| fullrecord |
[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Valkonen, Jasmin", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2021-03-12T10:14:12Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2021-03-12T10:14:12Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2021", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/74611", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tutkitaan pinta-alaa euklidisessa tasogeometriassa kahden eri l\u00e4ht\u00f6kohdan kautta. Perusideana on tarkastella kahden eri monikulmion samakokoisuutta pinta-alafunktion sek\u00e4 osittamisen n\u00e4k\u00f6kulmasta. Aluksi oletetaan pinta-alafunktion olemassaolo ja my\u00f6hemmin perehdyt\u00e4\u00e4n sen olemassaolon todistamiseen, joka on yksi ty\u00f6n p\u00e4\u00e4tuloksista.\n\nPinta-alalle m\u00e4\u00e4ritell\u00e4\u00e4n t\u00e4sm\u00e4llinen k\u00e4site, pinta-alafunktio, jolle p\u00e4tee nelj\u00e4 ominaisuutta. N\u00e4iden ominaisuuksien avulla pystyt\u00e4\u00e4n todistamaan aikaisemmin koulusta tuttuja geometrian tuloksia kuten esimerkiksi suorakulmaisen kolmion ja suorakulmion pinta-alojen kaavat sek\u00e4 Pythagoraan lause. \n\nKahden eri monikulmion samankokoisuutta voidaan tarkastella my\u00f6s toisella tavalla, monikulmioiden osittamisella. K\u00e4yt\u00e4nn\u00f6n ongelma osittamisessa on se, kuinka pilkkoa monikulmio osiin, uudelleen j\u00e4rjest\u00e4\u00e4 n\u00e4m\u00e4 osat ja muodostaa haluttu monikulmio, jolla on sama pinta-ala kuin alkuper\u00e4isell\u00e4 monikulmiolla. Ty\u00f6n toinen p\u00e4\u00e4tulos Wallacen, Bolyain ja Gerwienin lause kertoo, ett\u00e4 kaksi monikulmiota voidaan pilkkoa samanlaisiin osiin, joista voidaan muodostaa toinen monikulmio, jos ja vain jos monikulmioiden pinta-alat ovat yht\u00e4 suuret. Lauseen havainnollistamiseksi n\u00e4ytet\u00e4\u00e4n, miten tasasivuinen kolmio ositetaan neli\u00f6ksi ja miten Pythagoraan lause todistetaan osittamisella. \n\nPinta-alan tutkimisen lis\u00e4ksi ty\u00f6ss\u00e4 todistetaan pinta-alan olemassaolo. Pinta-alan olemassaolo tarkoittaa sit\u00e4, ett\u00e4 on todella olemassa pinta-alafunktio, joka t\u00e4ytt\u00e4\u00e4 m\u00e4\u00e4ritelm\u00e4n ehdot. Sekaannuksen v\u00e4ltt\u00e4miseksi m\u00e4\u00e4ritell\u00e4\u00e4n toinen k\u00e4site, kaava-ala (engl. formula area). Tarkoituksena on n\u00e4ytt\u00e4\u00e4, ett\u00e4 kaava-alafunktio on hyvin m\u00e4\u00e4ritelty ja sille p\u00e4tee samat ominaisuudet kuin pinta-alafunktiolle.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2021-03-12T10:14:12Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2021-03-12T10:14:12Z (GMT). No. of bitstreams: 0\n Previous issue date: 2021", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "43", "language": "", "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.title", "value": "Pinta-ala euklidisessa tasogeometriassa", "language": "", "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-202103121955", "language": "", "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": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "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": null, "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": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ositus", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "aksioomat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "pinta-ala", "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"}]
|