kalan

kalan (Calculus) ganit ka pramukh kshetr hai jismein raashiyon ke parivartan ka ganiteeya adhyayan kiya jaata hai. iski do mukhya shaakhaaeain hain- avakal ganit (difreinshiyl kailkulas) tatha samaakalan ganit (iteegral kailakulas) . kailakulas ke ye donon shaakhaaeain kalan ke moolabhoot prameya dvaara paraspar sambandhit hain. vartamaan samay mein vigyaan, injeeniyri, arthashaastr aadi ke kshetr mein kailkulas ka upayog kiya jaata hai.

Bhaarat mein kailkulas se sambandhit kai kaunsept 14veen shataabdi mein hi viksit ho gaye the.[1][2] kintu paramparaagat roop se yahi maanyata hai ki kailakulas ka prayog 17veen shataabdi ke uttaraardh mein aarambh hua tatha aaijak nyootan tatha laibneej iske janak the.

anukram

samaakalan

samaakalan (Integral Calculus) yeh ek vishesh prakaar ki yog kriya hai jismein ati-sookshm maan waali (kintu ginti mein atyadhik, anant) sankhyaaon ko joda jaata hai. kisi vakr tatha x-aksh ke beech ka kshetrafal nikaalne ke liye samaakalan ka prayog karna padata hai.

avakalan

avakalan (Differential Calculus) kisi raashi ke kisi anya raashi ke saapeksh tatkaalik badlaav ke dar ka adhyayan karta hai. is dar ko 'avakalaj' (en:Derivative) kehte hain.

kisi falan ke kisi char raashi ke saath badhne ki dar ko maapata hai. jaise yadi koi falan y kisi char raasi x par nirbhar hai aur x ka maan x1 se x2 karne par y ka maan y1 se y2 ho jaata hai to (y2-y1)/(x2-x1) ko y ka x ke sandarbh mein avakalaj kehte hain. ise dy/dx se niroopit kiya jaata hai. dhyaan rahe ki parivartan (x2-x1) sooshm se sookshmatam (tend to zero) hona chaahiye. isi liye seema (limit) ka avakalan mein bahut mahatvapoorn sthaan hai. kisi vakr (curve) ka kisi bindu par pravanata (slope) jaanane ke liye us bindu par avakalaj ki ganana karni padti hai.

itihaas

kailakulas ke vikaas ka mukhya shreya laibneej (Leibniz) aur aaijak nyootan ko diya jaata hai. kintu iski jadein bahut puraani hain.

Bhaarat ke Kerala ke mahaan ganitjnya maadhav ne chaudahaveen shataabdi mein kailakulas ke kai mahatvapoorn avayavon ki charcha ki aur is prakaar kailakulas ki neenv rakhi. unhone Taylor shreni, anant shreniyon ka sannikteekaran (infinite series approximations), abhisran (kanvarjeins) ka inteegral test, avakalan ka aarambhik roop, araikhik sameekaranon ke hal ka punaraavarti (itaretiv) hal, yeh vichaar ki kisi vakr ka kshetrafal usaka samaakalan hota hai, aadi vichaar (sankalpanaaen) unhone bahut pehle likh diya.[3][4][5][6]

farma tatha Japani ganitjnya seki kova ne bhi ismein yogadaan diya.

kailkulas ke vikaas mein Bhaarat ka yogadaan

aadhaarbhoot sankalpanaaen (concepts)

falan, seema, saatatya, shreni ka anant tak yog, atyanu (infinitesimal) aadi sankalpanaaon ki samajh aur vikaas ne kailakulas ko janm diya.

kalan ka moolabhoot prameya

'samaakalan aur avakalan ek doosare ke vyutkram kriyaayein hain'. is kathan ki pushti karne vaale do prameyon ko kalan ka moolabhoot prameya kaha jaata hai. in prameyon kee‌ khoj nyootan tatha leibnitj ne ki thi.

upayog

kailakulas ka upayog sabhi bhautik vijnyaaanon, injeeniyri, sanganak vigyaan, saankhyiki, arthashaastr, vaanijya, aayurvigyaan, evam anyaanya kshetron mein hota hai. jahaaain bhi kisi design samasya ka ganiteeya model banaaya ja sakta ho aur ishtatam (optimal) hal praapt karna ho, kalan ka upayog kiya jaata hai. kalan ki sahaayata se ham parivartan ke aniyt char daron (non-constant rates) ko bhi lekar aasaani se aage badh paate hain.

sandarbh

inhein bhi dekhein

baahari kadiyaaain