cross utpattula dwara vector kshetraalapai panichestundi.
marinta pratyaekamgaa, edainaa scalar feeled formula _ 14 mariyu edainaa vector feeled formula _ 15 kosam, okati "nirvachinchinatlayithe"
appudu suutram _ 21 yokka paina perkonna nirvachananni upayoginchi, okaru vraayavacchu.
mariyu
mariyu
delnu itara samanvaya vyavasthalalo kooda vyakteekarinchavachhu, udaaharanaku delnu sthuupaakaara mariyu golaakaara akshaamsaalalo chudandi.
sangnaatmaka upayogaalu.
aneka sudeergha ganita vyakteekaranalanu saralikrutam cheyadaaniki dell nuu sankshipta roopamgaa upayogistaaru. idhi saadhaaranamgaa pravanata, divergence, curl, directional derivative mariyu laplasian kosam vyakteekaranalanu saralikrutam cheyadaaniki upayoginchabadutundi.
pravanata.
scalar feeled formula _ 14 yokka vector utpannaanni pravanata antaaru, mariyu dheenini ilaa suuchimchavachchuh
idhi ellappuduu suutram _ 14 yokka atyadhika perugudala disalo suuchistumdi, mariyu idhi oka praamaanika utpannam vale, binduvu vadda garishta perugudala retuku samaanamaina parimaanaanni kaligi untundi. mukhyamgaa, oka kondanu samatala suutram _ 30 paie etthu functionga nirvachinchinatlayithe, icchina pradaesamloe pravanata xy-samatalamlo (patamlo baanam vale chudadaginadi) atyanta nitaruga unna disalo choopinche vector avutundi. pravanata yokka parimaanam eee atyanta nitaruga vaalu yokka viluva.
mukhyamgaa, eee sangnaamaanam saktivantamainadi endukante pravanata utpatti niyamam 1dii-utpanna kesutho chaalaa pooli untundih
ayithe, dott utpattula niyamaalu saralamgaa undavu, ivi vivarinchina vidhamgaah
vibhedaalu.
vector feeled yokka vaividhyam
suutram _ 33 anedi oka scalar feeled, dheenini ilaa suuchimchavachchuh
vichalanam anedi vector feeled adhi soochinche disalo perugudala yokka kolata; conny marinta khachitamgaa, idhi oka binduvu vaipu kalusukune ledhaa nundi veruchese aaa feeled yokka dhorani yokka kolata.
dell sangnaamaanam yokka shakti eee krindi utpatti niyamam dwara choopabadutundih
vector utpatti suutram konchem takkuva sahajamainadi, endukante eee utpatti parivartana kaaduh
curl.
vector feeled formula _ 37 yokka curl anedi vector function, dheenini ilaa suuchimchavachchuh
oka binduvu vadda curl anedi aann-axis tarkku anulomaanupaatamlo untundi, aaa samayamlo kendrikrutamai unte oka chinna pinveel lobadi untundi.
vector utpatti karyakalapalanu nakilee-nirnayakamgaa chudavachuhu
mallii sangnaamaanam yokka shakti utpatti niyamam dwara choopabadutundih
vector utpatti kosam niyamam saralamgaa maaradu.
disaatmaka utpannam.
disalo scalar feeled formula yokka disaatmaka utpannam _ 42
suutram _ 43 ilaa nirvachinchabadindih
idhi formula _ 46 yokka parimaanamtho scale cheyabadina formula _ 46 disalo feeled formula _ 14 yokka maarpu retunu istundi. operator sangnaamaanamlo, kundaleekaranamloni muulakaanni oke pondikaina unitga pariganinchavacchu; dravam dynamics eee sampradaayaanni vistrutamgaa upayogistundi, dheenini ushnaprasarana utpannam-dravam yokka "kadile" utpannam anie pilustundi.
formula _ 48 anedi scalarnu scalarga teesukune operator anie gamaninchandi. daani prati bhagampai vidigaa panicheyadam dwara, vektarpai panicheyadaaniki dheenini vistarinchavacchu.
laplasian.
laplace operator anedi vector ledhaa scalar feeldlaku vartimpajeyagala scalar operator; cartician coordinate systems kosam idhi ilaa nirvachinchabadindih
mariyu marinta saadhaarana samanvaya vyavasthalaku nirvachanam vector laplasianlo ivvabadindi.
aadhunika ganita bhouthika shaastram antataa laplasian sarvavyaapti chendindi, udaaharanaku laplace sameekaranam, poison sameekaranam, ushna sameekaranam, taranga sameekaranam mariyu shroadinger sameekaranamlo kanipistundi.
hessian matricks.
suutram _ 50 saadhaaranamgaa laplasian nuu suuchistumdi, konnisaarlu suutram _ 50 kooda hessian maatrikanu suuchistumdi. modatidi suutram _ 21 yokka antargata utpattini suuchistumdi, rendodi suutram _ 21 yokka diedic utpattini suchistundihi
kabatti suutram _ 50 anedi laplasian ledhaa hessian maatrikanu suuchistumdaa anedi sandarbham meeda aadharapadi untundi.
tensor utpannam.
delnu vector feeldku kooda vartimpajeyavachu, phalitamgaa tensor untundi. vector feeled formula _ 56 (moodu dimensionlalo) yokka tensor derivative anedi 9-turm rendava-rank tensor-antey, 3x3 matricks-conny dheenini kevalam formula _ 57 gaa suuchimchavachchu, ikkada formula _ 58 diedic utpattini suuchistumdi. eee parimaanam sthalaaniki sambandhinchi vector feeled yokka jacobian matricks yokka transposeku samanam. vector feeled yokka divergensenu eee matricks yokka tracega vyakteekarinchavachhu.
oka chinna sthaanabhramsham suutram _ 59 kosam, vector feeled loe maarpu ilaa ivvabadutundihi
utpatti niyamaalu.
vector calculus kosamh
maatrika calculus kosam (dheeni kosam suutram _ 62 nuu suutram _ 63 anie vraayavacchu):
aasakti yokka maroka sambandham (ii chudandi.zii. "euler sameekaranaalu") krindi vidhamgaa untundi, ikkada suutram _ 65 anedi bahya utpatti tensor
rendava utpannaalu.
dell oka scalar ledhaa vektarpai panichesinappudu, oka scalar ledhaa vector tirigi ivvabadutundi. vector utpattula (scalar, dott, cross) vaividhyam kaaranamgaa dell yokka oka anuvartanam ippatike moodu pradhana utpannaalaku daariteestundihi pravanata (scalar utpatti), divergence (dott utpatti) mariyu curl (cross utpatti). eee moodu rakaala utpannaalanu okadaanikokati mallee vartimpajeyadam valla scalar feeled "f" ledhaa vector feeled v kosam aidu saadhyamayye rendava utpannaalu labhistaayi; scalar laplasian mariyu vector laplasian vadakam maroe rendu istundih
ivi pradhanamgaa aasakti kaligistaayi endukante avi ellappuduu okadaanikokati pratyekamainavi ledhaa swatantramainavi kaavu. vidhulu baga pravartinchinanta varaku (chaalaa sandarbhaalalo suutram _ 68), vaatilo rendu ellappuduu sunnaga untayih
vaatilo rendu ellappuduu samanamga untayih
migilina 3 vector utpannaalu eee sameekaranam dwara sambandham kaligi untayih
mariyu vidhulu baga pravartinchinatlayite, vaatilo okadaanni tensor utpattitoe kooda vyakteekarinchavachhuh
jaagrattalu.
paina perkonna vector lakshanaalu chaalaa varaku (dell yokka avakalana lakshanaalapai spashtamgaa aadhaarapadevi minaha-udaaharanaku, utpatti niyamam) chihna punarvyavastheekaranapai matrame aadhaarapadataayi mariyu dell chihnanni edainaa itara vector bharti chesthe tappanisarigaa pattukoovaali. eee operator nuu vector gaa soochinchadamlo pondavalasina viluvalo idhi oka bhaagam.
dell nuu tarachugaa vector thoo bharti chesi, vector gurtimpunu pondagaliginappatiki, aaa gurtimpulanu ghnaapakasaktigaa marchinappatiki, dheeniki viruddhamgaa "kaadu" anedi nammadaginadi, endukante dell saadhaaranamgaa prayaaninchadu.
vaividhyam (suutram _ 73) mariyu advection operator (suutram _ 74) nuu pradarsinche prati udaaharana parivartana kaaduh
dell yokka avakalana lakshanaalapai aadhaarapade prati udaaharana
eee vyatyaasaalaku kendramgaa dell kevalam vector kaadu; idhi vector operator. vector anedi parimaanam mariyu dhisha rendintinee kaligi unna vastuvu ayithe, dell anedi oka functionpy panichese varaku parimaanam ledhaa disanu kaligi undadu.
aaa kaaranamgaa, dell thoo kuudina gurtimpulu vector gurtimpulu mariyu utpatti niyamam vanti "bhedam" gurtimpulu rendintinee upayoginchi jagrattagaa pondaali.

