Imave periods ekuTvnis evklides amocana

Sesavali
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adamianis moRvaweobis yvela sferoSi warmoiSoba miznis miRwevis arsebuli saSualebebidan saukeTesos, anu optimaluris arCevis problema. optimaluris povnis amocana ki daiyvaneba maqsimumis da minimumis anu eqstremumis povnis amocanaze. amitomac, aseT amocanebs eqstremaluri amocanebi ewodeba. aseTi tipis amocanebi pirvelad antikur epoqaSi daisva da isini maTematikis ganviTarebis mTeli istoriis manZilze ipyrobdnen yuradRebas. maT Soris uZvelesad iTvleba klasikuri izoperimetriuli amocana: mocemuli sigrZis Caketili wirebs Soris ipoveT iseTi, romelic Semowers udides farTobs. imave periods ekuTvnis evklides amocana (me-4s.Cv.w-de): mocemul samkuTxedSi CaxazeT udidesi farTobis mqone paralelogrami. antikuri civilizaciis ganadgurebasTan erTad, evropaSi mecnieruli azris ganviTareba minelda daaxloebiT me-15s-de. me-16s.-Si daisva algebruli Sinaarsis mqone eqstremaluri amocanebi. (erT-erTi maTgania tartalias amocana: ricxvi 8 gavyoT or nawilad ise, rom maTi namravlis namravli maTive sxvaobasTan iyos maqsimaluri). unda aRiniSnos, rom me-17s-de ar arsebobda eqstremaluri amocanis amoxsnis zogadi sqema. TiToeuli amocana ixsneboda misTvis Seqmnili xerxiT, da, rogorc wesi, atarebda misi amomxsnelis saxels. (evklides, tartalias, arqimedes da a.S. amocanebi) me-17s.-Si eqstre-maluri amocanebis kvlevas stimuli misca sabunebismetyvelo mecnierebebsa da praqtikaSi arsebulma moTxovnilebebma. 1615w. gamovida i. kepleris wigni `Rvinis kasrebis axali stereometria~, sadac is gamoTqvams varauds, rom `nebismieri maqsimumis siaxloves, cvlilebebi araa sagrZnobi~. am sityvebSia Cadebuli eqstremumebis Zebnis ZiriTadi algoriTmi, romelic Semdeg zusti Teoremis saxiT Camoayaliba jer fermam 1629w, (mravalwevrebisaTvis), xolo mogvianebiT niutonma da leibnicma, da romelic yvelasTvis cnobili fermas Teoremis saxels atarebs.

fermas saxels ukavSirdeba, agreTve pirveli variaciuli principi. sinaTlis gardatexis kanoni eqsperimentulad daadgina sneliusma, xolo fermamM wamoayena eqstremalur principi, romelic ambobs: araerTgvarovan garemoSi sinaTlis sxivi irCevs traeqtorias, romlis gaswvrivac misi erTi wertilidan meoremde gadasvlis dro minimaluria. am momentidan dawyebuli, `variaciuli principis~ idea (anu mtkiceba imis Sesaxeb, rom bunebis kanonebi SeiZleba gamoyvanilni iqnan raime eqstremaluri Sesabamisobidan) gaxda erT-erTi ZiriTadi idea bunebismetyvelebaSi. fermas variaciuli principis kvaldakval, aRmoCenil iqna variaciuli principebi, jer meqanikaSi, Semdeg fizikaSi. cnobilma mecnierma l. eilerma, ki aseTi mosazrebac gamoTqva: `qveyanaze ar xdeba araferi, raSic ar Cans raime maqsimumis an minimumis azri.~

me-17s-is bolos, kerZod, 1696 wels, iohan bernulim dasva axali amocana: vertikalur sibrtyeSi mocemulia ori da wertili. vipovoT traeqtoria , romlis gaswvrivac, sakuTari simZimis ZaliT moZravi sxeuli, -dan -mde miva umokles droSi. man maTe-matikosebs am amocanis amosaxsnelad erTi weli misca, Tumca amoxsnebi droze adre miiRo. es amocana amoxsnes leibnicma, iakob da iohan bernulebma, hiugensma, niutonma. aRmoCnda, rom uswrafesi daSvebis wiri, anu braqistoqroni, aris cikloidi. adre ganxiluli amocanebisgan (mag; evklides) gansxvavebiT, sadac SesaZlo variantebis arCeva xdeboda sasrulganzomilebiani simravlidan, am amocanisTvis, ori wertilis SemaerTebeli wirebis simravle usasruloganzomilebiania. braqistoq-.ronis amocaniT daiwyo klasikuri variaciuli aRricxvis istoria. variaciuli aRricxvis Teoriis ganviTarebaSi didi wvlili miuZRviT eilers, lagranJs, leJandrs, iakobis, vaierStrass, hamiltons da sxv. me-18s-Si eileris da lagranJis mier Seiqmna amoxsnis meTodebi eqstremaluri amocanebisTvis SezRudvebis gareSe da tolobis tipis SezRudvebiT. mogvianebiT, Cvens droSi, gamokvleuli iqna eqstremaluri amocanebi tolobis da utolobis tipis SezRudvebiT.

me-20 saukunis 30-ian wlebSi bevri Tvlida, rom eqstremaluri amocanebis Teoriis problematika praqtikulad amowuruli iyo. magram aRmoCnda, rom bevri ekonomikuri Sinaarsis mqone amocanis amoxsna praqtikulad SeuZlebeli iyo eqstremaluri amocanebis Teoriis arsebuli meTodebiT. me-20 saukunis 50-ian wlebSi kantoroviCma da dancigma Seqmnes axali mimarTuleba eqstremaluri amocanebis TeoriaSi, romelic wrfivi programirebis saxeliTaa cnobili.

eqstremaluri amocanebis Teoriis ganviTarebis Sedegad, Seiqmna rigi dargebisa: amozneqili programireba, romelic ikvlevs amozneqil obieqtebs. masSi centraluri adgili uWiravs kunisa da takeris Teoremas, romelic me-20 saukunis 50-ian wlebSi damtkicda. optimaluri marTvis Teoria, romelic sxva mimarTulebebis msgavsad, warmoiSva gamoyenebiTi xasiaTis amocanebTan, kerZod, avtomaturi regulirebis amocanebTan dakavSirebiT (SaxtSi liftis marTva misi uswrafesi gaCerebis mizniT, raketis moZraobis marTva da a.S.) optimaluri marTvis amocanebSi, sxva eqstremaluri amocanebisgan gansxvavebiT, monawileobs marTva. . simravle gansazRvravs adamianis gavlenas procesze. eqstremalurobis aucilebeli pirobebi am amocanisTvis aris pontriaginis maqsimumis principi.

yoveli konkretuli amocanis gamokvleva Sinaarsobrivad gansxvavebuli ramdenime etapisagan Sedgeba. esenia: amocanis maTematikuri modelis Sedgena, miRebuli eqstremaluri amocanisTvis amoxsnis (optimizaciis) meTodis SerCeva da amocanis amoxsna, Sedegebis analizi da interpretireba. sxva sityvebiT, yovel konkretul amocanaSi saWiroa miznisa da saSualebebis garkveva. magaliTad, Tu amocanaSi saqme exeba xarjebs, maSin bunebrivi mizania xarjebis minimizacia, Tu laparakia mogebaze, maSin mogebis maqsimizacia, Tu vangariSobT saukeTeso, anu umokles manZils, maSin isev minimizacia da a. S.. maTematikis enaze, amocanis mizani aris raime funqcionali (e. i. funqcia mniSvnelobebiT namdvil ricxvTa simravleSi), romelsac miznis funqcia ewodeba da romlis eqstremaluri mniSvnelobebic unda moiZebnos:

an orive erTad, anu

gasagebia, rom miznis funqciis mniSvnelobebi namdvili ricxvebia, radgan sxva SemTxvevaSi ver avarCevT eqstremalur (ukidures) mniSvnelobas. magram rogori unda iyos misi gansazRvris are? magaliTisTvis warmovidginoT, rom miznis funqcia aris da mis Sesaxeb mxolod is viciT, rom X raRac simravlea. am SemTxvevaSi, funqciis eqstremaluri mniSvnelobebis mosaZebnad ar gagvaCnia araviTari meTodi garda funqciis mniSvnelobebis gadarCevisa, rac araviTar Sedegs ar mogvcems, Tu funqciis saxe usasrulo raodenoba wertilebs Seicavs; am dros sasruli raodenoba moqmedebebiT amonaxsns ver vipoviT, SemTxveviT, erTxelac, eqstremaluri mniSvneloba rom Segvxvdes, amas mainc ver mivxvdebiT, radgan Sesadarebeli mniSvnelobebis raodenoba usasruloa.

umetes SemTxvevaSi, praqtikuli Sinaarsis mqone amocanebis maTematikuri modelis Seqmna ar amoiwureba miznis funqciis SerCeviT. am dros bunebrivad Cndeba SezRudvebi, romlebic gviCveneben rom eqstremumi unda veZioT ara argumentis yvela mniSvnelobisaTvis, aramed gansazRvris aris romeliRac konkretul qvesimravleze, romelsac dasaSvebi simravle ewodeba. dasaSvebi simravlis elementebs, Cveulebriv, dasaSvebi wertilebi ewodebaT. sailustraciod, ganvixiloT piradi moxmarebis Teoriis umartivesi amocana, rodesac bazarze aris erTaderTi produqti da erTaderTi myidveli (e.w. momxmarebeli), produqtis erTeulis fasia p, xolo momxmareblis Semosavali, anu mis xelT arsebuli Tanxa aris I. mocemulia agreTve zrdadi funqcia (am funqciis sxva Tvisebebi amjerad saintereso araa) U(x), romelic gamoxatavs x raodenoba produqtis sargeblianobas mocemuli myidvelisaTvis. amasTan, SesaZlebelia produqtis erTeulis danawileba nebismieri proporciiT (magaliTad, erTeuli SeiZleba iyos 1kg Saqari, an milioni wyvili fexsacmeli). piradi moxmarebis Teoriis umartivesi amocanas warmoadgens sargeblianobis maqsimizacia , imis gaTvaliswinebiT, rom unda Sesruldes x>0 (rogorc raodenoba) da agreTve biujeturi SezRudva , rom daxarjulma Tanxam ar gadaaWarbos Semosavals. sabolood, miiReba amocana:
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amocanas, romelSic mravali cvladis funqciis eqstremumebs veZebT da cvladebi garkveul pirobebs (SezRudvebs) akmayofileben, maTematikuri programirebis amocana ewodeba. maTematikuri programirebis amocanebi miznis funqciisa da SezRudvebis struqturis mixedviT iyofa wrfivi, arawrfivi, dinamiuri, diskretuli(mTelricxva), stoqasturi programirebis amocanebad. maTematikuri programirebis zemoCamoTvlil meTodebs Soris yvelaze ganviTarebuli da TiTqmis saboloo saxe aqvs miRebuli wrfivi programirebis meTodebs. wrfivi programirebis amocanebs, maTi Seswavlis meTodebis specifikurobis gamo, SeviswavliT operaciaTa gamokvlevis kursSi.

warmodgenil kursSi, misi auditoriis interesebis, meTodologiuri faqtorebis da koniunqturis gaTvaliswinebiT, Setanilia maTematikuri programirebis amocanebis amoxsnis SedarebiT gavrcelebuli meTodebi, romelTa efeqturoba dadasturebulia praqtikiT. isini sxvadasxva wyarodanaa aRebuli, magram gadamuSavebulia da gamoyenebulia aRniSvnebis erTiani sistema. gasaTvaliswinebelia erTi garemoebac. radgan maTematikuri programireba SedarebiT axalgazrda disciplinaa, zogjer erTi da igive cnebis gamosaxatavad gamoiyeneba sxvadasxva termini. Cven vimuSavebT SedarebiT gavrcelebuli terminebiT da aucileblobis SemTxvevaSi mivuTiTebT sinonimebs.

aRniSvnebi

_ n-ganzomilebiani namdvili veqtor _ striqonebis sivrce;

_ n-ganzomilebiani namdvili veqtor _ svetebis sivrce;

x,y,z, _ kaligrafiuli mcire asoebi aRniSnaven skalarul (ricxviT) sidideebs;

_ beWduri mcire asoebi aRniSnaven veqtorebs;

x _ veqtor_striqonis standartuli Canaweri;

xⁱ _ x veqtoris i_uri komponenti (koordinati);

_ skalaruli sidide i_ur xarisxSi;

da _ organzomilebiani da samganzomilebiani veqtoris tradiciuli Canawerebi, ZiriTadad gamoiyeneba magaliTebis amoxsnis procesSi;

locmin(i), an ubralod locmin (locmax, locextr) _lokaluri minimumis (maqsimumis, eqstremumis) wertilebis simravle (i) amocanaSi;

glmin(i), an ubralod glmin (glmax, glextr) _globaluri minimumis (maqsimumis, eqstremumis) wertilebis simravle (i) amocanaSi;

_ funqciis infimumi, anu am funqciis mniSvnelobaTa simravlis zusti qveda zRvari.

_ x veqtoris evkliduri norma: ;

_ Ria birTvi:

_ Caketili birTvi:
_ arauaryofiT namdvil ricxvTa simravle.

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