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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kosonen, Antti", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2016-10-30T17:58:56Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2016-10-30T17:58:56Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2016", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1579088", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/51720", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Differentiaalimuodot ovat oleellinen osa modernin matematiikan koneistoa. Niit\u00e4 k\u00e4ytet\u00e4\u00e4n paitsi geometrian tutkimuksessa, my\u00f6s teoreettisen fysiikan kent\u00e4ll\u00e4 muun muassa elektrostatiikassa, mekaniikassa ja termodynamiikassa. Differentiaalimuodot el\u00e4v\u00e4t luonnollisesti sileill\u00e4 monistoilla, jotka puolestaan esiintyv\u00e4t kaikkialla, miss\u00e4 on tarve puhua siisteist\u00e4 joukoista koordinaattien avulla.\r\n\r\nT\u00e4ss\u00e4 tutkielmassa tutustutaan differentiaalimuotojen perusteoriaan alkaen euklidisten avaruuksien alimonistoista. T\u00e4m\u00e4n j\u00e4lkeen m\u00e4\u00e4ritell\u00e4\u00e4n monistojen tangentti- ja kotangenttiavaruudet, k-muotojen ulkoinen tulo, differentiaalimuotojen ulkoinen derivaatta sek\u00e4 lopulta differentiaalimuodon integraali yli suunnistetun moniston. Tutkielman t\u00e4rkeimp\u00e4n\u00e4 yksitt\u00e4isen\u00e4 tavoitteena on esitt\u00e4\u00e4 ja todistaa Stokesin ja Cartanin lauseena tunnettu teoreema. T\u00e4m\u00e4 tulos mahdollistaa differentiaali- ja integraalilaskennasta tuttujen Gaussin, Greenin ja Stokesin lauseiden esitt\u00e4misen yhten\u00e4isell\u00e4 tavalla, samalla yleist\u00e4en niiden sanoman paitsi korkeampiin ulottuvuuksiin, my\u00f6s euklidisia avaruuksia abstraktimpiin joukkoihin.\r\n\r\nTutkielmassa k\u00e4sitteit\u00e4 ja konstruktioita pyrit\u00e4\u00e4n tarkastelemaan unohtamatta niihin liittyv\u00e4\u00e4 geometriaa. Lis\u00e4ksi ty\u00f6ss\u00e4 tarkastellaan edell\u00e4 mainittuihin fysiikan aihepiireihin liittyvi\u00e4 esimerkkej\u00e4 monistojen ja differentiaalimuotojen tarjoamassa kontekstissa. Kokonaisuutena ty\u00f6n on tarkoitus toimia johdatuksena n\u00e4ihin differentiaaligeometriaksi kutsutun matematiikan alan keskeisiin ty\u00f6kaluihin ja antaa v\u00e4l\u00e4hdys siit\u00e4, mit\u00e4 niill\u00e4 voi tehd\u00e4. N\u00e4m\u00e4 ty\u00f6kalut ovat v\u00e4ltt\u00e4m\u00e4tt\u00f6mi\u00e4 jatkettaessa aiheessa eteenp\u00e4in, lis\u00e4tt\u00e4ess\u00e4 monistoille edelleen rakenteita ja perehdytt\u00e4ess\u00e4 niihin syv\u00e4llisemmin.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Antti Kosonen (anpakako) on 2016-10-30 17:58:53.195227. 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 2016-10-30T17:58:55Z\r\nNo. of bitstreams: 2\r\nURN:NBN:fi:jyu-201610304497.pdf: 609306 bytes, checksum: e631eb29274a1d65594a8258a85ce8b3 (MD5)\r\nlicense.html: 4838 bytes, checksum: abdb0baba3149548a431b48fd7933cd6 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2016-10-30T17:58:56Z (GMT). No. of bitstreams: 2\r\nURN:NBN:fi:jyu-201610304497.pdf: 609306 bytes, checksum: e631eb29274a1d65594a8258a85ce8b3 (MD5)\r\nlicense.html: 4838 bytes, checksum: abdb0baba3149548a431b48fd7933cd6 (MD5)\r\n Previous issue date: 2016", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (82 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": "differentiaalimuodot", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "tangenttiavaruus", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ulkoinen tulo", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ulkoinen derivaatta", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "monistojen suunnistaminen", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "integrointi", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Stokesin ja Cartanin lause", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "differentiaalimuotojen sovellukset", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Differentiaalimuodot ja niiden integrointi euklidisten avaruuksien alimonistoilla", "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-201610304497", "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", 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