
polymer vidnyanatil path samakalan

polymer haah eka macromolecule aahe, joou anek samaan kinwa samaan punaravrutti upaghatakanni banalela asato. polymer heey sendriya madhyamanmadhye samanya aahet, parantu itakech maryadit naheet. teey parichit krutrim plastickpasoon teey dii. aine. aii. aanee prathine yasarkhya naisargik biopolymerparyant aahet. tyanchi adviteey vistarit aanvik rachana, kathorata, chikatpana aanee chashma aanee ardh-sfatikeey sanrachana tayar karanyachi pravrutti yasah adviteey bhautik gunadharm nirman karate. sahsanyojak bandhit sthool aanvik sanrachana mhanoon polymerchi aadhunik sankalpana herman staudingerne 1920 madhye prastavit kelee hotee.
polymerchya abhyasatil eka upkshetra mhanje polymer bhautikshastra. soft matter abhyasacha eka bhag mhanoon, polymer bhautikshastra heey yantrik gunadharmanchya abhyasashi sambandhit aahe aanee ghanroop padarth bhautikshastrachya drushtikonavar laksh kendrit karate.
polymer heey itake mothe renu asalyane, sthool pramananchya simevar, tyanche bhautik gunadharm samanyatah nirdharak paddhatincha vapar karoon sodavanyasathi khoop klisht asatat. tyamule, sambandhit parinam midvinyasathi sankhyikeey drushtikon anekada lagoo kele jaatat. yaa sapeksh yashache mukhya kaaran mhanje mothya sankhyene monomerspasoon tayar kelelya polymerche varnan, jaree pratyakshat teey aakarane nishchitach maryadit asale, taree, anant anek monomerschya thermodynamic maryadet karyakshamatene kele jaate.
aushnik chadhautar drav dravanatil polymerchya aakaravar satat parinam karatat aanee tyanchya prabhavache pratirupan karnyasathi sankhyikeey yantriki aanee gatishil tattvancha vapar karane aavashyak aahe. path avibhajya drushtikon yaa moolbhut aadharas anuroop aahe aanee tyache parvadanare parinam heey aviratpane sankhyikeey sarasari aahet. jevha polymerchya abhyasasathi paath integral lagoo kele jaate, tevha polymer changlya prakare paribhashit sambhavya aanee tapman paristhitinusar susangat asoo shakananya serve sambhavya sthanik sanrachaneche varnan, ganana aanee sankhyikiyadrushtya vajan karnyasathi moolatah eka ganiteey yantrana aahe. paath integralscha vapar karoon, aataparyant nirakaran naan jhalelya samasyanvar yashasviritya kaam kele geleh vagalalele khand, guntagunt, duve aanee gathee. yaa siddhantachya vikasat yogdan denanya pramukh yogdankartyanmadhye nobel paritoshik vijete pii.jee. daad gains, surr samm edwards, emm.doee, eff.doubleu. vigal aanee etch. clennert.
path avibhajya rachana.
paath integralsche suruvatiche prayatna 1918 salapasoon shodhale jaau shakatat. 1921 paryant eka uttam ganiteey aupcharikta sthapit jhaali navhati. yamule akheris richard fenmanne quantum mechanixasathi eka sootra tayar kele, jyala aata samanyatah fenman integrals mhanoon olakhale jaate.
path samakalananchya gabha madhye karyatmak samakalan sankalpana aahe. niyamit samakalananmadhye eka maryadit prakriya asate jithe karyanchya chalanchya jagevar karyanchi berries ghetali jaate. karyatmak ektrikaranat karyanchi berries karyanchya jagevar ghetali jaate. pratyek karyasathi karyatmak jodanisathi moolya parat karate. 
paath integralscha reshechya integralsshee gondhal houu naye jeey variablechya jaaget vakratesah moolyankan kelelya integrations niyamit integrals aahet.
phaar aashcharyakarakpane, karyatmak samakalan anekada vegale hotat, mhanoon bhautikdrushtya arthpurn parinam midvinyasathi path samakalanancha bhag ghetala jaato.
haah lekh fenman aanee hibbs yanni swikarlele sanket vaparoon, path avibhajya asey darshvelah
sootra _ 2 karyatmak aanee sootra _ 3 karyatmak vibhedak mhanoon.
aadarsh polymer.
polymerchi avkashiy rachana aanee sanrachaneche parimanatmak vishleshan karnyasathi eka atyant bhola parantu faldayi drushtikon mhanje mukt yadruchchik chalanyache model. polymerla rasayanik bandhanni ghatt bandhalelya ekak renunsarkhya bindunchi sakhali mhanoon chitrit kele aahe aanee mhanoon salag ekkanmadhil paraspar antar sthir asalyacha andaj lavala jaau shakato.
aadarsh polymer modelmadhye polymer upghatak ekmekanchya sandarbhat firanyasathi purnapane mukt aahet aanee mhanunach polymerisation prakriyekade yadruchchik trimiteey chaal mhanoon pahile jaau shakate, jyamadhye pratyek monomer purvanirdharit lambichya dusarya yadruchchik tappyashi sambandhit aahe. 
ganitiyadrushtya heey bandhanchya sthitee sadishasathi sambhavyata karyadvare aupcharik kele jaate, i.eae. jawalchya ekkanchya jodichi sapeksh sthiteeh
sutrasah _ 5 haah diarac tribhuj pradeshasathi ubha aahe. yethe lakshat ghenyasarkhi mahattvachi gosht mhanje bandh sthitee sadish heey trijya sootra _ 6 chya golavar eksaman vitaran aahe, jeey aaplya sthir bandh lambiche aahe.
aadarsh pratikrutiche dusare mahattvapoorn vaishishtya mhanje bandh vahak sootra _ 7 ekmekanpasun swatantra aahe, mhanje aapan sampoorn polymer rachnesathi vitaran karya asey lihu shakatoh
jethe aapan sootra 9 monomers gruheet dharale aanee sootra 10 haah dummy index mhanoon karya karato. kurale cons {} mhanje sootra _ 11 heey vector sootra _ 7 chya sanchache karya aahe
yaa modelchya thalak parinamanmadhye khaleel goshtincha samavesh aaheh
end too end vector square everage.
yadruchchik chalanyachya pratimananusar, samamitichya vicharanmule shevat teey shevat vector sarasari nahishi hote. mhanoon, polymerchya aakaracha andaj ghenyasathi, aapan shevat teey shevat vectorchya bhinnatekade valatoh sootra 13, shevat teey shevat vectors sootra 14 mhanoon paribhashit kele aahe.
asha prakare, polymerchya aakarasathi pahila kachcha andaj fakt sootra _ 15 aahe.
end too end vector sambhavyata vitaran.
namood kelyapramane, aaplyala samanyatah polymer sanrachanechya sankhyikeey vaishishtyanmadhye rus asato. tyamule madhyavarti praman heey shevat teey shevatparyantche vector sambhavyata vitaran aselah
lakshat ghya qee vitaran heey keval shevat teey shevatparyantchya vector parimanavar avalambun asate. tasech, vareel abhivyakti sootra _ 17 peksha mothya aakaransathi shunyetar sambhavyata dete, spashtapane eka avastav parinam joou tyachya vyutpattisathi ghetalelya sootra _ 18 chya maryadepasoon udbhavato.
niyantran vibhedak samikaran.
polymer sanrachanasathi gulgulit avkashiy bahyareshechi maryada ghetalyas, mhanje maryada sootra _ 19 aanee sootra _ 20 ghetalyas, nirbandh sootra _ 21 antargat sambhavyata vitaranasathi vibhedak samikaran yeteh
vastavik jagechya sandarbhat ghetalelya laplacian sutrasah _ 23. heey samikaran midvanyacha eka marg mhanje tellerchya vistarachya madhyamatoon sootra _ 24) aanee sootra _ 25 paryant vistarane.
aadhich vishleshanatmakaritya midavalelya karyasathichya vibhedak samikaranacha tras qaa karava assa prashna ekhadyala padala asel, parantu heey darshavilyapramane, heey samikaran aadarsh nasalelya paristhitinsathi dekhil samanyikrut kele jaau shakate.
path avibhajya abhivyakti.
gulgulit samocchatechya tyach gruhitkakhali, vitaran karya paath integral vaparoon vyakt kele jaau shakateh
jethe aapan sootra _ 27 paribhashit kele aahe
yethe sootra _ 28 polymersathi parametrization variable mhanoon karya karate, jeey tyachya avkashiy sanrachaneche kinwa samochchateche varnan karate.
ghatank heey polymer sanrachanechya sankhyechya ghanateche eka mojmap aahe jyamadhye polymercha aakar satat aanee bhinn karnyayogya vakrareshechya jawal asato.
sthanik adthale.
aataparyant, path avibhajya drushtikonacha aamhala konatahi naveen parinam midala nahee. tyasathi, ekhadyane aadarsh modelpeksha pudhe jaanyacha prayatna keela pahije. yaa maryadit pratimanapasun pahile prasthan mhanoon, aata aapan avkashiy adthalyanchya nirbandhancha vichar karato. aadarsh modelane pratyek atirikt monomerchya avkashiy sanrachanevar konatihi maryada gruheet dharali nahee, jyamadhye monomersmadhil shaktincha samavesh aahe, jeey spashtapane astitvat aahet, kaaran done monomers samaan jaaga vyapoo shakat naheet. yethe, aapan keval monomer-monomer parasparsanvadacha samavesh karnyasathich navhe tarr bhinti kinwa itar bhautik adthalyansarkhya dhool aanee seema paristhitichya upasthitimule udbhavananya nirbandhancha samavesh karnyasathi adthalyachi sankalpana gheu.
dhool.
abhedya kananni kinwa "dhool" neey bharalelya jaagecha vichar qaraa. monomercha antim bindu vagaloon jaagecha apurnank sootra _ 29 dware darshava jenekarun tyachi moolye sootra _ 30 paryant asoo shaktil.
sootra _ 25 sathee teller vistar tayar karoon, naveen prashasakeey vibhedak samikaranavar pohochata yeteh
jyasathi sambandhit paath integral aaheh
bhinti.
eka paripoorn kathor bhint tayar karnyasathi, bhint samochchatemule polymerchya aavakyabaherchya serve pradeshansathi fakt sootra _ 34 nishchit qaraa.
polymer sahasa jyaa bhintinshi sanvad sadhato tyaa jatil rachana asatat. samocch keval adthale aanee valane yanni bharalela asoo shakat nahee, tarr polymershy tyancha parasparsanvad varr darshavilelya kathor yantrik aadarshikaranashi door aahe. vyavharat, aakarshak aantaraanvik shaktinmule polymer anekada bhintivar "shoshala" jaeel kinwa ghanroop hoil. ushnatemule, hii prakriya entropy chalit prakriyedvare pratikar kelee jaate, jee tappyatil jaaget mothya pramananshi sambandhit asalelya polymer sanrachananna anukool asate. thermodynamic adhishoshan-shoshan prakriya udbhavate. yache eka samanya udaharan mhanje peshi patalamadhye maryadit asalele polymer.
aakarshan shaktincha hishob karnyasathi, prati monomer sambhavya ashee paribhashit karah sootra _ 35. sambhavya boltzmann ghatkadvare samavisht kele jaeel. sampoorn polymersathi heey aakar gheteh
jethe aapan tapman mhanoon sootra 38 saha sootra 37 aanee sootra 39 boltzmann sthirank vaparale. ujavya bajula, aamche nehamiche maryada sootra _ 40 ghetale gele.
nishchit antim bindunsah polymer sanrachananchi sankhya aata paath integraldware nirdharit kelee jaau shakateh
aadarsh polymer prakaranapramanech, yaa avibhajya bhagacha arth vibhedak samikaranasathi prasarak mhanoon lavala jaau shakatoh
yamule orthonormal eigen functions aanee eigenvalueschya drushtine sootra _ 43 sathee dvi-rekheey vistar hotoh
aanee tyamule aapali shoshan samasya eigenfunction samasyet qamee hote.
vishisht vihirinsarkhya (aakarshak) sambhavyatesathi heey shoshan ghatanesathi done vyavasthankade netat, jyat vishisht samasya mapdand sootra _ 46 dware nirdharit kelele nirnayak tapman sootra _ 45 aaheh
uchch tapman sootra _ 47 madhye, sambhavya vihiris konatyahi baddh avastha nasatat, mhanje serve eigenvalues sakaratmak asatat aanee sambandhit eigenfunction heey asymptotic form sootra _ 48 gheteh
chalanche pruthakkaran kelyanantar aanee sootra _ 51 varr prushthbhag gruheet dharalyanantar x nirdeshankasathi parinam darshavila jaato. 
hii abhivyakti polymersathi prushthbhagapasun door eka atishay khulee rachana darshavate, mhanje polymer nasht jhaala aahe.
qamee tapman sootra _ 52 sathee, nakaratmak eigen moolya asaleli kiman eka baddh sthitee astitvat aahe. aaplya "mothya polymer" maryadet, yacha arth assa aahe qee dvi-rekheey vistaravar jaminichya sthitiche varchasva asel, jeey _ 53 heey lakshanrahit sootra roop dharan karateh
yaveli polymerchi sanrachana prabhavi jaadi sutrasah prushthbhagajavalil arund tharamadhye sthanbaddh kelee jaate.
yaa paddhaticha vapar karoon, anek "bhint" bhoomiti aanee parasparsanvadachya kshamatevar garv balgananya vividh prakarchya adhishoshan samasyanche nirakaran kele jaau shakate.
parimanatmakdrushtya changlya prakare paribhashit parinam prapt karnyasathi ekhadyala punarprapt eigen functionscha vapar karava lagel aanee sambandhit sanrachana berries tayar karavi lagel.
sampoorn aanee kathor upayasathi pahaa.
khand vagalale.
aanakhi eka spasht adthala, jyakade aataparyant ughadpane durlaksh kele gele aahe, too mhanje tyach polymermadhil monomersmadhil parasparkriya. yaa atishay vastavadi nirbandhantargat sanrachananchya sankhyesathi achook upay adyap ekapeksha mothya konatyahi parimanasathi sapadala nahee. hii samasya aitihasikdrushtya vagalaleli aakarmanachi samasya mhanoon olakhali jaau lagali aahe. samasya adhik changlya prakare samajoon ghenyasathi, pratyek monomerchya antim binduvar eka lahan katheen golasah (varr namood kelelya "dhoolichya thipkyanpramane nahee"), purvi sadar kelyapramane, yadruchchik chalanyachya sakhalichi kalpana karata yete. yaa golanchi trijya anivaryapane sootra 56 chey palan karate, anyatha salag golank aachchhadit hotil.
path avibhajya drushtikon andaje upay midvinyasathi tulanene sopee paddhat pradan karatoh sadar kelele parinam trimiteey jagesathi asatat, parantu konatyahi aayamitesathi sahajpane samanyikrut kele jaau shakatat.
ganana done vajvi gruhitkanvar aadharit aaheh
purvi sadar kelelya sootra _ 58 sathichya path avibhajya abhivyaktinusar, sarvat sambhavya sanrachana vakr sootra _ 59 asel jee mool path avibhajyacha ghatank qamee karateh
abhivyakti qamee karnyasathi, bhinnatanche ganan vapara aanee yular-lagrange samikaran midvah
aapan sootra _ 62 nishchit karato.
yogya karya sootra _ 29 nirdharit karnyasathi, trijya sootra _ 64, jaadi sootra _ 65 aanee profile sootra _ 66 chaa vichar qaraa jeey polymerchya utpattibhovati kendrit aahe. yaa kavchatil monomerschi sarasari sankhya sootra _ 67 chya barobarichi asavi.
dusarikade, samaan sarasarine sootra _ 68 chya barobariche dekhil asave (lakshat thewa qee sootra _ 69 heey moolya sootra _ 70 saha parametrization ghatak mhanoon paribhashit kele gele hote). yaa samanatecha parinam assa hotoh
aaplyala sootra _ 72 aata asey lihile jaau shakateh
aapan punha bhinnata ganitacha vapar karoon tyavar pohochatoh
lakshat ghya qee aata aamchyakade konatehi sootra _ 76 avalambitva naan thevata sootra _ 75 sathee eka odd aahe.
jaree pahane khoop bhayanak asale taree yaa samikaranavar eka byanpaiki sopa upay aaheh
aapan yaa mahattvachya nishkarshavar pochalo qee vagalalelya aakarman asalelya polymersathi shevat teey shevatche antar n saha vadhateh
sootra _ 78, aadarsh pratikrutichya parinamache pahile prasthanah sootra _ 79.
gaushian sakhali.
sanrachanatmak vitaran.
aataparyant, gananet samavisht kelele ekmev polymer mapdand mhanje monomers sootra _ 80, jeey anantaparyant nele gele hote aanee sthir bandh lambi sootra _ 6. heey sahasa purese asate, kaaran polymerchya sthanik sanrachanecha samasyevar parinam karnyacha haah ekmev marg aahe. sthir bandh antar" andajapeksha thoda changla prayatna karnyasathi, aapan pudhil sarvat prathamik drushtikonache parikshan karuya; ekal bandh lambiche adhik vastavadi varnan heey eka gossian vitaran aselah
tyamule purvipramanech, aamhi nikal rakhoon thevatoh sootra _ 83. lakshat ghya qee purvipeksha thode adhik klisht asale taree sootra _ 84 madhye ajunahi ekach mapdand aahe-sootra _ 6.
aaplya naveen bandh sadish vitaranasathi sanrachanatmak vitaran karya asey aaheh
jithe aapan sapeksh bandh vector sootra _ 87 varoon nirpeksh sthitee vector farkankade valaloh sootra _ 88.
yaa rachanela gossian sakhali mhanatat. sootra 89 sathiche gossian andaj polymer sanrachanechya sookshm vishleshanasathi dharoon nahee parantu mothya pramanavaril gunadharmansathi achook parinam deil.
yaa pratimanacha arth lavnyacha eka antardnyani marg mhanje swarmeliy springdware kramikpane jodalele maniche yantrik pratiman. asha pratimanasathi sambhavya urja khalilpramane dilee jaateh
aushnik samatolamadhye boltzmann vitaranachi apeksha karata yete, jeey formula _ 91 sathee vareel parinam kharokharach punarprapt karate.
gausi sakhalicha eka mahattvacha gunadharm mhanje swa-samanata. mhanje konatyahi done ekkanmadhil sootra 92 chey vitaran punha gossian aahe, jeey fakt sootra _ 6 aanee ekak teey ekak antarachya sutravar avalambun aaheh
heey lagech sootra _ 96 kadey netat.
sthanik adthalyansathi vibhagat spashtapane kelyapramane, aapan pratyay sootra _ 97 laa satat maryadeparyant gheuun sootra _ 98 laa sootra _ 99 neey badalato. mhanoon aata, aapale sanrachanatmak vitaran yanusar vyakt kele jaateh
swatantra chull heey vectormadhun karyat rupantarit jhaale aahe, jyacha arth sootra _ 101 aata karyatmak aahe. yaa sutrala winer vitaran mhanatat.
bahya kshetrakhali sakhali rachana.
bahya sambhavya kshetra sootra _ 102 gruheet dharalyas, varr varnan kelele samtol sanrachanatmak vitaran boltzmann ghatkadvare sudharit kele jailah
gossian sakhali sanrachanatmak vitaranachya abhyasatil eka mahattvache sadhan mhanje harit karya, jeey path avibhajya bhagfaladware paribhashit kele jaateh
marg ektrikaranacha arth serve polymer vakr sootra 105 vareel berries mhanoon lavala jaato jeey sootra 106 pasoon suru hote aanee sootra 107 varr samapt hote.
sadhya shoonya kshetra cases sutrasathi _ 108 hirave karya qamee houn khalilpramane hoteh
adhik samanya babatit, sootra _ 110 serve sambhavya polymer sanrachanasathi sampoorn vibhajan karyat vajan ghatakachi bhoomika bajavateh
hiravya karyasathi eka mahattvachi olakh astitvat aahe jee thet tyachya vyakhyepasun udbhavate.
sootra _ 112
yaa samikaranala spasht bhautik mahatva aahe, jeey paath integralchi sankalpana spasht karnyasathi dekhil madat karoo shakateh
utpadan sootra _ 113 heey sakhalisathi vajan ghatak vyakt karate jeey sootra _ 114 pasoon suru hote, sootra _ 97 payyanmadhye sootra _ 115 madhoon jaate aanee sootra _ 80 payyannantar sootra _ 64 varr sampate. serve sambhavya madhyabindu sutravaril ektrikaran _ 115 heey sootra _ 114 pasoon suru honanya aanee sootra _ 64 varr samapt honanya sakhalisathi sankhyikeey vajan parat dete. aata heey spasht jhaale pahije qee paath integral hii done nishchit antim bindunmadhye polymer tayar karoo shakananya serve sambhavya shabdik marganchi berries aahe.
_ 122 yaa sootrachya madatine konatyahi bhautik praman sootra _ 123 chee sarasari mojali jaau shakate. sootra 124 gruheet dharane heey keval sootra 97 vya khandachya sthanavar avalambun asate, tarah
sootra _ 126
aaa heey ekapeksha jaast monomarvar avalambun asave heey kaaran aahe. aata asey gruheet dharalyas teey sootra _ 127 tasech sootra _ 128 varr avalambun aahe, sarasari heey roop gheteh
sootra _ 129
adhik monomerschya avalambitvasathi spasht samanyikaranasaha.
jarr konee vajvi seema atty ladalya tarah
mug _ 131 yaa sutrasathichya teller vistaranachya madatine, _ 132 yaa sutrasathiche vibhedak samikaran kadhale jaau shakateh
yaa samikaranachya madatine vividh samasyansathi sootra _ 122 chey spasht swaroop aadhalate. mug, vibhajan karyachya gananesah sankhyikeey pramananchi eka tolhee kadhali jaau shakate.
polymer kshetra siddhant.
vagalalelya aakarmanachya prabhavanmule nirman honare _ 135 heey urja avalambitva sootra shodhanyasathi eka vegala naveen drushtikon, purvi sadar kelelya paddhatipeksha shreshth manla jaato.
polymer bhautikshastratil kshetra siddhant drushtikon polymer chadhautar aanee kshetra chadhautaranchya jivhalyachya sambandhavar aadharit aahe. anek kann pranalichya sankhyikeey yantrikiche varnan ekach chadh-utaar kshetradvare kele jaau shakate. asha samoohatil kann yadruchchik polymer sakhalisarkhe disana paddhatine chadh-utaar karananya kakshet antaralatun firto. polymerchya mothya gatanche varnan ekach chadh-utaar kshetradvare kele jaau shakate, assa tatkal nishkarsh kadhala pahije. asey disun yete qee, ekach polymerbaddal dekhil asech mhatale jaau shakate.
sadar kelelya mool path avibhajya abhivyaktichya sadrushyat, polymerche shevat teey shevat vitaran aata asey roop dharan karateh
aamchya naveen path samakalanamadhye khaleel goshtincha samavesh aaheh
sootra _ 141
sutrasah _ 142 heey pariman aanee bandh lambidware nirdharit prabhavi vastuman mhanoon karya karate.
lakshat ghya qee antargat avibhajya dekhil aata eka path avibhajya aahe, mhanoon karyachya done jaaga ektrit kelya aahet-polymer conformation-formula 143 aanee scalar feeled formula 144.
yaa path samakalanancha bhautik arth aahe. krutee sootra _ 145 heey antaralavar avalambun asalelya yadruchchik sambhavya sootra _ 137 madheel kananchya kakshache varnan karate. sootra _ 143 vareel path avibhajya yaa sambhavyatetil chadh-utaar karananya polymerche shevat teey shevat vitaran karate. vajan sutrasah _ 148 yaa sutravaril dusra marg avibhajya _ 149 heey itar sakhali ghatakanchya vikarshanshil dhagasathi jababdar aahe. vichalan talhanyasathi, sootra _ 137 ekikaran kalpanik kshetra akshasah chalvave lagel.
chadh-utaar karananya polymersathi asha kshetrachya varnanacha mahattvacha fayda assa aahe qee too kshetra siddhantatil gambhir ghatananchya siddhantashi sambandh sthapit karato.
sootra _ 151 sathee upay shodhanyasathi, sahasa eka laplas transformcha vapar keela jaato aanee purvi varnan kelelya sankhyikeey sarasari sootra _ 152 sarkhe parasparsambandh karya vicharat ghetale jaate, jyat hiravya karyala chadh-utaar asalelya jatil kshetrane badalale jaate. mothya polymerchya (aine "1) samanya maryadet, end too end vector vitaranasathiche upay anek shareer pranalinmadhil gambhir ghatanansathi quantum feeled saiddhantik drushtikonat abhyasalelya suvikasit vyavastheshi sambandhit aahet.
anek-polymer pranali.
aataparyant sadar kelelya upcharanmadhye aanakhi eka sulabh dharana gruheet dharali gelee; serve models ekach polymerche varnan karatat. arthat, adhik bhautikdrushtya vastavadi varnan polymermadhil parasparsanvadachya shakyatesathi jababdar asel. thodakyat, haah vagalalelya aakarmanachya samasyecha vistar aahe.
heey chitratmak binduvarun pahanyasathi, eka kendrit polymer dravanachya snap shotchi kalpana karata yete. vagalalele aakarmanache parasparsambandh aata keval ekach sakhalimadhye hott naheet, tarr polymerche praman vadhalyane itar sakhalyanmadhun sampark bindunchi vadhati sankhya atirikt vagalalele aakarman dete. yaa atirikt samparkancha vaiyaktik polymerchya sankhyikeey vartanavar lakshaniy parinam houu shakato.
done vegvegalya lambichya pramananmadhye farak karane aavashyak aahe. eka vyavastha lahan toke teey shevat vector scale sootra 153 dware dilee jaeel. yaa scalever sakhali tukada keval swatahun parasparsambandh anubhavato, mhanje.eae., shastriy swayam-talhanare vartan. mothya pramanansathi sootra 154 swa-talhun jaanyache parasparsambandh mahattvapoorn bhoomika bajavat naheet aanee sakhalichi aakadevari gossian sakhalisarkhi disate. nirnayak moolya sootra _ 155 heey ekagrateche karya asane aavashyak aahe. antardnyanane, eka lakshaniy ekagrata aadhich aadhalu shakate. hii ekagrata sakhalyanmadhil parasparsanvadache vaishishtya darshavate. jarr polymer fakt kirkolpane overlap jhaale tarr eka sakhali tyachya swatachya aakarmanat vyapali jaate. heey deteh
sootra _ 156 jethe aapan sootra _ 157 vaparale
haah eka mahattvacha parinam aahe aanee ekhadyala lagech disate qee mothya sakhali lambisathi n, ativyapti
ekagrata khoop qamee asate. purvi varnan keleli swayam-talhanari chaal badalali jaate aanee mhanoon vibhajan karya aata ekal polymer volume vagalalelya margandware niyantrit kele jaat nahee, parantu urvarit ghanatechya chadhautarandware niyantrit kele jaate jeey polymer dravanachya ekoon ekagratedvare nirdharit kele jaate. jawaljawal purnapane bharalelya jaalidware kalpana kelelya khoop mothya sandratechya maryadet, ghanatechya chadhautar qamee aanee qamee mahatvache hotat.
suruvatila, aapan anek sakhalyansathi path avibhajya rachana samanya karuya.
vibhajan karyachya gananesathiche samanyikaran khoop sope aahe aanee serve sakhali vibhaganmadhil parasparsanvad lakshat ghene heey fakt karave lagelah
sootra _ 158
jethe tolalelya urjechya avasthanchi vyakhya ashee kelee jaateh
sootra _ 159
sutrasah _ 160 polymerchi sankhya darshavate.
heey samanyatah sope nasate aanee vibhajan karya achookpane mojale jaau shakat nahee.
eka saralikaran mhanje eksandhata gruheet dharane mhanje serve sakhalyanchi lambi samaan asate. kinwa, ganitiyadrushtyah sootra _ 161.
dusari samasya mhanje vibhajan karyamadhye anek degree auph friedam asatat. yaat samavisht asalelya sakhali sutranchi sankhya _ 160 khoop mothee asoo shakate aanee pratyek sakhalimadhye antargat degree auph friedam asate, kaaran tee purnapane lavchik asalyache manale jaate. yaa karanastav, samoohik chull sadar karane soyiche aahe, jeey yaa prakaranat polymer segmentchi ghanata aaheh
sootra _ 163 saha sootra _ 164 ekoon dravanache praman.
sootra _ 165 kadey sookshm ghanata prachalak mhanoon pahile jaau shakate jyache moolya _ 166 yaa aniyantrit bindu sutravar ghanata paribhashit karate.
parivartan sootra _ 167 heey kalpana karnyapeksha qamee kshullak aahe aanee teey achookpane kele jaau shakat nahee. antim parinam tathakathit yadruchchik tappyachya andaj (aar. pii. aii.) shhee sambandhit aahe, joou ghan-avasthechya bhautikshastrat varanvar vaparla jaato. vibhag ghanata vaparoon vibhajan karyachi spashtapane ganana karnyasathi ekhadyala paraspar jagekade valave lagel, chull badalave lagel aanee tyanantarach ektrikaran karyanvit karave lagel. tapshilvar vyutpattisathi pahaa. prapt vibhajan karyasah, purvi varnan kelyapramane vividh bhautik matra kadhalya jaau shakatat.