Minimalne plohe

master's thesis

• Title (english): Minimal surfaces
• Author: Maro Đanović
• Mentor: Marija Galić (mentor)
• Committee member: Marija Galić (predsjednik povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Aleksandra Čižmešija (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-03-03, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Kao što i sam naslov diplomskog rada kaže, minimalne plohe ključni su pojam ovog rada. Taj pojam ne obrađuje se temeljito u sklopu većine matematičkih programa diljem Hrvatske, ali itekako ima svoju važnost kroz povijest matematike. U ovom diplomskom radu, najprije smo uveli osnovne pojmove iz teorije krivulja i ploha, a zatim smo se posvetili minimalnim plohama i njihovim svojstvima, tj. pokazali smo da su minimalne plohe (one čija je srednja zakrivljenost jednaka nuli) upravo plohe minimalne površine. Također, uveli smo i pojam izotermnih koordinata te smo izveli Weierstrass–Enneperovu reprezentaciju minimalnih ploha. Navedena reprezentacija nam omogućuje generiranje minimalnih ploha iz holomorfnih funkcija. Na kraju smo dali pregled nekih poznatijih minimalnih ploha, a najveću pažnju smo posvetili helikoidu i katenoidu.

Abstract (english)

As we can see in the title, the key word of this thesis is minimal surfaces. That term is not too thoroughly studied in the majority of mathematics programs in Croatia but it has been very important throughout the history of mathematics. In the first part of this thesis, we recalled most important definitions and results from the theory of curves and surfaces and then we pointed out important properties regarding minimal surfaces. In another words, it was proven that minimal surfaces (mean curvature is equal to zero) are surfaces with the least area. We also introduced isothermal coordinates into our thesis after which we derived Weierstras–Enneper representation of minimal surfaces. Mentioned representation allows us to generate minimal surfaces from holomorphic functions. In the end, we gave an overview of some well-known minimal surfaces with emphasis on the two most famous minimal surfaces, helicoid and catenoid.

• Keywords: povijest matematike; teorija krivulja i ploha; izotermne koordinate; helikoid; katenoid; Weierstrass–Enneperova reprezentacija
• Keywords (english): history of mathematics; theory of curves and surfaces; isothermal coordinates; helicoid; catenoid; Weierstras–Enneper representation
• Language: croatian
• URN:NBN: urn:nbn:hr:217:310462
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-03-25 14:04:40