Uma história da definição rigorosa das integrais definidas de Cauchy, Riemann e Jordan
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Data
2022-11-09
Autores
Gambera, Artur Rezzieri [UNESP]
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
Tecemos uma história das integrais definidas sob o ponto de vista do rigor. Nossa pesquisa busca compreender como o rigor se manifestou no cálculo integral durante o século 19. Para isso, analisamos três fontes históricas de diferentes momentos do referido século que tratam da estrutura lógica da concepção de cálculo integral: o Résumé des leçons données a l'École Royale Polytechnique sur le calcul infinitésimal de Augustin-Louis Cauchy (1789 - 1857), o Ueber die Darstellbarkeit einer Fuction durch eine trigonometrische Reihe de Georg Friedrich Bernhard Riemann (1826 - 1866), e o Remarques sur les intégrales définies de Marie Ennemond Camille Jordan (1838 - 1922). Fazemos uma análise epistêmica e contextual de partes de cada uma das fontes, assistidos por historiografias que delas tratam. Brindamos traduções de algumas das lições do tratado de Cauchy, referentes a integrais definidas e elementos dela, e do artigo de Jordan na íntegra. Plotamos, e mostramos como fazer, o gráfico de uma função com infinitas descontinuidades apresentada por Riemann, identificando a construção do referido gráfico como um recurso didático para aulas de análise.
We have written a history of definite integrals form the point of view of rigor. Our research seeks to understand how rigor occurred in integral calculus in the 19th century. For this, we analyzed three historical sources that deal with the logical structure of the concept of integral calculus from different moments of that century: the Augustin-Louis Cauchy's Résumé des leçons données a l'École Royale Polytechnique sur le calcul infinitésimal, the Georg Friedrich Bernhard Riemann's Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe, and the Marie Ennemond Camille Jordan's Remarques sur les intégrales définies. We make an epistemic and contextual analysis of parts of each of the sources, assisted by historiographies that deal with them. We provide translations into Portuguese of some of the lessons from Cauchy's treatise, referring to definite integral and elements thereof, and Jordan's article. We plot, and show how to do it, the graph of a function with infinite discontinuities presented by Riemann, identifying it as a didactic resource for classes in analysis.
We have written a history of definite integrals form the point of view of rigor. Our research seeks to understand how rigor occurred in integral calculus in the 19th century. For this, we analyzed three historical sources that deal with the logical structure of the concept of integral calculus from different moments of that century: the Augustin-Louis Cauchy's Résumé des leçons données a l'École Royale Polytechnique sur le calcul infinitésimal, the Georg Friedrich Bernhard Riemann's Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe, and the Marie Ennemond Camille Jordan's Remarques sur les intégrales définies. We make an epistemic and contextual analysis of parts of each of the sources, assisted by historiographies that deal with them. We provide translations into Portuguese of some of the lessons from Cauchy's treatise, referring to definite integral and elements thereof, and Jordan's article. We plot, and show how to do it, the graph of a function with infinite discontinuities presented by Riemann, identifying it as a didactic resource for classes in analysis.
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Palavras-chave
História da análise, Rigor, Cálculo integral, History of analysis, Rigor, Integral calculus