Globalni izreki o sklenjenosti krivulj : diplomsko delo

Špringer, Tina (author), Slapar, Marko (mentor)

V prvem delu diplomske naloge najprej predstavimo osnovne definicije in lastnosti krivulj v prostoru, to je krivulj, ki ležijo v R3. Definiramo ločno dolžino in ukrivljenost. V nadaljevanju dokažemo nekaj globalnih izrekov o ukrivljenosti krivulj. Prvi izrek poveže ukrivljenost in skladnost krivulj. Fenchelov izrek nam pove, koliko se mora prostorska krivulja ukrivljati, da postane sklenjena, medtem ko nam Fary-Milnorjev izrek pove, koliko se mora prostorska krivulja vsaj še dodatno ukrivljati, da postane zavozlana. V zadnjem delu diplomske naloge pa zaključimo z ukrivljenostjo ravninskih krivulj, to je krivulj, ki ležijo v R2.

• Language: Slovenian
• Keywords: ukrivljenost krivulj v R3, skladnost krivulj, Fenchelov izrek, Fary-Milnorjev izrek
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Publisher: [T. Špringer]
• Year: 2017
• Number of pages: IV, 22 str.
• PID: 20.500.12556/RUL-95172
• UDC: 51(043.2)
• COBISS.SI-ID: 11699785
• Publication date in RUL: 19.09.2017

Secondary language

• Language: English
• Title: Global theorems of closed curves

Abstract

In this diploma thesis we first present basic definitions and properties of space curves, that is curves in R3. We define the length and local curvatures of space curves. Next we prove some global theorems concerning the curvatures. The first theorem connects congruence and curvatures of a space curves. Fenchel's theorem shows the lower estimate on total curvature of a closed curve while Frey-Milnor's theorem shows that a knotted curve must have an even larger total curvature. We conclude by discussing total curvature of plane curves.

• Keywords: mathematics, matematika