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Teorema de Euler em sala de aula / Euler’s theorem in classroonGontijo, Helen Kássia Coelho 25 July 2014 (has links)
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Previous issue date: 2014-07-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work is based on the study of the polyhedrons and the Euler's Theorem, by
applying strategies of teaching using the concrete material, provoking improvements
in the reasoning and in the geometrical perception the Euler's Theorem. Not mentioning
a bit of history of tracks already made by several mathematicians who have
contributed to the study of geometry, where the ideas previously applied by them teach
us and help every day. Going to the presentation of a few concepts and de nitions
about polyhedrons, as well as the demonstration that exist only ve polyhedrons of
Plato. We've tried to expose the demonstration of the Euler's Theorem, through two
researchers, Adrien Marie Legendre and of the professor Zoroastro Azambuja Filho,
considering them very interesting and easy to understand. However, in the perspective
that going from the concrete one is an alternative to improve the quality of teaching,
it has been selected the activity Geometry of cutting soaps , which is in an article of
Ana Maria Kale , see at [10], and Geometry of straws , at [9], which are based on
work experiences of the same author. Before the new technologies we have opted for
the mathematical software Poly, available on http://www.peda.com/poly which allows
a better visualization of polyhedrons of di cult construction. All these activities have
been presented to the students of the second grade in the Secondary Education to verify
the Euler's Theorem through concrete experiences, obtaining this way a useful and
creative geometrical knowledge, conquering the students' participation and interest. / Este trabalho baseia-se no estudo dos Poliedros e o Teorema de Euler, aplicando
estratégias de ensinar usando o material concreto, desencadeando melhoras no raciocínio e na percepção geométrica do Teorema de Euler. Não deixando de mencionar um
pouco da história de caminhos já trilhados por vários matemáticos que contribuíram
para o estudo da geometria, onde as ideias anteriormente aplicadas por eles nos ensinam
e ajudam no dia-a-dia. Partindo então para apresentação de alguns conceitos e
de nições sobre Poliedros, bem como a demonstração de que só existem cinco poliedros
de Platão. Buscamos expor a demonstração do Teorema de Euler, por dois pesquisadores,
Adrien Marie Legendre e do professor Zoroastro Azambuja Filho, considerando-as
bem interessantes e de fácil compreensão. Contudo, na perspectiva de que partir do
concreto é uma alternativa para melhorar a qualidade de ensino, foi selecionada a atividade
Geometria dos cortes de sabão , que se encontra em um artigo de Ana Maria
Kale , veja em [10] e Geometria de Canudos , em [9], que são fundamentados em
experiências de trabalho da mesma autora. Frente às novas tecnologias optamos pelo
uso do software matemático Poly, disponível em http://www.peda.com/poly, que permite
uma melhor visualização de poliedros de difícil construção. Todas estas atividades
foram apresentadas para os alunos do 2o ano do Ensino Médio para a veri cação do
Teorema de Euler através de experiências concretas, obtendo assim um conhecimento
geométrico criativo e útil, conquistando a participação e interesse dos estudantes.
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Cálculo das fórmulas de Euler e Pick no geoplano e no GeoGebra / Euler and pick’s numerical methods in calculus with geoplan and GeoGebraCarvalho, Wesley da Silva 09 December 2016 (has links)
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Previous issue date: 2016-12-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this dissertation, we first state Euler's polyhedral formula for a set of points with Euler characteristic 2. We address the two known ways to prove Euler's Theorem: beginning with the classical proof by using Euclidian Geometry and afterwards we take the advantage of Spherical Geometry to give another proof. Furthermore, we address a version of Euler's formula for planar polyhedron, as well as, Pick's formula and the equivalence between Euler and Pick's formula. In the end, we provide application of Euler and Pick's formula, via two pedagogy tools Geoplano and GeoGebra, by giving examples to teach in classroom. / Esta dissertação trata inicialmente da Fórmula de Euler e de sua validade para os conjuntos de pontos com característica de Euler igual a 2. São feitas duas demonstrações da Fórmula de Euler, uma utilizando conceitos de Geometria Euclidiana e uma outra via Geometria Esférica, além da apresentação de uma versão para poliedros planos da Fórmula de Euler. Posteriormente, é apresentada a Fórmula de Pick para o cálculo de áreas de polígonos simples reticulados e sua relação de equivalência com a Fórmula de Pick para poliedros planos. Finalmente mostramos duas possibilidades de trabalho com a Fórmula de Pick, no Geoplano e no software GeoGebra.
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Formas ponderadas do Teorema de Euler e partições com raiz : estabelecendo um tratamento combinatório para certas identidades de RamanujanSilva, Eduardo Alves da January 2018 (has links)
O artigo Weighted forms of Euler's theorem de William Y.C. Chen e Kathy Q. Ji, em resposta ao questionamento de George E. Andrews, matemático estadunidense, sobre encontrar demonstrações combinatórias de duas identidades no Caderno Perdido de Ramanujan, nos mostra algumas formas ponderadas do Teorema de Euler sobre partições com partes ímpares e partes distintas via a introdução do conceito de partição com raiz. A propositura deste trabalho é envolta à apresentação de resultados sobre partições com raiz de modo a posteriormente realizar formulações combinatórias das identidades de Ramanujan por meio deste conceito, procurando estabelecer conexões com formas ponderadas do Teorema de Euler. Em particular, a bijeção de Sylvester e a iteração de Pak da função de Dyson são elementos primordiais para obtê-las. / The article Weighted forms of Euler's theorem by William Y.C. Chen and Kathy Q. Ji in response to the questioning of George E. Andrews, American mathematician, about nding combinatorial proofs for two identities in Ramanujan's Lost Notebook shows us some weighted forms of Euler's Theorem on partitions with odd parts and distinct parts through the introduction of the concept of rooted partition. The purpose of this work involves the presentation of results on rooted partitions in order to make combinatorial formulations of Ramanujan's identities, seeking to establish connections with weighted forms of Euler's Theorem. In particular, the Sylvester's bijection and the Pak's iteration of the Dyson's map are primordial elements to obtain them.
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Pfaffian Differential Expressions and EquationsUnni, K. Raman 01 May 1961 (has links)
It is needless to point out the necessity and the importance of the study of Pfaffian differential expressions and equations. While it is interesting to consider from the pure mathematical point of view, their applications in many branches of applied mathematics are well known. To mention a few, one may observe that they arise in connection with line integrals (example, determination of work). They provide a more rational formulation of the foundations of thermodynamics as 'developed by the Greek mathematician Caratheodory. They also arise in the problem of determining the orthogonal trajectories. In many branches of engineering and other physical sciences they appear with problems concerning partial differential equations.
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Formas ponderadas do Teorema de Euler e partições com raiz : estabelecendo um tratamento combinatório para certas identidades de RamanujanSilva, Eduardo Alves da January 2018 (has links)
O artigo Weighted forms of Euler's theorem de William Y.C. Chen e Kathy Q. Ji, em resposta ao questionamento de George E. Andrews, matemático estadunidense, sobre encontrar demonstrações combinatórias de duas identidades no Caderno Perdido de Ramanujan, nos mostra algumas formas ponderadas do Teorema de Euler sobre partições com partes ímpares e partes distintas via a introdução do conceito de partição com raiz. A propositura deste trabalho é envolta à apresentação de resultados sobre partições com raiz de modo a posteriormente realizar formulações combinatórias das identidades de Ramanujan por meio deste conceito, procurando estabelecer conexões com formas ponderadas do Teorema de Euler. Em particular, a bijeção de Sylvester e a iteração de Pak da função de Dyson são elementos primordiais para obtê-las. / The article Weighted forms of Euler's theorem by William Y.C. Chen and Kathy Q. Ji in response to the questioning of George E. Andrews, American mathematician, about nding combinatorial proofs for two identities in Ramanujan's Lost Notebook shows us some weighted forms of Euler's Theorem on partitions with odd parts and distinct parts through the introduction of the concept of rooted partition. The purpose of this work involves the presentation of results on rooted partitions in order to make combinatorial formulations of Ramanujan's identities, seeking to establish connections with weighted forms of Euler's Theorem. In particular, the Sylvester's bijection and the Pak's iteration of the Dyson's map are primordial elements to obtain them.
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Poliedros Regulares no Ensino MédioSilva, Hércules do Nascimento 29 August 2014 (has links)
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Previous issue date: 2014-08-29 / In this work we present a study of the regular polyhedra, comparing and discussing
the concepts and de nitions given in the study of regular polyhedra in textbooks
most widely used in Brazilian high schools. We prove the theorem of Euler, we calculate
surface areas and volumes of regular polyhedra. Finally, we present some
mathematical software that can be used by students and mathematics teachers in
the spatial geometry classes as auxiliary material in the teaching and learning of
this subject in the classroom. / Neste trabalho apresentamos um estudo sobre os poliedros regulares, comparando
e discutindo os conceitos e as de nições que são dadas no estudo dos poliedros regulares
nos livros didáticos mais utilizados nas escolas brasileiras de Ensino Médio.
Provamos o teorema de Euler, calculamos áreas de superfícies e os volumes dos poliedros
regulares. Por m, apresentamos alguns softwares matemáticos que podem ser
utilizados pelos alunos e professores de Matemática nas aulas de geometria espacial
como material auxiliar no processo de ensino e aprendizagem deste tema em sala de
aula.
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A característica de EulerJustino, Gildeci José 24 September 2013 (has links)
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Previous issue date: 2013-09-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This dissertation is focused on the Euler's theorem for polyhedra homeomorphic
to the sphere. Present statements made by Cauchy, Poincaré and Legendre. As
a consequence we show that there are only ve regular convex polyhedra, called
polyhedra Plato. / Esta dissertação tem como tema central o Teorema de Euler para poliedros homeomorfos
à esfera. Apresentamos demonstrações feitas por Cauchy, Poincaré e
Legendre. Como consequência mostramos a existência de apenas cinco poliedros
convexos regulares, os chamados poliedros de Platão.
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Platonská a Archimédovská tělesa a jejich vlastnosti ve výuce matematiky na středních školách / Platonic and Archimedean solids and their properties in teaching of mathematics at secondary schoolsDohnalová, Eva January 2016 (has links)
Title: Platonic and Archimedean solids and their properties in teaching of mathematics at secondary schools Author: Eva Dohnalová Department: Department of Didactics of Mathematics Supervisor: doc. RNDr. Jarmila Robová, CSc. Abstract: This work is an extension of my bachelor work and it is intended for all people interested in regular and semiregular polyhedra geometry. It is a comprehensive text which summarizes brief history, description and classification of regular and semiregular polyhedra. The work contains proofs of Descartes' and Euler's theorems and proofs about number of regular and semiregular polyhedra. It can be also used as a didactic aid in the instruction of regular and semiregular solids at secondary schools. This text is supplemented by illustrative pictures made in GeoGebra and Cabri3D. Keywords: Regular polyhedra, platonic solids, Platon, semiregular polyhedra, Archimedean solids, Archimedes, dulaism, Descartes' theorem, Euler's theorem.
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