Licenciatura em Matemática (Sede)
URI permanente desta comunidadehttps://arandu.ufrpe.br/handle/123456789/24
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APP - Artigo Publicado em Periódico
TAE - Trabalho Apresentado em Evento
TCC - Trabalho de Conclusão de Curso
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Resultados da Pesquisa
Item O teorema da função inversa e sua relação com as superfícies regulares(2023-09-18) Bezerra, Bruna Vitória Borges; Carvalho, Gilson Mamede de; http://lattes.cnpq.br/0044877127514130; http://lattes.cnpq.br/7230765885728286This work’s main objective is to study and establish relationships between the inverse function theorem, which is presented in the Euclidean space Rn, with regular surfaces, in the context of Differential Geometry. We will show how a result arising from the context of mathematical Analysis can serve as a basis for introducing one of the main objects of study in Differential Geometry. For this construction, we will initially address basic concepts involving the topology of the Euclidean space Rn, which will be present throughout the text. Next, we will present the fundamental notions and results about continuity and differentiability in the n-dimensional Euclidean space and finally, we will introduce regular surfaces together with some relevant results to establish a natural and expected relationship with the topology of Euclidean spaces and o Inverse function theorem.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 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.