01. Universidade Federal Rural de Pernambuco - UFRPE (Sede)
URI permanente desta comunidadehttps://arandu.ufrpe.br/handle/123456789/1
Navegar
6 resultados
Resultados da Pesquisa
Item O Teorema Egregium(2024-02-29) Gomes, Heloisa Cardoso Barbosa; Gomes, Renato Teixeira; http://lattes.cnpq.br/0570606157057337; http://lattes.cnpq.br/8017333927762482During the development of differential geometry around the 17th century, an old problem occupied the minds of mathematicians at the time, which was determining whether the so-called 5th postulate of Euclid was in fact a postulate or a theorem. This postulate, which had an equivalent version published in 1795 by John Playfair (1748–1819), says that: through a point outside a given straight line it is possible to draw a single straight line parallel to the given straight line". There were many attempts to "prove"the fifth postulate, all of which failed. The answer to this question was given years later by Gauss, Lobachevski and Bolyai. In their work Disquisitiones generales circa superficies curves, Gauss shows that the curvature K(p) of a surface at the point p, initially calculated through the determinant of the differential of dNp which depends on the socalled first and second fundamental forms, actually depends only on the coefficients of the first fundamental form and their derivatives, and can be calculated using a formula that bears his name, the so-called Gauss formula. As a consequence of this formula we have the so-called Egregium Theorem which states that the Gaussian curvature of a surface is an invariant intrinsic, that is, it does not depend on the environment the surface is in and consequently, it is invariant due to local isometries. This discovery is closely related to non-Euclidean geometries, since the geometry of a surface with non-zero curvature is non-Euclidean. A consequence of this fact is that the 5th postulate is in fact a postulate and not a theorem. In this work, we will study the concepts necessary to understand Gauss’s Egregium theorem and its demonstration, as well as some applications of this important result.Item Um breve estudo sobre o transporte paralelo, geodésicas e a aplicação exponencial(2023-09-15) Costa, Matheus Rabelo Viana da; Gomes, Renato Teixeira; http://lattes.cnpq.br/0570606157057337; http://lattes.cnpq.br/3078665075835586Geodesics are curves on a regular surface that have the property of locally minimizing length, that is, if two points are close together, the curve that has the shortest length connecting these two points is a geodesic. They are roughly the "straight lines"of the surface, as they have a constant velocity vector norm, and are zero acceleration curves. We can arrive at these curves through the solution of a variational problem, or following the "path of Geometry"in which we define geodesics as a curve whose field of tangent vectors is parallel. The study of these curves on a surface leads us to the knowledge of several important geometric properties, in addition to the development of new machinery, such as special coordinate systems, for example, which facilitate the study of surfaces and help in the calculation of their important geometric structures. In this work we will make a brief study about parallel transport, Geodesics and the exponential map and its properties. We will study the notion of a covariant derivative, and how we parallel transport vectors along curves. With this idea of parallelism, we will define geodesics as a curve that has a field of parallel tangent vectors, we will study some properties of these curves and the geodesic curvature of curves on surfaces. Finally, we will study the exponential map, the normal coordinate system and the geodesic polar coordinate system and we will use this one to, among other things, show that geodesics have the property of locally minimizing the length.Item Estudo teórico de faixas de Möbius moleculares(2023-09-18) Nascimento, Murilo Assunção do; Bastos, Cristiano Costa; Silva, Luiz Carlos Barbosa da; http://lattes.cnpq.br/1481188647500485; http://lattes.cnpq.br/6385190604693576; http://lattes.cnpq.br/6391945509254633Several mathematical applications have been studied in many contexts. One of these cases is the application of differential geometry to molecular systems. The study of nanostructures and their use has gained more and more space. In this work, we apply Da Costa’s formalism, in which by applying a confining potential dependent on the mean curvature and Gaussian curvature of the structure studied, we restrict the movement of the particle on the surface, to the Möbius strip to study the variation of electronic and structural properties based on geometric variations and compare them with computational results. We found that as the size of a strip decreases, the greater the variations in properties will be for lower stability, lower conductivity, and lower reactivity, the greater the number of twists.Item Simulação do confinamento eletrônico em sistemas moleculares utilizando átomos fantasmas(2021-12-23) Cavalcante, Hisla da Silva; Bastos, Cristiano Costa; http://lattes.cnpq.br/6385190604693576; http://lattes.cnpq.br/1092905152020419Realizing about diversity of nanostructures use in generating compounds with diversified and very precise geometries, making use of approaches that include differential geometry components seems a reasonable path. From this perspective, we employ electronic confinement in nanostructures through computational, intrinsic and extrinsic approaches, which allow preliminary studies in the development of new electronic devices of greater sophistication. This work, therefore, seeks to study the confinement of 1 and 2 electrons in one-dimensional structures (linear, circular, linear with curves and similar to paraboles), analyze their energy spectra, charge distributions from computer simulations and compare them with theoretical models. We present load distribution results that proved to be adequate for theoretical models.Item Um breve estudo sobre a geometria diferencial de superfícies em R3(2021-07-23) Santos, Túlio José de Souza; Gomes, Renato Teixeira; http://lattes.cnpq.br/0570606157057337; http://lattes.cnpq.br/5181696493328012The purpose of this work is to make a brief study on the differential geometry of surfaces in R3, with the objective of demonstrating the Gauss-Bonnet theorem in its local and global version. This relevant result relates the geometry and topology of surfaces in R3 and has very interesting consequences. Through it, it is possible to give an answer to an ancient problem of determining whether Euclid’s fifth postulate is an axiom or a theorem. In fact, what is obtained is that there is no harm in denying the fifth postulate, that is, to suppose that there may be more than one or no parallel line to a line r passing through a point p outside of r. What is found are "brave new worlds"that have different geometries from the Euclidean one.Item Modelos de química quântica aplicados à nanoestruturas(2018-08-13) Silva Filho, Franklin Ferreira da; Bastos, Cristiano Costa; Silva, Luiz Carlos Barbosa da; http://lattes.cnpq.br/1481188647500485; http://lattes.cnpq.br/6385190604693576; http://lattes.cnpq.br/0671142012284329Considering the remarkable versatility of nanostructures on generating compounds with varied and well-defined geometry, to study models that include elements of differential geometry seems to be a natural path. On this perspective, one of the possible analysis is the use of extrinsic Hamiltonians, which incorporate elements of the ambient that contains the physical system in the form of a potential term. This work aim to study the confinemnt on bidimensional structures (planes, cylinders, spheres and cones), to explore its Hamiltonians, energy spectra and seeks relationships with physical and chemical properties of equivalent nanomaterials via computational simulation. We present computational simulated electron density results that are consistent with the theoretical models, and we explore an application in the acidity-basicity control of nanostructures using their curvature.