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[{"key": "dc.contributor.advisor", "value": "Onninen, Jani", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kunelius, Joni", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2017-09-05T10:26:46Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2017-09-05T10:26:46Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2017", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1721298", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/55276", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4 tutkielma k\u00e4sittelee variaatiolaskentaa. Variaatiolaskenta on saanut alkunsa matemaattisesta analyysist\u00e4 1700-luvun vaihteessa Johann Bernoullin esitt\u00e4m\u00e4n Brachistochrone-ongelman vaikutuksesta. T\u00e4t\u00e4 matematiikan alaa voidaan pit\u00e4\u00e4 yleistyksen\u00e4 analyysin ongelmaan funktioiden \u00e4\u00e4riarvopisteiden l\u00f6yt\u00e4misest\u00e4.\r\nReaaliarvoisten funktioiden sijaan variaatiolaskenta k\u00e4sittelee funktionaaleja.\r\nFunktionaalit ovat kuvauksia funktioavaruudesta reaaliluvuille.\r\nFunktionaaleilla mallinnetaan ongelmaa, johon variaatiolaskennalla etsit\u00e4\u00e4n ratkaisu. Variaatiolaskenta keskittyykin funktion, jolla funktionaali saa suurimman tai pienimm\u00e4n arvonsa, etsimiseen.\r\nYksinkertaisin esimerkki on kahden pisteen, P1 ja P2, v\u00e4lisen lyhimm\u00e4n et\u00e4isyyden ratkaiseminen.\r\nT\u00e4ll\u00f6in k\u00e4sitelt\u00e4v\u00e4 funktionaali on pisteit\u00e4 yhdist\u00e4v\u00e4n jatkuvasti differentioituvan k\u00e4yr\u00e4n P:[0,1]->Rn pituus /|P'(t)dt ja funktionaalia minimoidaan reunaehtot P(0)=P1 ja P(1)=P2 toteuttavien k\u00e4yrien luokassa.\r\nMinimointiongelmiin voidaan liitt\u00e4\u00e4 my\u00f6s muunkinlaisia reunaehtoja, esimerkiksi k\u00e4yr\u00e4n rajaamaa pinta-alaa voidaan minimoida annetun pituisten k\u00e4yrien joukossa. T\u00e4llaisiin ongelmiin etsit\u00e4\u00e4n ratkaisu reunaehtojen toteutuessa k\u00e4ytt\u00e4en variaatiolaskennan ty\u00f6kaluja. Niist\u00e4 yksi t\u00e4rkeimmist\u00e4 on Eulerin yht\u00e4l\u00f6, joka antaa analyysin derivaatan nollakohtaa vastaavan ehdon funktioille, joilla funktionaalin \u00e4\u00e4riarvo saavutetaan.\r\nNykyisin variaatiolaskentaa sovelletaan monien eri tieteenalojen, kuten kemian, tietotekniikan, biologian ja taloustieteiden ongelmiin.\r\nSen sijaan alkuper\u00e4iset variaatiolaskennan kysymykset ovat yleens\u00e4 per\u00e4isin fysiikasta tai geometrian ongelmista. N\u00e4ist\u00e4 esimerkiksi Brachistochrone-ongelmassa tutkitaan kappaleen liukumiseen kuluvaa aikaa ja pyrit\u00e4\u00e4n l\u00f6yt\u00e4m\u00e4\u00e4n sen minimi. Toinen esimerkki klassisista ongelmista on isoperimetrinen ongelma, jossa etsit\u00e4\u00e4n pinta-alan maksimia, kun alueen rajaavan k\u00e4yr\u00e4n pituus on kiinnitetty.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Joni Kunelius (jokrjaku) on 2017-09-05 10:26:45.798290. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2017-09-05T10:26:46Z\r\nNo. of bitstreams: 2\r\nURN:NBN:fi:jyu-201709053662.pdf: 847548 bytes, checksum: 77d8857143322beecdd7ca118cb62061 (MD5)\r\nlicense.html: 4783 bytes, checksum: 2e5dda738af974d57f34ae96092819bf (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2017-09-05T10:26:46Z (GMT). No. of bitstreams: 2\r\nURN:NBN:fi:jyu-201709053662.pdf: 847548 bytes, checksum: 77d8857143322beecdd7ca118cb62061 (MD5)\r\nlicense.html: 4783 bytes, checksum: 2e5dda738af974d57f34ae96092819bf (MD5)\r\n Previous issue date: 2017", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (42 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "brakistokroni", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "isoperimetrinen ongelma", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Klassisia variaatio-ongelmia", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201709053662", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2017-09-05T10:26:46Z", "language": "", "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "variaatiolaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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