About my Ph.D. program at Virginia Tech, when I got about 50% (or perhaps a little more) of its total time achieved, I realized to be completely convinced any successful approach on duality theory should be based on the works of W.R. Bielski, J.J Telega and John Toland, indeed on a combination of such approaches.

Since I also realized my adviser at that moment did not agree with such an approach, I asked the Head of Department for a change in the supervision.

At that time the thesis work was at the beginning and was almost all developed under supervision of Professor Robert C. Rogers.

I can guarantee there was no other kind of incident, allegation or circunstance concerning such an event.

During such a Ph.D. program I got a GPA of about 9.75/10.0 and published a paper in an excellent european Journal of Pure and Applied Mathematics.

The published paper is this one

8. Fabio Botelho, Dual Variational Formulations for a Non-linear Model of Plates. JOURNAL OF CONVEX ANALYSIS, v. 17, p. 131-158, 2010.

With kind regards,

Fabio Silva Botelho

https://twitter.com/FabioBotelho367

Recent Pre-Prints

arXiv:2205.15910 [pdf, ps, other]

math.GM

An approximate proximal numerical procedure concerning the generalized method of lines

Authors: Fabio Silva Botelho

Abstract: This article develops an approximate proximal approach for the generalized method of lines. The present results are extensions and applications of previous ones which have been published since 2011, in books and articles such as [3,4,5,6]. We also recall that in the generalized method of lines, the domain of the partial differential equation in question is discretized in lines (or in curves) and the concerning solution is developed on these lines, as functions of the boundary conditions and the domain boundary shape. △ Less

Submitted 31 May, 2022; originally announced May 2022.

Comments: 9 pages

MSC Class: 65N40; 65N06

0.0-arXiv:2112.05189 [pdf, ps, other]

math.GM

An approximate numerical method for ordinary differential equation systems with applications to a flight mechanics model

Authors: Fabio Silva Botelho

Abstract: This short communication develops a new numerical procedure suitable for a large class of ordinary differential equation systems found in models in physics and engineering. The main numerical procedure is analogous to those concerning the generalized method of lines, originally published in the here referenced books of 2011 and 2014, [3,5], respectively. Finally, in the last section, we apply the method to a model in flight mechanics. △ Less

Submitted 9 December, 2021; originally announced December 2021.

Comments: 6 pages

MSC Class: 65L10

0.1- arXiv:2110.00994 [pdf, ps, other]

math.OC

A convex dual formulation for a large class of non-convex models in variational optimization

Authors: Fabio Silva Botelho

Abstract: This short communication develops a convex dual variational formulation for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field.

Submitted 30 March, 2022; v1 submitted 3 October, 2021; originally announced October 2021.

Comments: 7 pages, a substantial part of this article has been re-written, some corrections implemented

MSC Class: 49N15

0.2-arXiv:2109.01662 [pdf, ps, other]

math.OC

A convex dual variational formulation for non-convex optimization applied to a non-linear model of plates

Authors: Fabio Silva Botelho

Abstract: This article develops duality principles applicable to the non-linear Kirchhoff-Love model of plates. The results are obtained through standard tools of convex analysis, functional analysis, calculus of variations and duality theory. The main duality principle concerns a convex (in fact concave) dual variational formulation and related new optimality conditions for the model in question. Finally, in the last section we develop some global existence results for a similar model in elasticity. △ Less

Submitted 4 September, 2021; originally announced September 2021.

Comments: 23 pages. arXiv admin note: text overlap with arXiv:1712.01595

MSC Class: 35J58; 49N15

arXiv:2108.10170 [pdf, ps, other]

math.OC

Dual variational formulations for a large class of non-convex models in the calculus of variations

Authors: Fabio Silva Botelho

Abstract: This short communication develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg-Landau type system in superconductivity in the absence of a magnetic field. Finally, in the last section, we develop a new general primal dual variational formulation also suitable for the non-convex global optimization for a large class of models in physics and engineering. △ Less

Submitted 4 July, 2022; v1 submitted 18 August, 2021; originally announced August 2021.

Comments: 17 pages, some corrections implemented, a new section added

MSC Class: 49N15

Last update (This last up-date presents a general convex dual variational formulation for a global optimization of a general non-convex primal variational formulation)

8. Fabio Botelho, Dual Variational Formulations for a Non-linear Model of Plates. JOURNAL OF CONVEX ANALYSIS, v. 17, p. 131-158, 2010.

Tópicos em Cálculo Variacional e Otimização em Espaços de Banach – 2022-1

Cálculo Variacional (Notas de Aula)

Observação: As vídeo-aulas desse curso serão postadas também no meu canal do You Tube

https://www.youtube.com/channel/UCGe1piu6tU12zZArIxZ3xzw

Aula 1 – Espaços de Banach – O Espaço das Funções Contínuas é de Banach com a Norma do Máximo

https://youtu.be/Zv_1tSIJUjc

Aula 2 – O Espaço das Funções C¹([a,b]) é de Banach

https://youtu.be/wgv5Cm4nqSo

Aula 3 – Funcionais em Espaços de Banach e Pontos de Mínimos Globais, Definições e Exemplos

https://youtu.be/WRq6E-ofZvQ

Aula 4 – A Primeira Variação à Gâteaux

https://drive.google.com/file/d/1RKCi13nIsutdeZiXExoaJ9vWiA5stMkN/view?usp=sharing

Aula 5 – Minimização de Funcionais Convexos – Primeira Parte: Funções Convexas

https://drive.google.com/file/d/1BoW1rSfCsqdZGYatmyQsTyPWn70lFBtD/view?usp=sharing

Aula 6 – Minimização de Funcionais Convexos – Segunda Parte – Definições e Exemplos

https://drive.google.com/file/d/1ZBxz0CGxveMEGiD6ffW2G1Bifobgh5RQ/view?usp=sharing

Aula 7 – Condições Suficientes de Otimalidade no Caso Convexo

https://drive.google.com/file/d/1yUywTBqzN_2PTNOBgbN22F5aW3bWJ0U5/view?usp=sharing

Aula 8 – Um Exercício Sobre a Otimização de um Funcional Convexo

https://drive.google.com/file/d/1uTqF-he85X6yhkDeqGl2u7HXbAgz_nrN/view?usp=sharing

Aula 9 – Condições Suficientes de Otimalidade no Caso Convexo, Caso de um Extremo Livre

https://drive.google.com/file/d/1tolWzASpV83Om5fFGsYUksS66AZHCbLd/view?usp=sharing

Aula 10 – Condições Suficientes de Otimalidade no Caso Convexo, Caso de Dois Extremos Livres

https://drive.google.com/file/d/14eOc_MiMXUmGef4V4njal99Bzp13mzgC/view?usp=sharing

Aula 11 – Exercícios sobre a Otimização de Funcionais Convexos

https://drive.google.com/file/d/1KyhY1MCN33yNy2ega2nb6CheFmVbec34/view?usp=sharing

Aula 12 – Condições de Otimalidade para um Funcional Convexo Envolvendo Derivadas de Segunda Ordem

https://drive.google.com/file/d/1_Y_bCrafeNx_6j7JpQ0BaMs0nycX22uV/view?usp=sharing

Aula 13 – O Lema de du Bois-Reymond

https://drive.google.com/file/d/1xENdphcroBvFrUIhI79lKUdN8cDFC097/view?usp=sharing

Aula 14 – O Lema Fundamental do Cálculo Variacional para o Caso Uni-Dimensional

https://drive.google.com/file/d/1v_oMpXE9Vzujx23WP6wCiStJHchnZdXv/view?usp=sharing

Aula 15 – Cálculo Variacional, o Caso de Espaços de Funções Escalares no Rn

https://drive.google.com/file/d/19e2iDxAQbe7aHp7TAdWS5JaePz__7xLj/view?usp=sharing

Aula 16 – Condiçoes Suficientes de Otimalidade no Caso Convexo, Caso de Espaços de Funções Escalares no Rn

https://drive.google.com/file/d/1xFQOo_RS-yP8UASLaiOJY_MpHyDSxUSi/view?usp=sharing

Aula 17 – A Segunda Variação à Gâteaux

https://youtu.be/pQ_jpLcYQW4

Aula 18 – Condições Necessárias de Primeira Ordem e Suficientes de Segunda Ordem para um Mínimo Local

https://youtu.be/UzD_feYss7I

Aula 19 – Funcionais Contínuos em Espaços de Banach

https://youtu.be/fPnMWesOwpk

Aula 20 – Variação à Gâteaux – Resultados Formais

https://youtu.be/OJ_2rMK2zbs

As seguintes vídeo-aulas do curso de Análise Funcional são partes fundamentais desse curso de Cálculo Variacional

Aula 36 – O Teorema da Hahn-Banach

https://drive.google.com/file/d/1QMv-7uOwWKAOBdgwCbmAcDpZ8rmkFeJX/view?usp=sharing

Aula 37 – Aula de Exercícios – Preparação para a Primeira Avaliação

https://drive.google.com/file/d/1UNRLD0sYD83vbec2fSLgj9p9CTrDiEWn/view?usp=sharing

Aula 38 – Espaço Dual Topológico e Corolários do Teorema de Hahn-Banach

https://drive.google.com/file/d/1KP-O28MZg52NB1faElqDMhubl0zlxoFs/view?usp=sharing

Aula 39 – O Teorema de Hahn-Banach, Forma Geométrica

https://drive.google.com/file/d/136wEF4xfOXwYFtZjTGxSA523aisF9Nu4/view?usp=sharing

Correções para a Aula 39

https://drive.google.com/file/d/1TZO4jOdSc19zFcvIKc4zNMeWR_2UBjpw/view?usp=sharing

Aula 40 – Teorema de Hahn-Banach, Segunda Forma Geométrica

https://drive.google.com/file/d/1lkWwqDpXHGD1CE39UCmrXY88yIYkxvOH/view?usp=sharing

Aula 41 – Em um Espaço de Banach, um Conjunto Convexo A é Fortemente Fechado se, e somente se, é Fracamente fechado

https://drive.google.com/file/d/114ITf1FN8UIaVC0GCkHlVX5AqOSQvnIP/view?usp=sharing

Aula 42 – A Topologia Fraca-Estrela ( a ser postada em breve)

Aula 43 – O Teorema de Banach- Alaoglu – Compacidade Fraca-Estrela no Espaço Dual

https://drive.google.com/file/d/1S6kI0A82_5gYvOWFz28wfg56xWC5UIVd/view?usp=sharing

Aula 44 – O Teorema de Kakutani – Um Espaço de Banach é Reflexivo se, e somente se, a Bola Fechada Unitária é Fracamente Compacta

https://drive.google.com/file/d/1Ra9nIpWtOohCqCMWLekdsQTxg8rd5O_s/view?usp=sharing

Aula 21 -B – Variação à Gâteaux, O Caso Vetorial no Cálculo Variacional

https://drive.google.com/file/d/1gy2pCLLCYAyRKLNr2PaWurXRcueulvBu/view?usp=sharing

https://youtu.be/FXt9iO--2LQ (Aula 21 -B - no You Tube)

Aula 22 – B – Funcionais Fracamente Semi-Contínuos Inferiormente (f.s.c.i.) em Espaços de Banach

https://drive.google.com/file/d/1j9NK3C0NGYZZ0rZ0UQIEqmDI_HDYvuWe/view?usp=sharing

https://youtu.be/4V3YkDBpTi4 (Aula 22 – B no You Tube)

Aula 23 – B – Funcionais Polares, Bi-Polares e o Envelope Convexo

https://youtu.be/VuKLEZUuDoE

Aula 24 -B – Subgradientes e Continuidade de Funcionais Convexos

https://youtu.be/DbptG8p7BZU

Aula 25 – B- O Conjunto dos Subgradientes de um Funcional Convexo e Contínuo num Ponto é Não-Vazio nesse Ponto

https://youtu.be/5f1zNELFx1Q

Aula 26 – B – A Transformada de Legendre e Propriedades

https://youtu.be/HDWjd-nZEnM

Aula 27 - Equivalência entre os Pontos Críticos dos Funcionais Primal e Dual no Contexto da Transformada de Legendre

https://youtu.be/wQjRoMURl0M

Aula 30 – Teoria da Dualidade no Caso Convexo

https://youtu.be/GINHTtN9crI

Aula 31 – O Teorema Mín-Máx em Espaços de Banach

https://youtu.be/uVETYzYoB8c

Cálculo 4 – 2021 - 2

Aula 1 – Sequências Reais, Primeiras Definições, Limite de uma Sequência Real

https://drive.google.com/file/d/1kU6u95VP7E7mxVtK9rw8Syrnrlb5cdER/view?usp=sharing

Aula 2 – Unicidade do Limite de uma Sequência Real, Exercícios

https://drive.google.com/file/d/1IEv8OUztJpZQO6JotJFsHNaOmQzCp0qj/view?usp=sharing

Aula 3 – Sequências Reais Monótonas e Limitadas são Convergentes

https://drive.google.com/file/d/1hOl4OXKh1vASjns-5Qab7pxRIdTR5l2M/view?usp=sharing

Aula 4 – Propriedades dos Limites de Sequências Reais, Limite da Série Geométrica

https://drive.google.com/file/d/1TaTZpT4Vi-gIZhtb9rDqZkJjAV1vmXxh/view?usp=sharing

Aula 5 – O Teorema do Confronto para Sequências Reais

https://drive.google.com/file/d/1UJhc1CHVlrTnk8XvcLri9X_2ugWx6JWV/view?usp=sharing

Aula 6 – Limites Superior e Inferior de uma Sequência Limitada (Aula Assíncrona)

https://drive.google.com/file/d/1y8dzMxI1lzv0JYLpEiA43UezRyKkBI1x/view?usp=sharing

Aula 7 – Exercícios sobre Limites Inferiores e Superiores para uma Sequência Real

https://drive.google.com/file/d/1qLYBgx25J3uCOD66NMgt9uN6niIott3n/view?usp=sharing

Aula 8 – Sequências de Cauchy

https://drive.google.com/file/d/1X47EnHnKt9xozKRa9ermAxfPa6KG-hjq/view?usp=sharing

Aula 9 – Uma Classe Especial de Sequências de Cauchy

https://drive.google.com/file/d/18jB2FkMwunwxdiiZ-9qRYKq-yAz026Jo/view?usp=sharing

Aula 10 – Séries Numéricas Reais, o Critério de Comparação

https://drive.google.com/file/d/1ouWUDD7n-KOTqTCC8D0ErwlKKkZLtiM4/view?usp=sharing

Aula 11– (Revisada, uma Correção no Último Exercício)

Critério de Cauchy para a Convergência de Séries Reais

https://drive.google.com/file/d/1Ny3ify7HJAVeQ4oYfBU8hDkR_BMIjn3M/view?usp=sharing

Aula 11 A – Critério de Convergência para Séries Alternadas

https://drive.google.com/file/d/1O6Bh5DuVeBS-LdwfOiOXdZXEzGn3LI5d/view?usp=sharing

Aula 12 – Teste da Razão para Convergência de Séries Reais (Aula Assíncrona)

https://drive.google.com/file/d/1229Fz0QZgeebMI4b3y2bag1G2ptNBg1r/view?usp=sharing

Aula 13 – Teste da Raiz para Convergência de Séries Reais

https://drive.google.com/file/d/1PtaRiNXPlNE7HBx9TuvUB7d3C0xsSgch/view?usp=sharing

Aula 14 – Critério de Comparação das Razões para Convergência de Séries Reais

https://drive.google.com/file/d/1b6FM0PgJJPvNoip82mk4CkMdUpvJQr-L/view?usp=sharing

Aula 15 - Critério da Integral para a Convergência de Séries Reais

https://drive.google.com/file/d/12v_9Yb8T9rSPFKEWLANQELQeJ_G0LX2R/view?usp=sharing

Aula 16 – Critério de Comparação dos Limites para Convergência de Séries Reais

https://drive.google.com/file/d/1Ea1AOw6PB3UIhmY9_2MZiMn8Q_B476QE/view?usp=sharing

Aula 17 – Critério de Cauchy para a Convergência de Séries Reais

https://drive.google.com/file/d/1Ny3ify7HJAVeQ4oYfBU8hDkR_BMIjn3M/view?usp=sharing

Aula 18– Lema de Abel e Critério de Dirichlet para a Convergência de Séries Reais

https://drive.google.com/file/d/1T5oLPUuLFK1cw5WLqccLcQlaGCKFvJzp/view?usp=sharing

Conteúdo para a segunda avaliação a ser postada dia 25 de março:

Aula 19 – Sequências de Funções (Primeira Parte)

https://drive.google.com/file/d/16so3zARFbYfJtFZROP6S5BWg7JW0rs3t/view?usp=sharing

Aula 20 – Integrais e Derivadas de Sequências de Funções Uniformemente Convergentes

https://drive.google.com/file/d/1DNRk1fgikRgIKvhKs8ByhTKjNMw_Gb1v/view?usp=sharing

Aula 21– Séries de Funções, Definições e Primeiros Resultados

https://drive.google.com/file/d/1l87B08DaRGDVpli9zraKeQz9vZ8HCPbx/view?usp=sharing

Aula 22 – Critério M de Weierstrass para Convergência Uniforme de Séries de Funções Reais

https://drive.google.com/file/d/1NbJTuVYacAKUfb2zMBgdLxt3oiEsZ_vy/view?usp=sharing

Aula 23 – Integrais e Derivadas de Séries de Funções Uniformemente Convergentes

https://drive.google.com/file/d/1dIrTsWMS0wHNANQpla1jWf090TaLlWJE/view?usp=sharing

Aula 24 – Séries de Potências, Primeiras Definições e Resultados

https://drive.google.com/file/d/1ddrx_ggnHVgmyMx6IMGWKOXfTHCa0cg-/view?usp=sharing

Aula 25 – Séries de Potências – Raio de Convergência

https://drive.google.com/file/d/1Mvxtd4bkdOl9Q2NMaOECLR38EtSnEn5O/view?usp=sharing

Aula 26 – Integrais e Derivadas de Séries de Potências

https://drive.google.com/file/d/19m0PrdmVth7NMsAK0-beAXjRHpqfaNM8/view?usp=sharing

Aula 27– Séries de Fourier – Primeiros Resultados

https://drive.google.com/file/d/1vMnh3SOz_NkmC_bXCz_6zs54KdfbDEr9/view?usp=sharing

Aula 28 – Convergência Uniforme da Série de Fourier de uma Função de Classe C² num intervalo fechado

https://drive.google.com/file/d/1mPX9wdARHwIDeK1jvDGJ83aXfoadGchR/view?usp=sharing

Aula 29 – A Série de Fourier de uma Função de Classe C² Converge Uniformemente para tal Função num Intervalo Fechado

https://drive.google.com/file/d/1yWj3jDkim5GyGuZfVb3hR2QexPoxWpSH/view?usp=sharing

Aula 30 – Um Exercício sobre o Cálculo da Série de Fourier de uma Função

https://drive.google.com/file/d/1oKVxRIyNIbJoOSG2RDVghe2nu6jtbzpr/view?usp=sharing

Aula 31 – Revisão – Teorema do Valor Médio – Preparando a Rota para a Prova da Fórmula de Taylor

https://drive.google.com/file/d/1e45ZD0zYDwGGsc9kEyQf5I_NGZwL4Cqn/view?usp=sharing

Aula 32 – Fórmula de Taylor – Preparando a Rota para as Séries de Taylor

https://drive.google.com/file/d/1R64AwA5RVEneU6Eji0dJwQKH4_d8jGEq/view?usp=sharing

Aula 33 – Método de Separação de Variáveis para EDPs – Equação do Calor

https://drive.google.com/file/d/1nT3IoTb48Lek5_bkcR1ftghdLHjV_0_E/view?usp=sharing

Aula 34 – Obtenção da Equação da Onda mediante o Cálculo Variacional

https://drive.google.com/file/d/1FZBVS9MZ8vGeHPY3MwsyPWvSFn1Ef0ih/view?usp=sharing

Correções para a Aula 34

https://drive.google.com/file/d/1hn7bKEF5jjBvsJU2r5QMrACs0Q4g78Nw/view?usp=sharing

Aula 35 – Equação da Onda – Solução pelo Método de Separação de Variáveis

https://drive.google.com/file/d/1ev858heD3AQUoMi7Y4heet37ybgDpD37/view?usp=sharing

Aula 36 – Aula Síncrona de 24/09/2021 – Preparação para a Segunda Avaliação

https://drive.google.com/file/d/18vD6wEa6A2Z81wmKoKeMDjgp0XXaqqSe/view?usp=sharing

Análise 1 – 2021-2

Aula 1 – Espaços Métricos – Primeiras Definições e Resultados

https://drive.google.com/file/d/18uIqJYHJq9n_SKjdpjrtkiZX1suPtG7a/view?usp=sharing

Aula 2 – Espaços Métricos, Propriedades dos Conjuntos Abertos e Fechados

https://drive.google.com/file/d/1lz4O24djdYUhMjmug_fEqCwsOtIs7CuY/view?usp=sharing

Aula 3 – O Fecho de um Conjunto em um Espaço Métrico

https://drive.google.com/file/d/1kjpbaiGKH5AFALvlpjTI6sNJ6JXlxW9z/view?usp=sharing

Aula 4 – Conjuntos Compactos, Primeiras Definições e Resultados

https://drive.google.com/file/d/1JNvRInnoTSOABorH3IuJsiwA6Yq1_Hk0/view?usp=sharing

Aula 5 – Conjuntos Compactos, Outros Resultados

https://drive.google.com/file/d/1U2X3CLQoJK70Eg11_FP_MkMhteuKpTMe/view?usp=sharing

Aula 6 – O Teorema de Heine-Borel

https://drive.google.com/file/d/1w0949K_9Qg8tmQ6-62k4dZPhcVhpb-Bj/view?usp=sharing

Aula 7 – O Espaço Rn, Topologia para o Rn

https://drive.google.com/file/d/16Ea6fULb1N3ud0D3vyW2uYZhODJnFIMO/view?usp=sharing

Aula 8 – Desigualdade de Cauchy-Schwartz

https://drive.google.com/file/d/1z-YZSF2GhteNhNmhaXyQxeU14F7e7qvc/view?usp=sharing

Aula 9 – Funções Escalares de Várias Variáveis e Respectivos Limites

https://drive.google.com/file/d/1fbQrGZ31WsbfmpwEX7XejqyDwYdZlovH/view?usp=sharing

Aula 10 – Unicidade do Limite para Uma Função de Várias Variáveis

https://drive.google.com/file/d/1Dq6HRdPgDnDyNziccSj7d8kxffLgaYEK/view?usp=sharing

Aula 11 – O Teorema do Confronto e Outros Resultados

https://drive.google.com/file/d/1_fkcG9mf_eWZ0U2dMlPPlhG5snvNTCS-/view?usp=sharing

Aula 12 – Propriedades dos Limites

https://drive.google.com/file/d/1YsJEwejPSecymJHnkojQmV7CTEcvhY4W/view?usp=sharing

Análise 1 - Aulas reiniciando em Fevereiro/2022

Aula 1 – B – Limite da Função Composta

https://drive.google.com/file/d/155D6vj9Ja4bJVw4ZxuN_YN0kNKhBH2gM/view?usp=sharing

Aula 2 – B- Tipos de Descontinuidades

https://drive.google.com/file/d/1UEKIUjtQWg_5aAuLvJE6q19Yq44uTrIT/view?usp=sharing

Aula 3 – B - Derivadas Direcionais

https://drive.google.com/file/d/1GTWL841H29Agh_Q34ZGrjWuyydZoVMlZ/view?usp=sharing

Aula 4 – B – Diferenciabilidade no Rn

https://drive.google.com/file/d/14RB2ubfbCD6rYEt6Y6Q5Z38-ZPOSunrZ/view?usp=sharing

Aula 5 – B – Mais Detalhes Sobre a Diferenciabilidade no R2

https://drive.google.com/file/d/1jdYpqvdZEj1iHOGxuRBedCv3YEAOtiwl/view?usp=sharing

Aula 6 -B – Exemplo de Função Diferenciável Utilizando a Definição de Diferenciabilidade

https://drive.google.com/file/d/1zv1gq8EsQNt7xjrD9foLLDJNfzG0sSij/view?usp=sharing

Aula 7 – B – Interpretação da Diferenciabilidade e o Conceito de Diferencial

https://drive.google.com/file/d/1aETiYyHTuQHUaWMm3gN2Y3u_GpJFH4fX/view?usp=sharing

Aula 8 – B- Condições Suficientes para a Diferenciabilidade no Rn

https://drive.google.com/file/d/1h8rr2x-z2D3gF1qd_9Hu-M9ZW0GzRbCr/view?usp=sharing

Aula 9- B – Regra da Cadeia no Rn

https://drive.google.com/file/d/1N2G1Bo1cw5CmGe1JefmXvieMnu6lsLlW/view?usp=sharing

Aula 10 – B – Regra da Cadeia no R2

https://drive.google.com/file/d/1iBTpHQiAteknSQ-kIlSZ0LQhclWQrSEY/view?usp=sharing

Aula 11 – B – Derivadas de Ordem Mais Alta

https://drive.google.com/file/d/1CQ8P8tLKjeK0-PZd8TAtT13zKOpw2eiv/view?usp=sharing

Aula 12 – B- Sobre a Igualdade das Derivadas Mistas de Segunda Ordem

https://drive.google.com/file/d/1NverGUn00LGDodfRF-pZ1D1gL_k40hgx/view?usp=sharing

Aula 13 – B – O Teorema de Taylor no Rn

https://drive.google.com/file/d/1_aUIFHmTMgUHNb31jH3qJ4WHiyUEgIDk/view?usp=sharing

Aula 14 – B- F órmula de Taylor para o R2 com Resto de Lagrange de Segunda Ordem

https://drive.google.com/file/d/1Xf_dCZPQLNyo99yoL5NGa557dWol446O/view?usp=sharing

Aula 15 – B – Fórmula de Taylor para o R2, uma Estimativa para o Resto de Lagrange

https://drive.google.com/file/d/10mBePUnYsmWNjT7X1B-hpy2zLPfRlC9p/view?usp=sharing

Aula 16 – B – Extremos Locais e Globais para Funções no Rn

https://drive.google.com/file/d/1_2SAasYy0CUxc1HFWNwMZiSUVrPh54Od/view?usp=sharing

Aula 17 – B – Extremos Locais , Teste das Derivadas Parciais de Segunda Ordem

https://drive.google.com/file/d/1WfcO6JHcm1HXvEvnrFPdS8qGj4YW-KJ_/view?usp=sharing

Aula 18 – B- Teorema da Função Implícita, Caso Escalar

https://drive.google.com/file/d/16uKzDgWTUjOTn0c_dH3sxdv8XD5-TK1C/view?usp=sharing

Aula 19 – B- Funções Vetoriais no Rn e Respectivos Limites

https://drive.google.com/file/d/1vKSHpaFOZTH4lBwVXiCsHBC7TywfrzPE/view?usp=sharing

Aula 20 – B – Funções Vetoriais no Rn, Propriedades dos Limites

https://drive.google.com/file/d/1FRPaLCF1V0tHSdzEXvgIE5Dd26YzCLoR/view?usp=sharing

Aula 21 – B – Continuidade, Derivadas Direcionais, Diferenciabilidade e Matriz Jacobiana para Funções Vetoriais

https://drive.google.com/file/d/1gG8CwtjPWYwVl2fLfc9_hDv34NS5kARu/view?usp=sharing

Aula 22 – B – Uma Função Vetorial é Diferenciável Se, e Somente Se, Cada Função Coordenada é Diferenciável

https://drive.google.com/file/d/1CoPJedfk5fP8vzyab41M692nuksXFpGp/view?usp=sharing

Aula 23 – B – Desigualdades do Valor Médio para Funções Vetoriais a uma e n Variáveis

https://drive.google.com/file/d/19mWf2VSG1taJmTIuJWIlL61Xd3wE8Qem/view?usp=sharing

Aula 24 – B – O Teorema do Ponto Fixo de Banach no Rn

https://drive.google.com/file/d/1x92JUVemfVShubYNiqokZLGOqTOBjDX9/view?usp=sharing

Aula 25 – B – O Teorema da Função Implícita, Caso Vetorial

https://drive.google.com/file/d/1TkTA6DjVWBCHVV_dELbYqdfUZDny12Sa/view?usp=sharing

Aula 25 – B-1-O Teorema da Função Implícita, Caso Vetorial – Revisão

https://drive.google.com/file/d/1U_L7UcoHC_oRgagYlN4V89sBwKWdwOyA/view?usp=sharing

Aula 26 – B – Multiplicadores de Lagrange, Primeira Parte, Uma Restrição no R3

https://drive.google.com/file/d/1xkoh3k_g875uj5HuoswrcZIiTuARH6ub/view?usp=sharing

Aula 26 – B -1 – Multiplicadores de Lagrange, Primeira Parte, Revisão

https://drive.google.com/file/d/1zfjyTnnJoVCAg7XkjgYRwyAQFBuN9C4B/view?usp=sharing

Aula 27 – B – Multiplicadores de Lagrange, Segunda Parte, Duas Restrições no R4

https://drive.google.com/file/d/1khlikZZ1JnO2r-Rw69egNfP5ul1hy390/view?usp=sharing

Aula 28 – B – Multiplicadores de Lagrange, Terceira Parte, Caso Geral no R(n+m) (m restrições)

https://drive.google.com/file/d/1-5X5OjJM1xQz55hZ79wX8-osCvkDf0qn/view?usp=sharing

Aula 29 – B – O Teorema da Função Inversa no Rn

https://drive.google.com/file/d/1ljvMs9h-Ftv-v71O36vGq0PQeAXJDU7z/view?usp=sharing

Obs.: Aulas 30 a 34, Material de Cálculo 4, mas com padrão de Análise Real, uma atualização será postada em breve

Aula 30 – Sequências de Funções (Primeira Parte)

https://drive.google.com/file/d/16so3zARFbYfJtFZROP6S5BWg7JW0rs3t/view?usp=sharing

Aula 31 – Integrais e Derivadas de Sequências de Funções Uniformemente Convergentes

https://drive.google.com/file/d/1DNRk1fgikRgIKvhKs8ByhTKjNMw_Gb1v/view?usp=sharing

Aula 32– Séries de Funções, Definições e Primeiros Resultados

https://drive.google.com/file/d/1l87B08DaRGDVpli9zraKeQz9vZ8HCPbx/view?usp=sharing

Aula 33 – Critério M de Weierstrass para Convergência Uniforme de Séries de Funções Reais

https://drive.google.com/file/d/1NbJTuVYacAKUfb2zMBgdLxt3oiEsZ_vy/view?usp=sharing

Aula 34 – Integrais e Derivadas de Séries de Funções Uniformemente Convergentes

https://drive.google.com/file/d/1dIrTsWMS0wHNANQpla1jWf090TaLlWJE/view?usp=sharing

Aula 35 – B - O Teorema de Arzela-Ascoli

https://drive.google.com/file/d/1n6TjpiFDajCEMRg_MB8gXno0KocE5wnf/view?usp=sharing

Aula 36 – B- O Teorema de Stone-Weierstrass – Densidade dos Polinômios no Espaço das Funções Contínuas

https://drive.google.com/file/d/1MhNrX92x6Ms-US7d7P7mQTVW0kNLzkSa/view?usp=sharing

Primeira Atualização:

Aula 37 – B – Convergência Uniforme e Continuidade em Espaços Métricos

https://drive.google.com/file/d/1F6doljfJ2Sb4-K1V9jPmV3Z9jJfLrjQN/view?usp=sharing

Aula 38 – B - Espaços de Banach

https://drive.google.com/file/d/1VRf6-_Ruk_gxHHxEFRPh7FuvtLH3hGzT/view?usp=sharing

Aula 39 - B – O Espaço das Funções C¹([a,b]) é de Banach

https://drive.google.com/file/d/10gKKLowHsSr_jKpgC9v4ztjxL1ivFcuZ/view?usp=sharing

Análise Funcional – 2021-2

Notas de Aula – Listas de Exercícios

Aula 1 – Espaços Métricos – Revisão – Primeiras Definições e Exemplos

https://drive.google.com/file/d/153R3C1v4ERP4ZdN6_BPraURAIgnTpQoL/view?usp=sharing

Aula 2 – Uma Métrica para o Espaço das Sequências Complexas

https://drive.google.com/file/d/1BmgrfxNi3OTn-M8Tybsqa6zsSISftAxk/view?usp=sharing

Aula 3 – Espaços lp – Desigualdades de Young, Hölder e Minkowiski

https://drive.google.com/file/d/1vf_uoJcNgRBzLGbYP8DPtqPOXZZOss9D/view?usp=sharing

Aula 3 – Correções e Revisão da Desigualdade de Minkowiski

https://drive.google.com/file/d/1R8LeVjnSdQ2T_p-X5LIBJEOtdYBJ8cbm/view?usp=sharing

Aula 4 – Parte Final da Aula Síncrona de 08/09/2021 – Definições Fundamentais em Espaços Métricos

https://drive.google.com/file/d/10dYRuDFqb0159krMmSQvp8V_ncDlJr5C/view?usp=sharing

Aula 4 – Correções

https://drive.google.com/file/d/1okysIn5HdaD_crN79sGpaXSP_D3zqELB/view?usp=sharing

Aula 5 – Propriedades de Conjuntos Abertos e Fechados em Espaços Métricos

https://drive.google.com/file/d/1KHjn-HAaQOCjL5tGY7ZiaRfHAw6lkLzf/view?usp=sharing

Aulas 6 a 10, Revisão de alguns conceitos fundamentais em Análise Real

Aula 6 – O Fecho de um Conjunto em um Espaço Métrico

https://drive.google.com/file/d/12YOyKgidrYIE6gCLNaHmNQrDTkanO5J5/view?usp=sharing

Aula 7 – Conjuntos Compactos, Definições e Propriedades

https://drive.google.com/file/d/1dbhI0hZtB2ZxCHBNb8DHG2fjguYdbBj2/view?usp=sharing

Aula 8 – Conjuntos Compactos, Propriedades – Segunda Parte

https://drive.google.com/file/d/1F7dVAOyCF4zvTozkt8-Ckq4E2muaqH1f/view?usp=sharing

Aula 9 – Todo Intervalo Real Fechado e Limitado é Compacto

https://drive.google.com/file/d/1uexHRUQqzzMiYVIcWiwkFMnHR7qwYQBw/view?usp=sharing

Aula 10 – O Teorema de Heine-Borel

https://drive.google.com/file/d/1OaO9S27Du4y5Vscltmu6CJXoWaLtJA8p/view?usp=sharing

Aula 11 – Espaços Métricos Separáveis e Completos

https://drive.google.com/file/d/13FdncBK8iwrnI1EmV4z-Xox_XAq9xcRo/view?usp=sharing

Aula 12 – Completamento de um Espaço Métrico

https://drive.google.com/file/d/1-96HeG7HOHQMgoA4-GXgND2aMGTfaPoI/view?usp=sharing

Aula 12 – Uma Correção para a Parte Final da Aula

https://drive.google.com/file/d/1A3N3WOONYDoSH-tDcnNHZX5aJRq-ySAI/view?usp=sharing

Aula 13 – Compacidade em Espaços Métricos

https://drive.google.com/file/d/1IcZtq1WIOSp2o7NvxbFM0LFFYNCSaI6d/view?usp=sharing

Aula 14 – Em Espaços Métricos, um Conjunto A é Compacto Se, e Somente Se, A é Sequencialmente Compacto (Revisão)

https://drive.google.com/file/d/1ChkufNWci9GxSXRzedawYmbxSu-R-1Gn/view?usp=sharing

Aula 15 – Em um Espaço Métrico Completo, um Conjunto A é Relativamente Compacto Se, e Somente Se, é Totalmente Limitado

https://drive.google.com/file/d/13hFBBLPNLmAaQqCHMckH0kYENdM3OD0R/view?usp=sharing

Aula 16 – O Teorema de Arzela-Ascoli

https://drive.google.com/file/d/1n6TjpiFDajCEMRg_MB8gXno0KocE5wnf/view?usp=sharing

Aula 17 – Espaços Topológicos, Primeiras Definições e Resultados

https://drive.google.com/file/d/144M6y19mx8RztsohbG9YlRWo2ke7kblF/view?usp=sharing

Aula 18 – Todo Espaço VETORIAL Topológico é Espaço de Haussdorff

https://drive.google.com/file/d/1RCwbCjfyz3-PIRDYZCY1jbFk2THWRSVP/view?usp=sharing

Aula 19 – Vizinhanças Balanceadas e Convexas em Espaços Vetoriais Topológicos

https://drive.google.com/file/d/1mdmH8oEYOmWsLUuN8uHpImPUHzHeH8fs/view?usp=sharing

Aula 20 – Nets e Convergência em Espaços Topológicos

https://drive.google.com/file/d/18yXyDR5bMmEBwrP5wcVU0bYFYpXoNuZp/view?usp=sharing

Aula 21 – Compacidade em Espaços Topológicos

https://drive.google.com/file/d/1xBna4iKVgCIsUMFfyr6MvqGD4ZforpEp/view?usp=sharing

Aula 22 - O Envelope Convexo de um Conjunto em Espaços Vetoriais Topológicos

https://drive.google.com/file/d/1RmcX1GFmXw8iT0U-zWteFENKuuFLRBNn/view?usp=sharing

Aula 23 – Linearidade e Continuidade em Espaços Vetoriais Topológicos

https://drive.google.com/file/d/1RvLSnb3eItjDm0PJmWhiU9UTxi48hTGA/view?usp=sharing

Aula 24 – Linearidade e Continuidade em Espaços Vetoriais Topológicos, Operadores Lineares Limitados

https://drive.google.com/file/d/1XpD5EAXjjHIFWyATeP3tOTfee-JR2oGg/view?usp=sharing

Aula 25 – O Teorema da Categoria de Baire

https://drive.google.com/file/d/1_CCup85O48KDVuP6W8Y6MjA0cLVCbA9e/view?usp=sharing

Aula 26 – O Princípio da Limitação Uniforme (P.L.U.)

https://drive.google.com/file/d/1kn88VKrhJ30tAdtfU-Oo1RJVZkjG_wQw/view?usp=sharing

Aula 27 – O Teorema da Aplicação Aberta

https://drive.google.com/file/d/1FohsXY8MrfjYrCnvlqCyWRZNdp7Vl5Qr/view?usp=sharing

Uma Correção – Aula 27

https://drive.google.com/file/d/1GAFV6Utl_8XozZ9qTcjTzcwE4MzXIAcw/view?usp=sharing

Aula 28 – O Teorema do Gráfico Fechado e Alguns Resultados sobre Subespaços de Dimensão Finita

https://drive.google.com/file/d/1LcTS_f9QYTyqr2hITHEnoJK780YxEY2b/view?usp=sharing

Aula 29 – Espaços de Hilbert, Primeiras Definições e Resultados

https://drive.google.com/file/d/1rSBCS-nhwzq6NS-WmJCMFYYCesjBD3bq/view?usp=sharing

Aula 30 – Espaços de Hilbert, Ortogonalidade e Projeção Ortogonal em um Subsespaço Vetorial Fechado

https://drive.google.com/file/d/1yACCnE6VLIO06s0cd-UUWHrV-mo3gdSN/view?usp=sharing

Aula 31 – O Lema de Riesz ( Representação do Espaço Dual a um Espaço de Hilbert)

https://drive.google.com/file/d/1thlz_vKcW8MAccwC8XQfPPoeEMwRLaiL/view?usp=sharing

Aula 32 – Ortonormalidade e Bases Ortonormais em Espaços de Hilbert

https://drive.google.com/file/d/1RTrD22baH-78y5UigKYD11ITZQVOnhSF/view?usp=sharing

Aula 33 – Projeção em um Conjunto Convexo em um Espaço de Hilbert

https://drive.google.com/file/d/1gJ7pL19WASb69UVgTpeyopLluHZ0LF39/view?usp=sharing

Aula 34 – Projeção em um Conjunto Convexo, Segunda Parte

https://drive.google.com/file/d/1MhzRSHLifPnW3TKbb4dzi-Imfpda320t/view?usp=sharing

Aula 35 - Os Teoremas de Stampacchia e Lax-Milgram

https://drive.g oogle.com/file/d/1D2IVjbjYZ6hLD7WmWKTiZLSa37hP_Ioy/view?usp=sharing

Aula 36 – O Teorema da Hahn-Banach

https://drive.google.com/file/d/1QMv-7uOwWKAOBdgwCbmAcDpZ8rmkFeJX/view?usp=sharing

Aula 37 – Aula de Exercícios – Preparação para a Primeira Avaliação

https://drive.google.com/file/d/1UNRLD0sYD83vbec2fSLgj9p9CTrDiEWn/view?usp=sharing

Aula 38 – Espaço Dual Topológico e Corolários do Teorema de Hahn-Banach

https://drive.google.com/file/d/1KP-O28MZg52NB1faElqDMhubl0zlxoFs/view?usp=sharing

Aula 39 – O Teorema de Hahn-Banach, Forma Geométrica

https://drive.google.com/file/d/136wEF4xfOXwYFtZjTGxSA523aisF9Nu4/view?usp=sharing

Correções para a Aula 39

https://drive.google.com/file/d/1TZO4jOdSc19zFcvIKc4zNMeWR_2UBjpw/view?usp=sharing

Aula 40 – Teorema de Hahn-Banach, Segunda Forma Geométrica

https://drive.google.com/file/d/1lkWwqDpXHGD1CE39UCmrXY88yIYkxvOH/view?usp=sharing

Aula 41 – Em um Espaço de Banach, um Conjunto Convexo A é Fortemente Fechado se, e somente se, é Fracamente fechado

https://drive.google.com/file/d/114ITf1FN8UIaVC0GCkHlVX5AqOSQvnIP/view?usp=sharing

Aula 42 – A Topologia Fraca-Estrela

Aula 43 – O Teorema de Banach- Alaoglu – Compacidade Fraca-Estrela no Espaço Dual

https://drive.google.com/file/d/1S6kI0A82_5gYvOWFz28wfg56xWC5UIVd/view?usp=sharing

Aula 44 – O Teorema de Kakutani – Um Espaço de Banach é Reflexivo se, e somente se, a Bola Fechada Unitária é Fracamente Compacta

https://drive.google.com/file/d/1Ra9nIpWtOohCqCMWLekdsQTxg8rd5O_s/view?usp=sharing

Aula 45 – Se um Espaço de Banach V é tal que V* é Separável, então V é Também Separável

https://drive.google.com/file/d/1pU5os8e0Yn1MVsau6QqeD2HasQB1mwkh/view?usp=sharing

Aula 46 – Um Espaço de Banach V é Separável se, e somente se, a Bola Unitária em V* é Metrizável com Relação à Topologia Fraca-Estrela

https://drive.google.com/file/d/11Jhbh-U4hEZ2b6cD_yKQu89P7Si5z3pM/view?usp=sharing

Aula 47 – Espaços de Banach Uniformemente Convexos são Reflexivos, O Teorema de Milman Pettis

https://drive.google.com/file/d/1HvYrLD4R4Z9p3Eh6IV9lAusL1xWQAGN6/view?usp=sharing

Aula 48 – Exercícios de Preparação para a Segunda Avaliação e Alguns Outros Tópicos

https://drive.google.com/file/d/1TBJjS8in2ZS1KWpuKby3WogX9Bxl29xz/view?usp=sharing

Medida e Integração – 2021 -1

Aula 1 – A Medida de Lebesgue na Reta – Medida Exterior

https://drive.google.com/file/d/192U1zBEzySFrAhPqWoco9M-YuIeSPrvz/view?usp=sharing

Aula 2 – Medida Exterior de um Intervalo Fechado

https://drive.google.com/file/d/1k9-B21yD5TPfMXOhyIj7N6F_aELYhP_C/view?usp=sharing

Aula 3 – Conjuntos Mensuráveis e a Medida de Lebesgue

https://drive.google.com/file/d/1vkOB_WrMTy-UzUu3h_BxyPpoTwWzKVFE/view?usp=sharing

Aula 4 – A União Enumerável de Conjuntos Mensuráveis é Mensurável

https://drive.google.com/file/d/1nL0B6K8HAVdfRRIVqohrVLflAgH1vVyA/view?usp=sharing

Aula 5 – Intervalos Reais são Mensuráveis

https://drive.google.com/file/d/1-f35F4xH3X5bcbaRVwRrznj120rW0BhV/view?usp=sharing

Aula 6 – Aditividade Enumerável da Medida de Lebesgue para Conjuntos Mensuráveis e Disjuntos

https://drive.google.com/file/d/1zCdH0AkyX2Atd9NsgaD36iL84eca2Vog/view?usp=sharing

Aula 7 – Aula Síncrona de Exercícios de 30/06/2021

https://drive.google.com/file/d/15HdkDqx30rjzFhtJANdb2AeY08mr7v7i/view?usp=sharing

Aula 7 – Correções e Complementos

https://drive.google.com/file/d/14IKQN8ZvTwGKJ7TzWStgpyeo9vMP1HRm/view?usp=sharing

Aula 8 – Funções Mensuráveis

https://drive.google.com/file/d/16FFeUUZmCAvoIbRNWAvAAKVy9M6s0PYo/view?usp=sharing

Aula 9- Propriedades das Funções Mensuráveis

https://drive.google.com/file/d/1wgRP322Q5iy5Evj0j_hRqeWRy4ThsQIr/view?usp=sharing

Correções e um exercício sobre a Aula 2

https://drive.google.com/file/d/1HG0dk5H_e6ZQPdJtZz2GG_9eLgNwRslf/view?usp=sharing

Aula 10 – Propriedades do Ínfimo, Supremo e Limites de Sequências de Funções Mensuráveis

https://drive.google.com/file/d/1hzNqI_7LMW6bJLnESqrNAG7rKPaSXn6Q/view?usp=sharing

Aula 11– Medidas, Definições e Propriedades

https://drive.google.com/file/d/12F_rxVEBcFYW74YZN61sr2ZMFLRTRLSz/view?usp=sharing

Aula 12– Funções Simples, Propriedades e Integrais

https://drive.google.com/file/d/1Ka_FGCzN7TucP2QhgfE7lwvo5JbKWZkv/view?usp=sharing

Aula 13– A Integral de Lebesgue, o Teorema da Convergência Monótona de Lebesgue

https://drive.google.com/file/d/1PZriu67bahRRpQxLzjxIAmYV8fQInZw1/view?usp=sharing

Aula 14 – Revisão – Teorema da Convergência Monótona de Lebesgue

https://drive.google.com/file/d/1g9SP4wwG045rv6A1dG8hsi2Tx_MjL7Gm/view?usp=sharing

Aula 15– Lema de Fatou

https://drive.google.com/file/d/1eonudNpY1tDA_YCaNFLvlRogL0PL_MKT/view?usp=sharing

Aula 16 – Integral e Conjuntos de Medida Zero, Teorema de Lebesgue da Convergência Monótona num Contexto mais Geral

https://drive.google.com/file/d/13iQbeMKufB-kuTgKa7Lw3BfZFbjYNwmC/view?usp=sharing

Aula 17 – Aula Síncrona de Exercícios de 14/07/2021

https://drive.google.com/file/d/1xZ6edbEmn66aBkBMzxJCi_5811gotLPJ/view?usp=sharing

Aula 18 – Aula Síncrona de Exercícios de 21/07/2021

https://drive.google.com/file/d/10fFbc6R7hghAM_bUhx4XAXsTqstYEdUK/view?usp=sharing

Aula 19– Funções Integráveis à Lebesgue, Propriedades da Integral de Lebesgue

https://drive.google.com/file/d/1FAiHxwyltpy6SsvFWna9vFAinsdpZYBW/view?usp=sharing

Aula 20 – Teorema de Lebesgue da Convergência Dominada

https://drive.google.com/file/d/1ZD_b3N383sgWo64VHnz73U6dHXHxcr3F/view?usp=sharing

Aula 21– Revisão - Integrais que Dependem de um Parâmetro

https://drive.google.com/file/d/1xjS7T1r4uTtCQBZWJWK_kxG9-hF0ujzL/view?usp=sharing

Aula 22 A – Integrais que Dependem de um Parâmetro, Parte Final

https://drive.google.com/file/d/1-1bBgAf3UGXD7a6asxgRp82N45zvT9lB/view?usp=sharing

Aula 23 – Espaços Lp – Desigualdade de Hölder

https://drive.google.com/file/d/1wQpMcNgGfwldVSpoBK5kmCM933-Wzj8L/view?usp=sharing

Aula 24 – Espaços Lp – Desigualdade de Minkowiski

https://drive.google.com/file/d/1_0mwUJZJjyVY3em-4s-QAr2LeWKfvE6s/view?usp=sharing

Aula 25 – Espaços Lp são Espaços de Banach , onde 1≤ p<∞

https://drive.google.com/file/d/1JdHu3PNV1eJRPe8NspPUSQQ49e3_NlZ8/view?usp=sharing

Aula 26 – Espaços L∞ são Espaços de Banach

https://drive.google.com/file/d/1FDzmXVt78DMyDDBOrzeT1H9apUKMN8SQ/view?usp=sharing

Aula 27 – Aula Síncrona de Exercícios de 28/07/2021

https://drive.google.com/file/d/1qEi-C5hSR6B-9C9D_AsSFTufAICR3_HO/view?usp=sharing

Aula 28 – Modos de Convergência – Primeira Parte

https://drive.google.com/file/d/1P0y-xtmTCcAOz4T0utqGSlCPeOmvoV_8/view?usp=sharing

Aula 29 – Convergência em Medida

https://drive.google.com/file/d/1DtclFy1YDEZ8v1uaPZG5yiMSHghDQlRj/view?usp=sharing

Aula 30 – Convergência em Medida Implica para uma Sub-Sequência Convergência Simples em Quase Todo o Domínio

https://drive.google.com/file/d/16mmVAfV1rHOvujid6E1JASTYKJD1oASa/view?usp=sharing

Aula 31- Exercício 3Q – Capítulo 3 – Bartle

https://drive.google.com/file/d/17h9lBMRIGZ0OeOqWOvE8va_CJK-WjM_8/view?usp=sharing

Aula 32 – Sequências de Cauchy em Medida Convergem em Medida

https://drive.google.com/file/d/1RaQk4RZEclSS3O6vu3lPfZw8CFdJErIP/view?usp=sharing

Aula 33 – Sequências Convergentes em Medida Dominadas em Módulo Pontualmente por uma Função no Lp Convergem em Lp

https://drive.google.com/file/d/1Uc-35v-yWi7wxXGIsjf-oVw7q-WZCgjo/view?usp=sharing

Aula 34 – Sequências Quase Uniformemente de Cauchy Convergem Quase Uniformemente e Convergem Pontualmente em Quase Todo o Domínio

https://drive.google.com/file/d/1drWUq3bNWmJtz5dNwuK3q9lnqbs_mnvr/view?usp=sharing

Aula 35 – Relações entre Convergência Quase Uniforme e Convergência em Medida (Revisada)

https://drive.google.com/file/d/15f1l44p87f7m8HqP8D3hlhNnBetfUQHx/view?usp=sharing

Aula 36 – Correções

https://drive.google.com/file/d/1yR0uFyxPvdDgayvqJYfukgW0Jf5dl3PD/view?usp=sharing

Aula 37 – Aula Síncrona de Exercícios de 04/08/2021

https://drive.google.com/file/d/1Jg7BGPCFnkqntYeGI8OjSz3JGztiRgPd/view?usp=sharing

Aula 38 – Aula Síncrona de Exercícios de 11/08/2021

https://drive.google.com/file/d/193mAN2LYofCsc9bBeD7_qphPeggBI_-m/view?usp=sharing

Aula 39 – Teorema de Egoroff

https://drive.google.com/file/d/1Z55LvI7Sph-oRDUTe-j43UJ0I3mBwO8Z/view?usp=sharing

Aula 40 – Correções (na conclusão da aula)

https://drive.google.com/file/d/1dceMhA6OZgVtKxNE0Yf3WyPMmqdRyVTu/view?usp=sharing

Aula 41 – Teorema de Vitali (Primeira Parte)

https://drive.google.com/file/d/1d3yBs4MpC9flm1ZRt0gZSpm8mDi1LKvk/view?usp=sharing

Aula 42 – Teorema de Vitali (Segunda Parte)

https://drive.google.com/file/d/1JNvXB5HRcNCQNvuk7lDeScR1Er6iarqA/view?usp=sharing

Aula 43 – Teorema de Vitali (Terceira Parte – Recíproca das Primeira e Segunda Partes)

https://drive.google.com/file/d/1vVhQQtaPM9IIvMt5NGeuDNgGjuLizArT/view?usp=sharing

Aula 44 – Exercício 5.O – Capítulo 5 – Bartle

https://drive.google.com/file/d/1An38Ala_XmkUU1twNFxDmHT47EToVTsO/view?usp=sharing

Aula 45 – Exercício 5.D – Capítulo 5 – Bartle

https://drive.google.com/file/d/1HwOSdc0xCXleQh0TIQB1CH1w5la0SLiG/view?usp=sharing

Correções – Aula 30 – Exercício 5.D – Bartle

https://drive.google.com/file/d/1sX7h_WJ34rFponnD2SdBO_ek3wpU3AXB/view?usp=sharing

Aula 46 – Exercício 5.C – Capítulo 5 – Bartle

https://drive.google.com/file/d/1izVpvR0iG6y3ybcwDOznMaBkQC6uKroK/view?usp=sharing

Aula 47 – Medidas com Sinal, Primeiras Definições e Resultados

https://drive.google.com/file/d/1tZUxvCDBPudiCnDpDdCfM6H0kK5hT7H7/view?usp=sharing

Aula 48 – Medidas com Sinal, Conjuntos Positivos e Negativos, Propriedades e Resultados

https://drive.google.com/file/d/1V_v_MR-Bm6LnMggyOIXU1iaxIGDKumUp/view?usp=sharing

Aula 49 – Decomposição de Hahn e Respectiva Decomposição de Jordan de uma Medida com Sinal

https://drive.google.com/file/d/1xS6yjPFCsR9CMLApo4jTgxQZPDuHAahD/view?usp=sharing

Aula 50 – Exercício Adicional de Preparação para a Primeira Avaliação

https://drive.google.com/file/d/1Okqlqw-ATceT2lwhwaHfBzT-X1VVEuZG/view?usp=sharing

Aula 51 – Exercício 4 S – Capítulo 4 – Bartle

https://drive.google.com/file/d/1cNFe-QcgALRBUGWy_vrz88tMpGiPHfhG/view?usp=sharing

Aula 52 – Exercício 4 T – Capítulo 4 – Bartle

https://drive.google.com/file/d/1a1opcnJY-9DwfrDyk0qdF7iJ8RoPxvda/view?usp=sharing

Aula 53 – Teorema de Radon-Nikodym

https://drive.google.com/file/d/1AV3G7jwigXIWsJjtnMLpppU0GNri_8XH/view?usp=sharing

Aula 54 – Teorema da Decomposição de Lebesgue

https://drive.google.com/file/d/1lJnjJjVuN4FtCMqVwOVUCiwcNtyYpK5c/view?usp=sharing

Aula 55 – Densidade das Funções Simples e Mensuráveis no Espaço Lp, onde 1≤ p< +∞.

https://drive.google.com/file/d/1yevqmoLbGx0eyiThMFckvRu-h5VyiwXA/view?usp=sharing

Aula 56 – Preparando a Rota para a Representação do Espaço Dual ao Lp, onde 1<p<+∞

https://drive.google.com/file/d/1BlLy3EDwDmDkC37JaCbWUTSk0ckF678x/view?usp=sharing

Aula 57 – (Revisada) Um Lema Auxiliar sobre uma Classe Especial de Séries de Funções no Lp

https://drive.google.com/file/d/1lM1CEYpmstXMBu-yedbcvIqoe1NTYwi7/view?usp=sharing

Aula 58 – Representação do Espaço Dual ao Lp – Teorema de Riesz – Parte 1 – Medida Finita

https://drive.google.com/file/d/1Ztrg3bDB8fsGkDSxXirs8HF2s7e-fqqo/view?usp=sharing

Aula 59 – Representação do Espaço Dual ao Lp – Teorema de Riesz – Parte 2 – Medida Sigma-Finita

https://drive.google.com/file/d/1eIMTzE0qF-TyW3slWOmnXhRg3Pwm_y4J/view?usp=sharing

Aula 60 – Representação do Espaço Dual ao Lp – Teorema de Riesz – Parte 3 – Caso Geral

https://drive.google.com/file/d/1e7Mvyre6GceGPd4hbwLmDYehBJ9sTCkZ/view?usp=sharing

Aula 61 – Definições de Álgebra de Conjuntos e Medida Sobre uma Álgebra – Exemplos na Reta Real

https://drive.google.com/file/d/1K7fWHv-fPfbDXOyA0dIZstoGxp0leIBj/view?usp=sharing

Aula 62 – Medida Exterior, Definição e Propriedades

https://drive.google.com/file/d/1S5VRUwTKwcuWViC-BeSx7ESM90FM05Ks/view?usp=sharing

Correções para a Conclusão da Aula 47

https://drive.google.com/file/d/1r_PBchbOxJavxhbJ-JkN4Fnrj4Zqbkcl/view?usp=sharing

Aula 63 – Medida Exterior - Teorema de Carathéodory

https://drive.google.com/file/d/1_4-dK8zlly2lX3IKh-ddPWQncAPYqYDJ/view?usp=sharing

Aula 64 – Teorema da Extensão de Hahn, Medida de Lebesgue e Medida de Lebesgue-Stieltjes

https://drive.google.com/file/d/1gIfFN9IQVw9ezMlW-H9izhc2ddFUhtWa/view?usp=sharing

Aula 65 – Representação do Espaço Dual ao Espaço das Funções Contínuas C([a,b]), Teorema de Riesz

https://drive.google.com/file/d/1zve01-RVyIpln090-Fxynnks-EN4kwlk/view?usp=sharing

Aula 66 – Aula Síncrona de 14/04/2021 – Exercícios – Capítulo 6 – Bartle

https://drive.google.com/file/d/1FqCi1nMF4aPZJKKXlHRXKMKbZs4eWcBw/view?usp=sharing

Aula 67 – Espaços de Medida Produto , Primeiras Definições

https://drive.google.com/file/d/1oWy_HsROuHhtZh6_x1yHSDsVV-IcPMFJ/view?usp=sharing

Aula 68 – Sobre a Mensurabilidade de Seções x e y de Conjuntos e Funções Mensuráveis no Espaço Produto X x Y.

https://drive.google.com/file/d/1DTrS2Shk_ZVn5S6UwwgNBX-40Cbgruvd/view?usp=sharing

Aula 69 – Classes Monótonas em Espaços Mensuráveis

https://drive.google.com/file/d/1kvFr5-4wOzotetu6hoy2T2N8Jb3R29UJ/view?usp=sharing

Aula 70 – Integração de Funções Simples no Espaço Produto

https://drive.google.com/file/d/1mtwGmVcle9PFX8p1U6NVRLlmdpiOLY0X/view?usp=sharing

Aula 71 – Teoremas de Tonelli e Fubini sobre Integração Iterada em Espaços Produto

https://drive.google.com/file/d/1dyIXbKi8nIR9Zm2O2g2aY0pOGIQXfs72/view?usp=sharing

Aula 72 – A Medida de Lebesgue no Rn – Medida Exterior e Propriedades

https://drive.google.com/file/d/1PKSeFZPV0XXiCbZhLlc4enwTpVXZTtyS/view?usp=sharing

Cálculo 4 – 2021 – 1

Aula 1 – Sequências Reais, Definições, Propriedades e Limites

https://drive.google.com/file/d/1aYi9EUBgmQZZOhMkpELppgSzaQcoL3Ue/view?usp=sharing

Aula 1 – Uma Correção e Complementos

https://drive.google.com/file/d/1nXb6_bWNvtz8AwK77nE2Mu8ySYIXfIbq/view?usp=sharing

Aula 2 – Teorema do Confronto, Exemplos e Exercícios

https://drive.google.com/file/d/1SMqssRs_cz1iFy4HVk8HABRQ8kbA-mF2/view?usp=sharing

Aula 2 – Correções e Complementos

https://drive.google.com/file/d/1BZlWL8UA7nZfMrDQFN9JjX6MQNC2trhG/view?usp=sharing

Aula 3 – Sequências Monótonas, Definições e Exercícios

https://drive.google.com/file/d/1d2flG1HKJOM1Iq0vFKzuCJ7xALfvT0hd/view?usp=sharing

Aula 3-1- Complementos sobre Supremo e Ínfimo, Sequências Reais Monótonas e Limitadas são Convergentes

https://drive.google.com/file/d/1Dj4_67spTX0HcYEcRouC3ToElMhKnzMj/view?usp=sharing

Aula 4 – Outros Resultados sobre Sequências Reais e Exercícios

https://drive.google.com/file/d/1yYtxhYYI2skZ6MRwfLGTFqgemykVh8Qg/view?usp=sharing

Aula 5 – Limites Superior e Inferior de uma Sequência Limitada

https://drive.google.com/file/d/1y8dzMxI1lzv0JYLpEiA43UezRyKkBI1x/view?usp=sharing

Aula 6 – Sequências Reais de Cauchy

https://drive.google.com/file/d/1axmNPJGYRo_GkXm2lkUW-Y4G1_LqNYdh/view?usp=sharing

Aula 7 – Uma Classe Especial de Sequências de Cauchy

https://drive.google.com/file/d/1iZ77Z272O1cjqHfS1lIpIqwUwr4hFTNZ/view?usp=sharing

Aula 8 – Séries Numéricas Reais – O Critério de Comparação para Convergência de Séries Reais

https://drive.google.com/file/d/1ixk-ilb90DpjuFD4QCKl-zp03k8gtbd7/view?usp=sharing

Aula 9 – Critério de Convergência para Séries Alternadas

https://drive.google.com/file/d/1zmSmxFIsGptDvnIA0TV8afvYUt9vdDGe/view?usp=sharing

Aula 10 – Teste da Razão para Convergência de Séries Reais

https://drive.google.com/file/d/1229Fz0QZgeebMI4b3y2bag1G2ptNBg1r/view?usp=sharing

Aula 11 – Teste da Raiz para Convergência de Séries Reais

https://drive.google.com/file/d/1PtaRiNXPlNE7HBx9TuvUB7d3C0xsSgch/view?usp=sharing

Aula 12 – Critério de Comparação das Razões para Convergência de Séries Reais

https://drive.google.com/file/d/1b6FM0PgJJPvNoip82mk4CkMdUpvJQr-L/view?usp=sharing

Aula 13 - Critério da Integral para a Convergência de Séries Reais

https://drive.google.com/file/d/12v_9Yb8T9rSPFKEWLANQELQeJ_G0LX2R/view?usp=sharing

Aula 14 – Critério de Comparação dos Limites para Convergência de Séries Reais

https://drive.google.com/file/d/1Ea1AOw6PB3UIhmY9_2MZiMn8Q_B476QE/view?usp=sharing

Aula 15 – Critério de Cauchy para a Convergência de Séries Reais

https://drive.google.com/file/d/1Ny3ify7HJAVeQ4oYfBU8hDkR_BMIjn3M/view?usp=sharing

Aula 16 – Lema de Abel e Critério de Dirichlet para a Convergência de Séries Reais

https://drive.google.com/file/d/1T5oLPUuLFK1cw5WLqccLcQlaGCKFvJzp/view?usp=sharing

Aula 16 A– Aula de Exercícios de Preparação para a Primeira Avaliação

https://drive.google.com/file/d/11ULQ0ajXFDFXr3X9wngaRiZVhrJBGwTw/view?usp=sharing

Aula 17 – Aula Síncrona de Exercícios de 04/08/2021 – Preparação para a Primeira Avaliação

https://drive.google.com/file/d/1O-hc5HO9MTT6XT7PLiJo8V0f_cU6W27b/view?usp=sharing

Aula 18– Aula Síncrona 2020-2, dia 26/03/2021 – Exercícios, Preparação para a Primeira Avaliação

https://drive.google.com/file/d/10x1hs46cq0biNh2S90KHsIe64UuAO6hD/view?usp=sharing

Aula 19 – Sequências de Funções (Primeira Parte)

https://drive.google.com/file/d/16so3zARFbYfJtFZROP6S5BWg7JW0rs3t/view?usp=sharing

Aula 20 – Integrais e Derivadas de Sequências de Funções Uniformemente Convergentes

https://drive.google.com/file/d/1DNRk1fgikRgIKvhKs8ByhTKjNMw_Gb1v/view?usp=sharing

Aula 21– Séries de Funções, Definições e Primeiros Resultados

https://drive.google.com/file/d/1l87B08DaRGDVpli9zraKeQz9vZ8HCPbx/view?usp=sharing

Aula 22 – Critério M de Weierstrass para Convergência Uniforme de Séries de Funções Reais

https://drive.google.com/file/d/1NbJTuVYacAKUfb2zMBgdLxt3oiEsZ_vy/view?usp=sharing

Aula 23 – Integrais e Derivadas de Séries de Funções Uniformemente Convergentes

https://drive.google.com/file/d/1dIrTsWMS0wHNANQpla1jWf090TaLlWJE/view?usp=sharing

Aula 24 – Séries de Potências, Primeiras Definições e Resultados

https://drive.google.com/file/d/1ddrx_ggnHVgmyMx6IMGWKOXfTHCa0cg-/view?usp=sharing

Aula 25 – Séries de Potências – Raio de Convergência

https://drive.google.com/file/d/1Mvxtd4bkdOl9Q2NMaOECLR38EtSnEn5O/view?usp=sharing

Aula 26 – Integrais e Derivadas de Séries de Potências

https://drive.google.com/file/d/19m0PrdmVth7NMsAK0-beAXjRHpqfaNM8/view?usp=sharing

Aula 27– Séries de Fourier – Primeiros Resultados

https://drive.google.com/file/d/1vMnh3SOz_NkmC_bXCz_6zs54KdfbDEr9/view?usp=sharing

Aula 28 – Convergência Uniforme da Série de Fourier de uma Função de Classe C² num intervalo fechado

https://drive.google.com/file/d/1mPX9wdARHwIDeK1jvDGJ83aXfoadGchR/view?usp=sharing

Aula 29 – A Série de Fourier de uma Função de Classe C² Converge Uniformemente para tal Função num Intervalo Fechado

https://drive.google.com/file/d/1yWj3jDkim5GyGuZfVb3hR2QexPoxWpSH/view?usp=sharing

Aula 30 – Um Exercício sobre o Cálculo da Série de Fourier de uma Função

https://drive.google.com/file/d/1oKVxRIyNIbJoOSG2RDVghe2nu6jtbzpr/view?usp=sharing

Aula 31 – Revisão – Teorema do Valor Médio – Preparando a Rota para a Prova da Fórmula de Taylor

https://drive.google.com/file/d/1e45ZD0zYDwGGsc9kEyQf5I_NGZwL4Cqn/view?usp=sharing

Aula 32 – Fórmula de Taylor – Preparando a Rota para as Séries de Taylor

https://drive.google.com/file/d/1R64AwA5RVEneU6Eji0dJwQKH4_d8jGEq/view?usp=sharing

Aula 33 – Método de Separação de Variáveis para EDPs – Equação do Calor

https://drive.google.com/file/d/1nT3IoTb48Lek5_bkcR1ftghdLHjV_0_E/view?usp=sharing

Aula 34 – Obtenção da Equação da Onda mediante o Cálculo Variacional

https://drive.google.com/file/d/1FZBVS9MZ8vGeHPY3MwsyPWvSFn1Ef0ih/view?usp=sharing

Correções para a Aula 34

https://drive.google.com/file/d/1hn7bKEF5jjBvsJU2r5QMrACs0Q4g78Nw/view?usp=sharing

Aula 35 – Equação da Onda – Solução pelo Método de Separação de Variáveis

https://drive.google.com/file/d/1ev858heD3AQUoMi7Y4heet37ybgDpD37/view?usp=sharing

Aula 36 – Aula Síncrona de 24/09/2021 – Preparação para a Segunda Avaliação

https://drive.google.com/file/d/18vD6wEa6A2Z81wmKoKeMDjgp0XXaqqSe/view?usp=sharing

Introdução à Análise – 2021 -1

Aula 1 – Conjuntos e Funções, Revisão

https://drive.google.com/file/d/1qdTAjyODlZU_LXXMu_Uv0MskAS5MTAaT/view?usp=sharing

Aula -1 - Correção sobre a Inclusão Própria

https://drive.google.com/file/d/1LEmEWpR5dvSOVHydiWMg481Asv-24Olb/view?usp=sharing

Aula 2 – Números Naturais – Primeiras Definições, Soma e Respectivas Propriedades

https://drive.google.com/file/d/1sffRXukcYPz_epkWHoxVfZTvzrLfT_Mf/view?usp=sharing

Aula 3 – Números Naturais – Multiplicação e Respectivas Propriedades

https://drive.google.com/file/d/1ZI8D2pAnCzVK9vIaglb0vJ_F_kJbEwJj/view?usp=sharing

Aula 4 – Princípio da Boa Ordenação

https://drive.google.com/file/d/16esbWWXqoj57se9t_CLEyVqCAKJdFIbF/view?usp=sharing

Aula 5 – O Conjunto dos Inteiros e o Conjunto Racional

https://drive.google.com/file/d/1QZGCSbdvTujk2ZLABfFZ-5NaAAzEZAoB/view?usp=sharing

Aula 6 – Conjuntos Limitados Superior e Inferiormente, Limitantes Superiores e Inferiores, Supremo e Ínfimo

https://drive.google.com/file/d/1zJjgSYb4DOTrxdBSrHVh6WzTioPgGL_8/view?usp=sharing

Aula 6 – Uma Correção sobre a Propriedade do Menor Limitante Superior e Supremos

https://drive.google.com/file/d/1c_VQyDtuCqMCCnUCmZnfhIHBufZh1ow6/view?usp=sharing

Aula 7 – Aula Síncrona de Exercícios de 20/05/2021

https://drive.google.com/file/d/1V8HSyAzJnGdLtUvCarKnaU5yJoAfiQLF/view?usp=sharing

Aula 8 – Corpos, Definições e Propriedades

https://drive.google.com/file/d/1qc3szWpurMNIoHPaW2BKW98eVDF-6qyc/view?usp=sharing

Aula 9 – Corpos Ordenados, Preparando a Rota para a Construção do Corpo Real

https://drive.google.com/file/d/1oDldZEprNzZtDJT0Js86eGFB6vNjOzgh/view?usp=sharing

Aula 10 – Existência do Corpo Real – Primeira Parte – Definição de Cortes

https://drive.google.com/file/d/14jf2A3HNSCx8DiPWh1hmcQvHRGqybnva/view?usp=sharing

Aula 11 – Existência do Corpo Real – Segunda Parte – Definição da Soma

https://drive.google.com/file/d/1qbYEdVKmYq05elVP4uSEMZoofDl75-b3/view?usp=sharing

Aula 12 – Aula Síncrona de Exercícios de 27/05/2021

https://drive.google.com/file/d/1JrQB4-7fHEzV_pHX6JH5u6uD_5a4yYF8/view?usp=sharing

Aula 13 – Existência do Corpo Real – Terceira Parte – Elemento Inverso em Relação à Soma

https://drive.google.com/file/d/1dKlWZRJRQv6hfLsKJjRgEu1ajvt-A17d/view?usp=sharing

Aula 14 – Existência do Corpo Real – Quarta Parte – Definição da Multiplicação

https://drive.google.com/file/d/1i9lcbFxIBcjsY8LkgNjitCLxAZqrq4zA/view?usp=sharing

Aula 15 – Existência do Corpo Real – Quinta Parte - Elemento Inverso em Relação à Multiplicação

https://drive.google.com/file/d/1nUmyMEydN_vlIm9Zf0fKr-TLm9EQVonU/view?usp=sharing

Aula 16 – Propriedade Arquimediana e Densidade do Conjunto Racional no Conjunto Real.

https://drive.google.com/file/d/1MQ-mLfrK4yOtzmtNnK5Ez9gbSnvD2N5G/view?usp=sharing

Aula 17 – A Raiz Quadrada de 2 é um Número Irracional – Prova Formal

https://drive.google.com/file/d/1nGyjfQNBRosM5XPJ1DogAGYbj3UAHW6s/view?usp=sharing

Aula 18 – Conjuntos Enumeráveis e Não-Enumeráveis

https://drive.google.com/file/d/1N0KX1fOm5DC0PxKyDwg0jtqKz0PvGk98/view?usp=sharing

Aula 18 – Uma Correção

https://drive.google.com/file/d/1uxTuj9wvACp0RYxVw_kanJAq7QZeNps5/view?usp=sharing

Aula 19 – Toda União Enumerável de Conjuntos Enumeráveis é Enumerável. O Conjunto Racional é Enumerável

https://drive.google.com/file/d/1oQ55qyWSI4rMlurv091_PluLuRwvgc-r/view?usp=sharing

Aula 20 – O Conjunto Real é Não-Enumerável

https://drive.google.com/file/d/1ob2fGGi1M7MuoaE3WPRxy0vtdAHUPuqs/view?usp=sharing

Aula 21 – Espaços Métricos – Primeiras Definições

https://drive.google.com/file/d/1Kayr7VV18pheX07LfvYIKLT9Y8mwdxK9/view?usp=sharing

Aula 22 – Definições Fundamentais em Espaços Métricos – Conjuntos Abertos, Fechados e Outras.

https://drive.google.com/file/d/17pjuu4hksC5fqqsZimpX3vQuypDzP9l7/view?usp=sharing

Aula 23 – Propriedades de Conjuntos Abertos e Fechados em um Espaço Métrico

https://drive.google.com/file/d/1v9B5g-VfZKKrrj57oA0-UA5qhKx58yP8/view?usp=sharing

Aula 24 – Propriedades de Conjuntos Abertos e Fechados em um Espaço Métrico – Segunda Parte

https://drive.google.com/file/d/1PSBAdCCBEWJ0YSYjBkCQ3ad8ODYqKgt8/view?usp=sharing

Aula 24 – Uma Correção

https://drive.google.com/file/d/1w0aqFCsyHK_Gs-BYZCV6PTvnGLUtR1gp/view?usp=sharing

Aula 25 - Aula Síncrona de Exercícios de 10/06/2021

https://drive.google.com/file/d/1dsHc0Bu9enMtHNCD_xcEomAaV8EL7Ac9/view?usp=sharing

Aula 26 – O Fecho de um Conjunto em um Espaço Métrico

https://drive.google.com/file/d/12YOyKgidrYIE6gCLNaHmNQrDTkanO5J5/view?usp=sharing

Aula 27 – Aula Síncrona de Exercícios de 17/06/2021

https://drive.google.com/file/d/1g5jGQEl9oBoiLXnyN9TgcfOBBE8KQhsJ/view?usp=sharing

Aula 28 – Conjuntos Compactos, Definições e Propriedades

https://drive.google.com/file/d/1dbhI0hZtB2ZxCHBNb8DHG2fjguYdbBj2/view?usp=sharing

Aula 29 – Aula Síncrona de Exercícios de 24/06/2021

https://drive.google.com/file/d/1UfSzVPK_fpnPXoQabjFmQNxsB-plEmQt/view?usp=sharing

Aula 30 – Conjuntos Compactos, Propriedades – Segunda Parte

https://drive.google.com/file/d/1F7dVAOyCF4zvTozkt8-Ckq4E2muaqH1f/view?usp=sharing

Aula 31 – Todo Intervalo Real Fechado e Limitado é Compacto

https://drive.google.com/file/d/1uexHRUQqzzMiYVIcWiwkFMnHR7qwYQBw/view?usp=sharing

Aula 32 – O Teorema de Heine-Borel

https://drive.google.com/file/d/1OaO9S27Du4y5Vscltmu6CJXoWaLtJA8p/view?usp=sharing

Aula 33 – Aula Síncrona de Exercícios de 01/07/2021

https://drive.google.com/file/d/17UjHVcqAKhKa24B9j_4FeCf_L7vjiX3L/view?usp=sharing

Aula 34 – Sequências Reais, Primeiras Definições e Limites

https://drive.google.com/file/d/1W8vngdGaQTUPyx-lRAr66zbtkJk6fE7d/view?usp=sharing

Aula 35 – Unicidade do Limite, Sequências Reais Convergentes são Limitadas

https://drive.google.com/file/d/1f-GccYCx1I2hCXix-2YPFwi9o7jddCXs/view?usp=sharing

Aula 36 – Sequências Reais Monótonas e Limitadas são Convergentes

https://drive.google.com/file/d/1XiiogI8NC9QvY6mO_6eNRZJ85GT9-q7o/view?usp=sharing

Aula 37 – Propriedades dos Limites de Sequências Reais

https://drive.google.com/file/d/1C27A_HVIBFd5-lOQnrD3slROsdJhpf7s/view?usp=sharing

Aula 38 – Teorema do Confronto para Sequências Reais

https://drive.google.com/file/d/1qC8EwzXcnXuD9P0qU66yBbS7V4LXSXlh/view?usp=sharing

Aula 39 – Condição Necessária e Suficiente para um Número Real Ser Limite de uma Subsequência

https://drive.google.com/file/d/14PyFJSfQDq-R4ONNvnpTV_RJr1tE_2Hh/view?usp=sharing

Aula 40 – Limites Superior e Inferior de uma Sequência Real Limitada

https://drive.google.com/file/d/1rZa-YSk_3j6ZgY3YINoA3x8Rqe9UiXan/view?usp=sharing

Aula 41 – Aula Síncrona de Exercícios de 15/07/2021

https://drive.google.com/file/d/1fPZkLy8juX2n6N7_7x5I0h-5SavtTCKH/view?usp=sharing

Aula 42 – Uma Sequência Real Limitada é Convergente se, e somente se, Seus Limites Inferior e Superior são Iguais

https://drive.google.com/file/d/16KA_Ym59W5laEbnUaN3ZTQ5lC-qQJNps/view?usp=sharing

Aula 43 – Sequencias Reais de Cauchy – Sequências Reais são de Cauchy se, e somente se, são Convergentes

https://drive.google.com/file/d/1amG5w0QK3cfnSjtuHemqsJs9EncAFoWC/view?usp=sharing

Aula 44 – Uma Classe Especial de Sequências de Cauchy

https://drive.google.com/file/d/1KNO-M39m73QCegIXrWlkD8AAUn0SC0v_/view?usp=sharing

Aula 45 – Limites Infinitos para Sequências e Primeiras Definições e Resultados sobre Séries Numéricas Reais

https://drive.google.com/file/d/1Ouyo2p0JJ7JoR9zBjNl-TTKZzdoZ7iDb/view?usp=sharing

Aula 46 – O Critério de Comparação para Séries Reais

https://drive.google.com/file/d/1rSGQvdv__rPqvHMXGp7A0vldhTsF3EzB/view?usp=sharing

Aula 47 – Critério de Cauchy para Séries Reais e o Teste da Raiz

https://drive.google.com/file/d/1e4CahL25rNOvezpXUD6ndkaq08_dYT8L/view?usp=sharing

Aula 48 – Teste da Razão – Primeira Parte

https://drive.google.com/file/d/1Wqoi5LJyzYIN0wKAaicG_UKQ66jAahNJ/view?usp=sharing

Aula 49 – Um Resultado sobre a Comparação entre os Testes da Raiz e da Razão

https://drive.google.com/file/d/1x1Piq5o8ugYmUfMM2KBP5achiH_5LZLA/view?usp=sharing

Aula 50 – Aula Síncrona de Exercícios de 29/07/2021

https://drive.google.com/file/d/1hGiPmR-FAA8Z9GN5ABY3vHKQyHjnVrxN/view?usp=sharing

Aula 51 – Limites de Funções Reais. Primeiras Definições, Resultados e Exemplos

https://drive.google.com/file/d/1uAsDKuI0SqpY_hBNnBwifkpe8aEDd43N/view?usp=sharing

Aula 52 – Teorema do Confronto para Limites de Funções Reais

https://drive.google.com/file/d/1G57PIMYc41FpY_i09ctXvqwVUsi0CUri/view?usp=sharing

Aula 53 – Uma Condição Necessária e Suficiente para um Número Real Ser Limite de uma Função Mediante Sequências

https://drive.google.com/file/d/1fad2jcp-JAJghfVfwJZQ4GabVYMqOdi6/view?usp=sharing

Aula 54 – Uma Condição Suficiente para que o Limite de uma Função Exista Mediante Sequências

https://drive.google.com/file/d/1XlgV5ELR-d5ph0V4L0pmTgPAhMKFtwii/view?usp=sharing

Aula 55 – Propriedades dos Limites de Funções Reais

https://drive.google.com/file/d/1lsABs18ddocJu5CFFPqahpm1IXucKomN/view?usp=sharing

Aula 56 – Limites para Funções Compostas

https://drive.google.com/file/d/1UMbfm59_2XUtbbqSuR_T57FHZVk9BOeb/view?usp=sharing

Aula 57 – Funções Monótonas e Limites Laterais

https://drive.google.com/file/d/1Z-xTgk5wUwg4PmeVxpPF9FVlCL3_9qmH/view?usp=sharing

Aula 58 – Aula Síncrona de Exercícios de 05/08/2021

https://drive.google.com/file/d/1TpCqUsaRIe0w0h7GV-HXdV1bd6xXOULw/view?usp=sharing

Aula 59 – Valores de Aderência para Funções Reais

https://drive.google.com/file/d/1EM2oR43m36aiUNisjkSozd2vjugfI1MC/view?usp=sharing

Aula 60 – O Conjunto dos Valores de Aderência é Compacto para uma Função Limitada numa Vizinhança do Ponto em Questão

https://drive.google.com/file/d/1md510RFCAB45Mf5OXkAMK-eAtV8JjloJ/view?usp=sharing

Aula 61 – Limites Superiores e Inferiores para Funções Reais

https://drive.google.com/file/d/1n1awt1gLu3xJEjOcbmyEN30D1NV-0Np4/view?usp=sharing

Aula 62 – Aula Síncrona de 12/08/2021 – Limites Infinitos

https://drive.google.com/file/d/1SCpUqrQ2qx6laLXZU7FYESCTfn1mF3jh/view?usp=sharing

Aula 63 – Limites Reais no Infinito

https://drive.google.com/file/d/1pWMT6H3cAmNIYg4NT26mkP51yJyfjy8B/view?usp=sharing

Aula 64 – Limites Infinitos no Infinito

https://drive.google.com/file/d/1qegopjPQ86I1VWpzeT85dL77QhhqNNwQ/view?usp=sharing

Aula 65 – Funções Contínuas, Primeiras Definições e Resultados

https://drive.google.com/file/d/1Cm3hvyp2-VDLLlPpKfRAGUqWiemg4vRx/view?usp=sharing

Aula 66 – Uma Função Real em um Aberto é Contínua Se, e Somente Se, sua Pré-Imagem é Aberta para todo Aberto Real

https://drive.google.com/file/d/1eqQckKv2vQ1IARJdjUeBVhokCgGKIKTU/view?usp=sharing

Aula 67 – Propriedades das Funções Contínuas, Continuidade da Função Composta

https://drive.google.com/file/d/1CI-kwuPnDkurJ5qfnNJzx3O3dbVsN-Yl/view?usp=sharing

Aulo 68 – Descontinuidades de Primeira e Segunda Espécies para Funções Reais

https://drive.google.com/file/d/1qWKxhZWRVQ6-fNZU6B5cEhWiUFtRsHNO/view?usp=sharing

Aula 69 – Funções Contínuas em Conjuntos Compactos

https://drive.google.com/file/d/1A8iScNj13satu_Rge6ttWLP2OVmpxJdW/view?usp=sharing

Aula 70 – Aula Síncrona de 19/08/2021 – Teorema do Valor Intermediário

https://drive.google.com/file/d/17jkRkATqeIYBpAQzTOl_ovsP7OSiJRV_/view?usp=sharing

Cálculo 4 – 2020-2

Aula 1 – Sequências Reais, Definições, Propriedades e Limites

https://drive.google.com/file/d/1aYi9EUBgmQZZOhMkpELppgSzaQcoL3Ue/view?usp=sharing

Aula 2 – Teorema do Confronto, Exemplos e Exercícios

https://drive.google.com/file/d/1SMqssRs_cz1iFy4HVk8HABRQ8kbA-mF2/view?usp=sharing

Aula 3 – Sequências Monótonas, Definições e Exercícios

https://drive.google.com/file/d/1d2flG1HKJOM1Iq0vFKzuCJ7xALfvT0hd/view?usp=sharing

Aula 4 – Outros Resultados sobre Sequências Reais e Exercícios

https://drive.google.com/file/d/1yYtxhYYI2skZ6MRwfLGTFqgemykVh8Qg/view?usp=sharing

Aula 5 – Limites Superior e Inferior de uma Sequência Limitada

https://drive.google.com/file/d/1y8dzMxI1lzv0JYLpEiA43UezRyKkBI1x/view?usp=sharing

Aula 6 – Sequências Reais de Cauchy

https://drive.google.com/file/d/1axmNPJGYRo_GkXm2lkUW-Y4G1_LqNYdh/view?usp=sharing

Aula 7 – Uma Classe Especial de Sequências de Cauchy

https://drive.google.com/file/d/1iZ77Z272O1cjqHfS1lIpIqwUwr4hFTNZ/view?usp=sharing

Aula 8 – Séries Numéricas Reais – O Critério de Comparação para Convergência de Séries Reais

https://drive.google.com/file/d/1ixk-ilb90DpjuFD4QCKl-zp03k8gtbd7/view?usp=sharing

Aula 9 – Critério de Convergência para Séries Alternadas

https://drive.google.com/file/d/1zmSmxFIsGptDvnIA0TV8afvYUt9vdDGe/view?usp=sharing

Aula 10 – Teste da Razão para Convergência de Séries Reais

https://drive.google.com/file/d/1229Fz0QZgeebMI4b3y2bag1G2ptNBg1r/view?usp=sharing

Aula 11 – Teste da Raiz para Convergência de Séries Reais

https://drive.google.com/file/d/1PtaRiNXPlNE7HBx9TuvUB7d3C0xsSgch/view?usp=sharing

Aula 12 – Critério de Comparação das Razões para Convergência de Séries Reais

https://drive.google.com/file/d/1b6FM0PgJJPvNoip82mk4CkMdUpvJQr-L/view?usp=sharing

Aula 13 - Critério da Integral para a Convergência de Séries Reais

https://drive.google.com/file/d/12v_9Yb8T9rSPFKEWLANQELQeJ_G0LX2R/view?usp=sharing

Aula 14 – Critério de Comparação dos Limites para Convergência de Séries Reais

https://drive.google.com/file/d/1Ea1AOw6PB3UIhmY9_2MZiMn8Q_B476QE/view?usp=sharing

Aula 15 – Critério de Cauchy para a Convergência de Séries Reais

https://drive.google.com/file/d/1Ny3ify7HJAVeQ4oYfBU8hDkR_BMIjn3M/view?usp=sharing

Aula 16 – Lema de Abel e Critério de Dirichlet para a Convergência de Séries Reais

https://drive.google.com/file/d/1T5oLPUuLFK1cw5WLqccLcQlaGCKFvJzp/view?usp=sharing

Aula 17 – Sequências de Funções (Primeira Parte)

https://drive.google.com/file/d/16so3zARFbYfJtFZROP6S5BWg7JW0rs3t/view?usp=sharing

Aula 18 – Integrais e Derivadas de Sequências de Funções Uniformemente Convergentes

https://drive.google.com/file/d/1DNRk1fgikRgIKvhKs8ByhTKjNMw_Gb1v/view?usp=sharing

Aula 19 – Séries de Funções, Definições e Primeiros Resultados

https://drive.google.com/file/d/1l87B08DaRGDVpli9zraKeQz9vZ8HCPbx/view?usp=sharing

Aula 20 – Aula Síncrona do dia 26/03/2021 – Exercícios

https://drive.google.com/file/d/10x1hs46cq0biNh2S90KHsIe64UuAO6hD/view?usp=sharing

Aula 21 – Critério M de Weierstrass para Convergência Uniforme de Séries de Funções Reais

https://drive.google.com/file/d/1NbJTuVYacAKUfb2zMBgdLxt3oiEsZ_vy/view?usp=sharing

Aula 22 – Integrais e Derivadas de Séries de Funções Uniformemente Convergentes

https://drive.google.com/file/d/1dIrTsWMS0wHNANQpla1jWf090TaLlWJE/view?usp=sharing

Aula 23 – Séries de Potências, Primeiras Definições e Resultados

https://drive.google.com/file/d/1ddrx_ggnHVgmyMx6IMGWKOXfTHCa0cg-/view?usp=sharing

Aula 24 – Séries de Potências – Raio de Convergência

https://drive.google.com/file/d/1Mvxtd4bkdOl9Q2NMaOECLR38EtSnEn5O/view?usp=sharing

Aula 25 – Integrais e Derivadas de Séries de Potências

https://drive.google.com/file/d/19m0PrdmVth7NMsAK0-beAXjRHpqfaNM8/view?usp=sharing

Aula 26 – Séries de Fourier – Primeiros Resultados

https://drive.google.com/file/d/1vMnh3SOz_NkmC_bXCz_6zs54KdfbDEr9/view?usp=sharing

Aula 27 – Convergência Uniforme da Série de Fourier de uma Função de Classe C² num intervalo fechado

https://drive.google.com/file/d/1mPX9wdARHwIDeK1jvDGJ83aXfoadGchR/view?usp=sharing

Aula 28 – A Série de Fourier de uma Função de Classe C² Converge Uniformemente para tal Função num Intervalo Fechado

https://drive.google.com/file/d/1yWj3jDkim5GyGuZfVb3hR2QexPoxWpSH/view?usp=sharing

Aula 29 – Um Exercício sobre o Cálculo da Série de Fourier de uma Função

https://drive.google.com/file/d/1oKVxRIyNIbJoOSG2RDVghe2nu6jtbzpr/view?usp=sharing

Aula 31 – Revisão – Teorema do Valor Médio – Preparando a Rota para a Prova da Fórmula de Taylor

https://drive.google.com/file/d/1e45ZD0zYDwGGsc9kEyQf5I_NGZwL4Cqn/view?usp=sharing

Aula 32 – Fórmula de Taylor – Preparando a Rota para as Séries de Taylor

https://drive.google.com/file/d/1R64AwA5RVEneU6Eji0dJwQKH4_d8jGEq/view?usp=sharing

Aula 33 – Método de Separação de Variáveis para EDPs – Equação do Calor

https://drive.google.com/file/d/1nT3IoTb48Lek5_bkcR1ftghdLHjV_0_E/view?usp=sharing

Aula 34 – Obtenção da Equação da Onda mediante o Cálculo Variacional

https://drive.google.com/file/d/1FZBVS9MZ8vGeHPY3MwsyPWvSFn1Ef0ih/view?usp=sharing

Correções para a Aula 34

https://drive.google.com/file/d/1hn7bKEF5jjBvsJU2r5QMrACs0Q4g78Nw/view?usp=sharing

Aula 35 – Equação da Onda – Solução pelo Método de Separação de Variáveis

https://drive.google.com/file/d/1ev858heD3AQUoMi7Y4heet37ybgDpD37/view?usp=sharing

Medida e Integração 2020-2

Aula 1 – Funções Mensuráveis

https://drive.google.com/file/d/16FFeUUZmCAvoIbRNWAvAAKVy9M6s0PYo/view?usp=sharing

Aula 2- Propriedades das Funções Mensuráveis

https://drive.google.com/file/d/1wgRP322Q5iy5Evj0j_hRqeWRy4ThsQIr/view?usp=sharing

Correções e um exercício sobre a Aula 2

https://drive.google.com/file/d/1HG0dk5H_e6ZQPdJtZz2GG_9eLgNwRslf/view?usp=sharing

Aula 3 – Propriedades do Ínfimo, Supremo e Limites de Sequências de Funções Mensuráveis

https://drive.google.com/file/d/1hzNqI_7LMW6bJLnESqrNAG7rKPaSXn6Q/view?usp=sharing

Aula 4 – Medidas, Definições e Propriedades

https://drive.google.com/file/d/12F_rxVEBcFYW74YZN61sr2ZMFLRTRLSz/view?usp=sharing

Aula 5 – Funções Simples, Propriedades e Integrais

https://drive.google.com/file/d/1Ka_FGCzN7TucP2QhgfE7lwvo5JbKWZkv/view?usp=sharing

Aula 6 – A Integral de Lebesgue, o Teorema da Convergência Monótona de Lebesgue

https://drive.google.com/file/d/1PZriu67bahRRpQxLzjxIAmYV8fQInZw1/view?usp=sharing

Aula 7 – Revisão – Teorema da Convergência Monótona de Lebesgue

https://drive.google.com/file/d/1g9SP4wwG045rv6A1dG8hsi2Tx_MjL7Gm/view?usp=sharing

Aula 8 – Lema de Fatou

https://drive.google.com/file/d/1eonudNpY1tDA_YCaNFLvlRogL0PL_MKT/view?usp=sharing

Aula 9 – Integral e Conjuntos de Medida Zero, Teorema de Lebesgue da Convergência Monótona num Contexto mais Geral

https://drive.google.com/file/d/13iQbeMKufB-kuTgKa7Lw3BfZFbjYNwmC/view?usp=sharing

Aula 10 – Funções Integráveis à Lebesgue, Propriedades da Integral de Lebesgue

https://drive.google.com/file/d/1FAiHxwyltpy6SsvFWna9vFAinsdpZYBW/view?usp=sharing

Aula 11 – Teorema de Lebesgue da Convergência Dominada

https://drive.google.com/file/d/1ZD_b3N383sgWo64VHnz73U6dHXHxcr3F/view?usp=sharing

Aula 12 – Revisão - Integrais que Dependem de um Parâmetro

https://drive.google.com/file/d/1xjS7T1r4uTtCQBZWJWK_kxG9-hF0ujzL/view?usp=sharing

Aula 12 A – Integrais que Dependem de um Parâmetro, Parte Final

https://drive.google.com/file/d/1-1bBgAf3UGXD7a6asxgRp82N45zvT9lB/view?usp=sharing

Aula 13 – Espaços Lp – Desigualdade de Hölder

https://drive.google.com/file/d/1wQpMcNgGfwldVSpoBK5kmCM933-Wzj8L/view?usp=sharing

Aula 14 – Espaços Lp – Desigualdade de Minkowiski

https://drive.google.com/file/d/1_0mwUJZJjyVY3em-4s-QAr2LeWKfvE6s/view?usp=sharing

Aula 15 – Espaços Lp são Espaços de Banach , onde 1≤ p<∞

https://drive.google.com/file/d/1JdHu3PNV1eJRPe8NspPUSQQ49e3_NlZ8/view?usp=sharing

Aula 16 – Espaços L∞ são Espaços de Banach

https://drive.google.com/file/d/1FDzmXVt78DMyDDBOrzeT1H9apUKMN8SQ/view?usp=sharing

Aula 17 – Modos de Convergência – Primeira Parte

https://drive.google.com/file/d/1P0y-xtmTCcAOz4T0utqGSlCPeOmvoV_8/view?usp=sharing

Aula 18 – Convergência em Medida

https://drive.google.com/file/d/1DtclFy1YDEZ8v1uaPZG5yiMSHghDQlRj/view?usp=sharing

Aula 19 – Convergência em Medida Implica para uma Sub-Sequência Convergência Simples em Quase Todo o Domínio

https://drive.google.com/file/d/16mmVAfV1rHOvujid6E1JASTYKJD1oASa/view?usp=sharing

Aula 20 - Exercício 3Q – Capítulo 3 – Bartle

https://drive.google.com/file/d/17h9lBMRIGZ0OeOqWOvE8va_CJK-WjM_8/view?usp=sharing

Aula 21 – Sequências de Cauchy em Medida Convergem em Medida

https://drive.google.com/file/d/1RaQk4RZEclSS3O6vu3lPfZw8CFdJErIP/view?usp=sharing

Aula 22 – Sequências Convergentes em Medida Dominadas em Módulo Pontualmente por uma Função no Lp Convergem em Lp

https://drive.google.com/file/d/1Uc-35v-yWi7wxXGIsjf-oVw7q-WZCgjo/view?usp=sharing

Aula 23 – Sequências Quase Uniformemente de Cauchy Convergem Quase Uniformemente e Convergem Pontualmente em Quase Todo o Domínio

https://drive.google.com/file/d/1drWUq3bNWmJtz5dNwuK3q9lnqbs_mnvr/view?usp=sharing

Aula 24 – Relações entre Convergência Quase Uniforme e Convergência em Medida (Revisada)

https://drive.google.com/file/d/15f1l44p87f7m8HqP8D3hlhNnBetfUQHx/view?usp=sharing

Aula 24 – Correções

https://drive.google.com/file/d/1yR0uFyxPvdDgayvqJYfukgW0Jf5dl3PD/view?usp=sharing

Aula 25 – Teorema de Egoroff

https://drive.google.com/file/d/1Z55LvI7Sph-oRDUTe-j43UJ0I3mBwO8Z/view?usp=sharing

Aula 25 – Correções (na conclusão da aula)

https://drive.google.com/file/d/1dceMhA6OZgVtKxNE0Yf3WyPMmqdRyVTu/view?usp=sharing

Aula 26 – Teorema de Vitali (Primeira Parte)

https://drive.google.com/file/d/1d3yBs4MpC9flm1ZRt0gZSpm8mDi1LKvk/view?usp=sharing

Aula 27 – Teorema de Vitali (Segunda Parte)

https://drive.google.com/file/d/1JNvXB5HRcNCQNvuk7lDeScR1Er6iarqA/view?usp=sharing

Aula 28 – Teorema de Vitali (Terceira Parte – Recíproca das Primeira e Segunda Partes)

https://drive.google.com/file/d/1vVhQQtaPM9IIvMt5NGeuDNgGjuLizArT/view?usp=sharing

Aula 29 – Exercício 5.O – Capítulo 5 – Bartle

https://drive.google.com/file/d/1An38Ala_XmkUU1twNFxDmHT47EToVTsO/view?usp=sharing

Aula 30 – Exercício 5.D – Capítulo 5 – Bartle

https://drive.google.com/file/d/1HwOSdc0xCXleQh0TIQB1CH1w5la0SLiG/view?usp=sharing

Correções – Aula 30 – Exercício 5.D – Bartle

https://drive.google.com/file/d/1sX7h_WJ34rFponnD2SdBO_ek3wpU3AXB/view?usp=sharing

Aula 31 – Exercício 5.C – Capítulo 5 – Bartle

https://drive.google.com/file/d/1izVpvR0iG6y3ybcwDOznMaBkQC6uKroK/view?usp=sharing

Aula 32 – Medidas com Sinal, Primeiras Definições e Resultados

https://drive.google.com/file/d/1tZUxvCDBPudiCnDpDdCfM6H0kK5hT7H7/view?usp=sharing

Aula 33 – Medidas com Sinal, Conjuntos Positivos e Negativos, Propriedades e Resultados

https://drive.google.com/file/d/1V_v_MR-Bm6LnMggyOIXU1iaxIGDKumUp/view?usp=sharing

Aula 34 – Decomposição de Hahn e Respectiva Decomposição de Jordan de uma Medida com Sinal

https://drive.google.com/file/d/1xS6yjPFCsR9CMLApo4jTgxQZPDuHAahD/view?usp=sharing

Aula 35 – Exercício Adicional de Preparação para a Primeira Avaliação

https://drive.google.com/file/d/1Okqlqw-ATceT2lwhwaHfBzT-X1VVEuZG/view?usp=sharing

Aula 36 – Exercício 4 S – Capítulo 4 – Bartle

https://drive.google.com/file/d/1cNFe-QcgALRBUGWy_vrz88tMpGiPHfhG/view?usp=sharing

Aula 37 – Exercício 4 T – Capítulo 4 – Bartle

https://drive.google.com/file/d/1a1opcnJY-9DwfrDyk0qdF7iJ8RoPxvda/view?usp=sharing

Aula 38 – Teorema de Radon-Nikodym

https://drive.google.com/file/d/1AV3G7jwigXIWsJjtnMLpppU0GNri_8XH/view?usp=sharing

Aula 39 – Teorema da Decomposição de Lebesgue

https://drive.google.com/file/d/1lJnjJjVuN4FtCMqVwOVUCiwcNtyYpK5c/view?usp=sharing

Aula 40 – Densidade das Funções Simples e Mensuráveis no Espaço Lp, onde 1≤ p< +∞.

https://drive.google.com/file/d/1yevqmoLbGx0eyiThMFckvRu-h5VyiwXA/view?usp=sharing

Aula 41 – Preparando a Rota para a Representação do Espaço Dual ao Lp, onde 1<p<+∞

https://drive.google.com/file/d/1BlLy3EDwDmDkC37JaCbWUTSk0ckF678x/view?usp=sharing

Aula 42 – (Revisada) Um Lema Auxiliar sobre uma Classe Especial de Séries de Funções no Lp

https://drive.google.com/file/d/1lM1CEYpmstXMBu-yedbcvIqoe1NTYwi7/view?usp=sharing

Aula 43 – Representação do Espaço Dual ao Lp – Teorema de Riesz – Parte 1 – Medida Finita

https://drive.google.com/file/d/1Ztrg3bDB8fsGkDSxXirs8HF2s7e-fqqo/view?usp=sharing

Aula 44 – Representação do Espaço Dual ao Lp – Teorema de Riesz – Parte 2 – Medida Sigma-Finita

https://drive.google.com/file/d/1eIMTzE0qF-TyW3slWOmnXhRg3Pwm_y4J/view?usp=sharing

Aula 45 – Representação do Espaço Dual ao Lp – Teorema de Riesz – Parte 3 – Caso Geral

https://drive.google.com/file/d/1e7Mvyre6GceGPd4hbwLmDYehBJ9sTCkZ/view?usp=sharing

Aula 46 – Definições de Álgebra de Conjuntos e Medida Sobre uma Álgebra – Exemplos na Reta Real

https://drive.google.com/file/d/1K7fWHv-fPfbDXOyA0dIZstoGxp0leIBj/view?usp=sharing

Aula 47 – Medida Exterior, Definição e Propriedades

https://drive.google.com/file/d/1S5VRUwTKwcuWViC-BeSx7ESM90FM05Ks/view?usp=sharing

Correções para a Conclusão da Aula 47

https://drive.google.com/file/d/1r_PBchbOxJavxhbJ-JkN4Fnrj4Zqbkcl/view?usp=sharing

Aula 48 – Medida Exterior - Teorema de Carathéodory

https://drive.google.com/file/d/1_4-dK8zlly2lX3IKh-ddPWQncAPYqYDJ/view?usp=sharing

Aula 49 – Teorema da Extensão de Hahn, Medida de Lebesgue e Medida de Lebesgue-Stieltjes

https://drive.google.com/file/d/1gIfFN9IQVw9ezMlW-H9izhc2ddFUhtWa/view?usp=sharing

Aula 50 – Representação do Espaço Dual ao Espaço das Funções Contínuas C([a,b]), Teorema de Riesz

https://drive.google.com/file/d/1zve01-RVyIpln090-Fxynnks-EN4kwlk/view?usp=sharing

Aula 51 – Aula Síncrona de 14/04/2021 – Exercícios – Capítulo 6 – Bartle

https://drive.google.com/file/d/1FqCi1nMF4aPZJKKXlHRXKMKbZs4eWcBw/view?usp=sharing

Aula 52 – Espaços de Medida Produto , Primeiras Definições

https://drive.google.com/file/d/1oWy_HsROuHhtZh6_x1yHSDsVV-IcPMFJ/view?usp=sharing

Aula 53 – Sobre a Mensurabilidade de Seções x e y de Conjuntos e Funções Mensuráveis no Espaço Produto X x Y.

https://drive.google.com/file/d/1DTrS2Shk_ZVn5S6UwwgNBX-40Cbgruvd/view?usp=sharing

Aula 54 – Classes Monótonas em Espaços Mensuráveis

https://drive.google.com/file/d/1kvFr5-4wOzotetu6hoy2T2N8Jb3R29UJ/view?usp=sharing

Aula 55 – Integração de Funções Simples no Espaço Produto

https://drive.google.com/file/d/1mtwGmVcle9PFX8p1U6NVRLlmdpiOLY0X/view?usp=sharing

Aula 56 – Teoremas de Tonelli e Fubini sobre Integração Iterada em Espaços Produto

https://drive.google.com/file/d/1dyIXbKi8nIR9Zm2O2g2aY0pOGIQXfs72/view?usp=sharing

Aula 57 – A Medida de Lebesgue no Rn – Medida Exterior e Propriedades

https://drive.google.com/file/d/1PKSeFZPV0XXiCbZhLlc4enwTpVXZTtyS/view?usp=sharing

Aula 58 – Aula Síncrona de 28/04/2021 – Exercícios - Bartle (Devido a uma falha na conexão comece o vídeo aos 14 minutos)

https://drive.google.com/file/d/1Pe4dzZ3q-9s0AMJyZkvZzXERP-IHSiYD/view?usp=sharing

Remark: My work on duality theory is a kind of extension and generalization of some results of J.J. Telega and W.R. Bielski combined with a specific D.C. optimization approach.

It is a great honour for me to have my work based on those of such exceptional researchers.

I am very grateful for their wonderful contributions to applied mathematics and to science as a whole.

My new book entitled

Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering, CRC Taylor and Francis, November 3rd, 2020

Last article:

arXiv:2012.03053 [pdf, ps, other]

math.CA math.AP math.FA

A note on the Korn inequality in a n-dimensional context

Authors: Fabio Silva Botelho

Abstract: In this short communication, we present a new proof for the Korn inequality in a n-dimensional context. The results are based on standard tools of real and functional analysis. For the final result the standard Poincaré inequality plays a fundamental role.

Submitted 5 December, 2020; originally announced December 2020.

Comments: 6 pages

MSC Class: 35Q74

Cálculo Avançado 2020-1

Resultados da terceira avaliação

Unidade I – Análise Diferencial no Rn

Aula 10 C - Teorema da Função Implícita no Rn - Caso Escalar (Revised)

https://drive.google.com/file/d/1gYzpEqsPElKZsNF3QzAotRtA0_W3rSzu/view?usp=sharing

Correções- Aula 10 C - O Teorema da Função Implícita no Rn, Caso Escalar

https://drive.google.com/file/d/1HsJA9h0DJ2BnZd0_fxira1UVG_VFhPmN/view?usp=sharing

Link Aula 11 - B - Funções Vetoriais no Rn

https://drive.google.com/file/d/1OvJlyA4u6dbxDfjuqELWQQobZq4VB6nV/view?usp=sharing

Link Aula 12 - B - Limites, Continuidade e Diferenciabilidade de Funções
Vetoriais no Rn

https://drive.google.com/file/d/1clMrPeeiyYVXZ8QIxwutoU3tAjpEIJAZ/view?usp=sharing

Link Aula 13 - B - Desigualdade do Valor Médio para Funções Vetoriais

https://drive.google.com/file/d/1MFD6KCJqdDaaww9yvu71XgeVznAon75l/view?usp=sharing

Aula 14 - O Teorema do Ponto Fixo de Banach no Rn

https://drive.google.com/file/d/1syoBLL877l5M5AMRWwazfExI_UROocrf/view?usp=sharing

Aula 15 - Teorema da Função implícita, Caso Vetorial no Rn (Links para as vídeo-aulas)

https://drive.google.com/file/d/1qIIsFGxSoXQnxc6xe_HtBCuWSuw-9880/view?usp=sharing

Aula 16 - O Método dos Multiplicadores de Lagrange (Parte 1)

https://drive.google.com/file/d/1DfALNu7KhKNle61sZct3ERem0bsaoiPi/view?usp=sharing

Aula 17 - Multiplicadores de Lagrange, Caso no R4 com Duas Restrições (Parte 2)

https://drive.google.com/file/d/1nuQz_aXfZRCJW_IFnieQmcp3Df1n3sjU/view?usp=sharing

Aula 18 - Multiplicadores de Lagrange, Caso Geral no R(n+m)

https://drive.google.com/file/d/1Fj72BhuPvzitB9u85M4-M1OU38q17uzS/view?usp=sharing

Aula 19 - Teorema da Função Inversa no Rn

https://drive.google.com/file/d/1DR5mt1xi7V6i7K3wQH6XGqKwBJllzbyd/view?usp=sharing

Unidade II - Integração no Rn

Aula 20 - Integração no Rn - Primeiras Definições e Resultados

https://drive.google.com/file/d/1k6iXydl_QfdTmoMEIILgfDWvjpz1BGhB/view?usp=sharing

Aula 21 - Critério de Integrabilidade no Rn e Propriedades da Integral de
Riemann

https://drive.google.com/file/d/1UYWxF1nh0-TISiL397DBMjyVlP-KVbZ4/view?usp=sharing

Aula 22 - Propriedades da Integral de Riemann no Rn

https://drive.google.com/file/d/1BhpHcnirYN-Ep47jnDt8Zsh7I9dZwAgY/view?usp=sharing

Aula 23 - A Integrabilidade à Riemann no Rn da Função Composta

https://drive.google.com/file/d/13cnXlxAAUNkRL3-h4IKmUXWMtEDOKlEH/view?usp=sharing

Aula 24 – Integrabilidade de Funções Contínuas em Blocos Compactos no Rn

https://drive.google.com/file/d/1mp-Vn1FAJiciGoSncgc9pAjKXO6B3Y1S/view?usp=sharing

Aula 25 - Propriedades da Medida Exterior no Rn

https://drive.google.com/file/d/135R3uynfSoeYbmhA1-pR7Gs9ZBYdasel/view?usp=sharing

Aula 26 - O Critério de Integrabilidade de Riemann Lebesgue - Primeira Parte

https://drive.google.com/file/d/1YrKr9NF-bIHdo-BrlCEwJEh85unM3pfN/view?usp=sharing

Aula 27 - Critério de Integrabilidade de Riemann-Lebesgue - Segunda Parte

https://drive.google.com/file/d/1JaeEeOhTJoQNYA1-T0PybUkPNhTwXAto/view?usp=sharing

Aula 28 - Somas de Riemann -Primeira Parte

https://drive.google.com/file/d/1HZe4pb8DnMbSE-zoYk-GLS0C1d4k4NFc/view?usp=sharing

Aula 29 - Somas de Riemann - Segunda Parte

https://drive.google.com/file/d/1SjXCe2ZkcAMdwZo-PdYY8ngSByTDK2PM/view?usp=sharing

Aula 30 Integração Dupla - Formalismo e Exemplos

https://drive.google.com/file/d/1fZFJPdWZrOBr9iAofrlMt5-YEd42EWhV/view?usp=sharing

Aula 31 - Mudança de Variáveis na Integral Dupla

https://drive.google.com/file/d/14BZN8z2dWlV0HspdXve_aIEhz5zHzSZe/view?usp=sharing

Aula 32 - Integração Tripla no R3 - Primeira Parte

https://drive.google.com/file/d/1rkCMyBFRRezI20ORUE0Po89kmyKi2Fr9/view?usp=sharing

Aula 33 - Mudança de Variáveis na Integral Tripla no R3

https://drive.google.com/file/d/1hZchIZM5RgpyBeeK00EsIJmh_iM_Kwtc/view?usp=sharing

Aula 34 - Integral Tripla no R3 em Coordenadas Esféricas

https://drive.google.com/file/d/1t8uvuYlDS0bHoanamRQlhQcIflKGOCtG/view?usp=sharing

Aula 35 - Integrais de Linha

https://drive.google.com/file/d/1UqRohi0adP6UQ9KmY0AdLV9lK041BPMR/view?usp=sharing

Aula 36 - Teorema de Green no Plano

https://drive.google.com/file/d/1u19Nz5VYZH4_DX-J3W78wxYyanM9cKrB/view?usp=sharing

Aula 36 - Teorema de Green no Plano - Correções

https://drive.google.com/file/d/1LMT4U-oYyqE-SsaTjHQKsM37zd0ElPfv/view?usp=sharing

Aula 37 - Formas Diferenciais no R3

https://drive.google.com/file/d/1JahXKqm7xxKZ-GJmb9NnDaA4gA8rFUSZ/view?usp=sharing

Aula 38 - Condições necessárias e suficientes para um campo vetorial ser um gradiente no Rn

https://drive.google.com/file/d/1HCQ3QTKyVvtXQV1d1s66WumC5bCZMzZl/view?usp=sharing

Aula 39 - Superfícies no R3 - Cálculo da Área

https://drive.google.com/file/d/1aCnTLtq5O14BKRnzTn5j9aWUq6ped8gn/view?usp=sharing

Aula 40 - Equações Paramétricas de uma Superfície no R3 - Cálculo da Área

https://drive.google.com/file/d/1S4gIHQbyzqg_ezh6P6NwBLj2in0nuu3_/view?usp=sharing

Aula 40 - Equações Paramétricas de Superfícies no R3 - Correções

https://drive.google.com/file/d/1wU8seT9W8Gz2wKhBvCqSWvzWv9fD1CBh/view?usp=sharing

Aula 41 - Teorema da Divergência no R3

https://drive.google.com/file/d/1RXXonWIrGO6i1jpnTBzib8GSnqKdNJNl/view?usp=sharing

Aula 42 - Teorema de Stokes no R3

https://drive.google.com/file/d/1fM3VhLfsBDysD2YYPiFDN2erETPW3P-T/view?usp=sharing

Aula 43 - A Forma Local das Submersões

https://drive.google.com/file/d/131hTJAWScJN5gE522yWBFztPBlX5simo/view?usp=sharing

Aula 44 - A Forma Local das Imersões

https://drive.google.com/file/d/1BVPtEHUJzVTGmpcIOjj49FIDLALFy7V4/view?usp=sharing

Correções para a Aula 44

https://drive.google.com/file/d/1JS9opmywD3P1mg1NWvyMswk_tM94-drM/view?usp=sharing

Aula 45 - Superfícies no Rn - Primeiros resultados

https://drive.google.com/file/d/1BYWHx9uVS9qU200VfIZ6LuSD1c-U7x5n/view?usp=sharing

Aula 46 - Superfícies Orientáveis no Rn

https://drive.google.com/file/d/1Gq20HJ6Huz0OPYLIExJhaUYgqEAQO547/view?usp=sharing

Aula 47 – Espaço Tangente para Superfícies no Rn

https://drive.google.com/file/d/1qIbmnVxHCFWMDB0HMq2-EnSatxLJRuK9/view?usp=sharing

Aula 48 - Superfícies no Rn com Bordo

https://drive.google.com/file/d/1LeJA8koOpZNjlwuhUm94AXiD1akAvfn4/view?usp=sharing

Aula 49 - Parametrizações para Superfícies no Rn com Bordo

https://drive.google.com/file/d/1H5uM7YEb6Wad6VHCcDZPmp6ol8mUDOFT/view?usp=sharing

Aula 50 – Espaço Tangente para uma Superfície no Rn com Bordo e a Parametrização Padrão

https://drive.google.com/file/d/1058oC_CtUxrrF4WEWciXxEDC2BXhIJds/view?usp=sharing

Aula 51 – Campo Normal Unitário Exterior ao Bordo de uma Superfície no Rn, Orientação do Bordo

https://drive.google.com/file/d/1HbMjNu7jRyZNoHWS5-Mit_dBUYCYlXtu/view?usp=sharing

Aula 52 - Espaço e Base Duais para um Espaço Vetorial m-Dimensional

https://drive.google.com/file/d/1f_6owfDEv1u9zEhrabhVx55fMvEzNIbR/view?usp=sharing

Aula 53 – Base para o Espaço dual ao Espaço Tangente a uma Superfície no Rn

https://drive.google.com/file/d/1VDgON8NjIalSF5huwhCQGXhuXT7NYrd6/view?usp=sharing

Aula 54 - Campos Vetoriais Tangenciais a uma Superfície no Rn e Colchete de Lie

https://drive.google.com/file/d/1Zn1DiIfOlxpwHyVLo9Gn-lsXDkofBtTQ/view?usp=sharing

Aula 55 - Campos Vetoriais Tangenciais a uma Superfície no Rn e Álgebras de Lie

https://drive.google.com/file/d/1ZWRqfzPmSCIsTTKCBGFITnhxp2XUmV7a/view?usp=sharing

Aula 56 – Sobre a Existência da Curva Integral para um Campo Vetorial em um Superfície no Rn

https://drive.google.com/file/d/1oPTLQm9Fg5xOGWNWP4k0Jdf_jf4p1O60/view?usp=sharing

Aula 57 - Desigualdade de Gronwall e Continuidade do Fluxo de um Campo Vetorial

https://drive.google.com/file/d/1UGLZ2Mtc88qMFA7HjcMXMjDo7tvJ15i5/view?usp=sharing

Aula 58 - Partições da Unidade

https://drive.google.com/file/d/1Sjr3JFov_wIlpkXH7Kc2q1XqJ2PqKdf7/view?usp=sharing

Aula 59 - Produto Exterior e Formas Diferenciais

https://drive.google.com/file/d/1Qg3ncooTYAWRn4-X7stcAQRFgKYmS5JL/view?usp=sharing

Aula 60 - O Teorema de Stokes para um superfície no Rn com Bordo

https://drive.google.com/file/d/1ICGblDgXEczfrSs_HoNW0-A8QGkQqcUx/view?usp=sharing

Aula 61 - Um Exemplo Simples Sobre a Integração de Formas Diferenciais em Superfícies no Rn.

https://drive.google.com/file/d/1OulfQtTWIZbvP5tQtM-DVG5Ozr7g78IL/view?usp=sharing

Aula 62- Volume de uma Superfície no Rn. Forma Diferencial Volume

https://drive.google.com/file/d/12UrLXA67pAeMdtwC7Z1L4b51QmkCPdS8/view?usp=sharing

Aula 63 - Mudança de Variáveis na Integral no Rn

https://drive.google.com/file/d/1H4V-DRK46IiGmo-VOMdoVzkN3ReEFgES/view?usp=sharing

Aula 64 - Re-obtendo os Teoremas Clássicos de Stokes e da Divergência no R3 Mediante o Teorema Geral de Stokes no Rn

https://drive.google.com/file/d/1-DCqv7_h6s_Jq2FDyyln_2GoRTtmuYB-/view?usp=sharing

Aula 65 - Derivada de Lie de Um Campo Vetorial e Exercícios para a Terceira
Avaliação
https://drive.google.com/file/d/1Mz6L9fn29WoC5XvPQllFHYoq9DSD001-/view?usp=sharing

Pre-prints 2020

1- arXiv:2008.00232 [pdf, ps, other]

math.AP math.OC

On duality principles for non-convex optimization with applications to superconductivity and some existence results for a model in non-linear elasticity

Authors: Fabio Silva Botelho

Abstract: This article develops duality principles applicable to the Ginzburg-Landau system in superconductivity. The main results are obtained through standard tools of convex analysis, functional analysis, calculus of variations and duality theory. In the second section, we present the general result for the case including a magnetic field and the respective magnetic potential in a local extremal context.… ▽ More

Submitted 3 September, 2020; v1 submitted 1 August, 2020; originally announced August 2020.

Comments: 27 pages

MSC Class: 49N15

2- arXiv:2007.02772 [pdf, ps, other]

math.FA On Lagrange multiplier theorems for non-smooth optimization for a large class of variational models in Banach spaces

Authors: Fabio Silva Botelho

Abstract: In this short communication, we present optimality conditions for a class of non-smooth variational problems. The main results are based on standard tools of functional analysis and calculus of variations. Firstly we address a model with equality constraints and, in a second step, a more general model with equality and inequality constraints, always in a general Banach spaces context.

Submitted 17 July, 2020; v1 submitted 29 June, 2020; originally announced July 2020.

Comments: 9 pages, some corrections implemented

MSC Class: 49K27

3-arXiv:2006.13042 [pdf, ps, other]

A generalization of the Ekeland variational principle

Authors: Fabio Silva Botelho

Abstract: In this short communication, we present a generalization of the Ekeland variational principle. The main result is established through standard tools of functional analysis and calculus of variations. The novelty here is a result involving the second Gâteaux variation of the functional in question.

Submitted 22 June, 2020; originally announced June 2020.

Comments: 5 pages

MSC Class: 49K27

4-arXiv:2006.11240 [pdf, other]

math.OC 4- Optimal control for the nonlinear Fisher-Kolmogorov system with applications to aquatic plant management

Authors: Alexandre Molter, Fabio Silva Botelho

Abstract: Spatiotemporal dynamics of populations may be described by the reaction-diffusion Fisher-Kolmogorov model. In this work we have proposed a new formulation for a control problem of aquatic plants in a temporal dynamics. The solution of this problem is extended to a spatiotemporal Fisher-Kolmogorov system with multiple species of plants interacting in the same place. The control consists on human in… ▽ More

Submitted 19 June, 2020; originally announced June 2020.

Comments: 15 pages

MSC Class: 49M15; 92B99

5-arXiv:2003.00325 [pdf, ps, other]

math.GM A general variational formulation for relativistic mechanics based on fundamentals of differential geometry

Authors: Fabio Botelho

Abstract: The first part of this article develops a variational formulation for relativistic mechanics. The results are established through standard tools of variational analysis and differential geometry. The novelty here is that the main motion manifold has a n+1 dimensional range. It is worth emphasizing in a first approximation we have neglected the self-interaction energy part. In its second part, th… ▽ More

Submitted 26 March, 2020; v1 submitted 29 February, 2020; originally announced March 2020.

Comments: 25 pages, minor corrections implemented, new sections added

MSC Class: 53Z05

About the first work to successfully apply convex analysis to the complementary energy concept for a model in non-linear mechanics, published in 1985.

W.R. Bielski and J.J. Telega – A contibution to contact problems for a class of solids and structures. Arch Mech 37: 303–320, 1985

My research on duality theory is, in some sense, an extension and generalization of the ideas of JJ Telega and WR Bielski contained in this article,

combined with some results on D.C. optimization.

Even if you desagree with the opinion and references of other people, from a spiritual perspective, it is important not to spread lies nor slandering such a person or his family.

Krishnamurti used to refer to such a serenity facing disagreement as the essence of a moral and spiritual maturity.

Análise Funcional – Curso de Verão – 2020 – Resultados da P-2

Cálculo Avançado – 2019 -2 - Notas de Aula

Cálculo Variacional

Análise Funcional – Curso de Verão – 2019 – Notas de Aula, Listas de Exercícios e resultados da segunda avaliação de 1/3/2019

Cálculo 1 – notas de aula

Livros:

Functional Analysis and Applied Optimization in Banach Spaces, Springer 2014.

Book review

Real Analysis and Applications, Springer 2018.

A Classical Description of Variational Quantum Mechanics and Related Models, Nova Science Publishing, 2017.

This work is dedicated in memoriam to David Bohm and Jiddu Krishnamurti

Pre-prints – Physics

0.1- arXiv:2003.00325 [pdf, ps, other]

math.GM

A general variational formulation for relativistic mechanics based on fundamentals of differential geometry

Authors: Fabio Botelho

Abstract: This article develops a variational formulation for relativistic mechanics. The results are established through standard tools of variational analysis and differential geometry. The novelty here is that the main motion manifold has a n+1 dimensional range. Finally, it is worth emphasizing in a first approximation we have neglected the self-interaction energy part.

Submitted 29 February, 2020; originally announced March 2020.

Comments: 6 pages

MSC Class: 53Z05

1. arXiv:1812.04097 [pdf, ps, other]

A variational formulation for relativistic mechanics based on Riemannian geometry and its application to the quantum mechanics context

Fabio Botelho

Comments: 15 pages, new results based on the Weinberg approach for relativistic mechanics

Subjects: Analysis of PDEs (math.AP)

2. A Variational Formulation for the Relativistic Klein-Gordon Equation

3- arXiv:1908.04611 [pdf, ps, other]

quant-ph

A variational formulation for relativistic mechanics, a new interpretation for the Bohr atomic model and some concerning applications

Authors: Fabio Botelho

Abstract: This article develops a variational formulation for the relativistic Klein-Gordon equation. The main results are obtained through an extension of the classical mechanics approach to a more general context, which in some sense, includes the quantum mechanics one. For the second part of the text, the definition of normal field and its relation with the wave function concept play a fundamental role in the main results establishment. Among the applications, we include a model with the presence of electromagnetic fields and also the modeling of a chemical reaction. Finally, in the last section, we present some results about the Spin operator in a relativistic context. △ Less

Submitted 16 October, 2019; v1 submitted 13 August, 2019; originally announced August 2019.

Comments: 38 pages, some minor mistakes and typos corrected

MSC Class: 81Q05

Pre-prints-Applied Mathematics

1-00 arXiv:1910.01118 [pdf, ps, other]

math.OC

On duality principles for one and three-dimensional non-linear models in elasticity

Authors: Fabio Botelho

Abstract: In this article, we develop duality principles applicable to primal variational formulations found in the non-linear elasticity theory. As a first application, we establish the concerning results in details for one and three-dimensional models. We emphasize such duality principles are applicable to a larger class of variational optimization problems, such as non-linear models of plates and shells and other models in elasticity. Finally, we formally prove there is no duality gap between the primal and dual formulations, in a local extremal context. △ Less

Submitted 3 October, 2019; originally announced October 2019.

Comments: 13 pages. arXiv admin note: text overlap with arXiv:1712.02701

MSC Class: 49N15

1- arXiv:1909.01790 [pdf, ps, other]

math.OC

A primal dual variational formulation suitable for a large class of non-convex problems in optimization

Authors: Fabio Botelho

Abstract: In this article we develop a new primal dual variational formulation suitable for a large class of non-convex problems in the calculus of variations. The results are obtained through basic tools of convex analysis, duality theory, the Legendre transform concept and the respective relations between the primal and dual variables. The novelty here is that the dual formulation is established also for the primal variables, however with a large domain region of concavity about a critical point. Finally, we formally prove there is no duality gap between the primal and dual formulations in a local extremal context. △ Less

Submitted 3 September, 2019; originally announced September 2019.

Comments: 8 pages

MSC Class: 49N15

Pre-print- On the solution of the Navier-Stokes system- announced, see it in the next line

arXiv:1908.09751 [pdf, ps, other]

math.GM

On the generalized method lines applied to the time-independent incompressible Navier-Stokes system

Authors: Fabio Botelho

Abstract: In the first part of this article, we obtain a linear system whose the solution solves the time-independent incompressible Navier-Stokes system for the special case in which the external forces vector is a gradient. In a second step we develop approximate solutions, also for the time independent incompressible Navier-Stokes system, through the generalized method of lines. We recall that for such a method, the domain of the partial differential equation in question is discretized in lines and the concerning solution is written on these lines as functions of the boundary conditions and boundary shape. Finally, we emphasize these last main results are established through applications of the Banach fixed point theorem. △ Less

Submitted 11 August, 2019; originally announced August 2019.

Comments: 21 pages

MSC Class: 65N40; 65N06

0.1- arXiv:1907.02503 [pdf, ps, other]

math.OC

A numerical method for an inverse optimization problem through the generalized method of lines

Authors: Fabio Silva Botelho

Abstract: This article develops a solution for an inverse problem through the generalized method of lines. We consider a Laplace equation on a domain with internal and external boundaries with standard Dirichlet boundary conditions. Also, we specify a third non-homogeneous Newmann type boundary condition for the external boundary, and consider the problem of finding the optimal shape for the internal boundary such that all the prescribed boundary conditions are satisfied. The novelty here presented is the application of the generalized method of lines as a tool to compute a solution for such an inverse optimization problem. △ Less

Submitted 4 July, 2019; originally announced July 2019.

Comments: 7 pages

MSC Class: 49N45; 65N40

1.0-0arXiv:1907.00200 [pdf, ps, other]

math.OC

A duality principle and related numerical method for a class of shape optimization problems in elasticity

Authors: Fabio Botelho, Alexandre Molter

Abstract: In this article we develop a duality principle and concerning numerical method for a shape optimization problem in elasticity. We consider the problem of finding the optimal shape for an elastic solid which minimizes its structural inner energy resulting from the action of external loads to be specified. The main results are obtained through standard tools of convex analysis and duality theory. We emphasize our algorithm do not include a filter to process the results, so that the result obtained is indeed a critical point for the original optimization problem. Finally, in the last section, we present some numerical examples concerning applications of the theoretical results established. △ Less

Submitted 29 June, 2019; originally announced July 2019.

Comments: 7 pages

MSC Class: 49N15; 49Q10

1.0-arXiv:1906.07758 [pdf, ps, other]

math.OC

Duality suitable for a class of non-convex optimization problems

Authors: Fabio Botelho

Abstract: In this article we develop a duality principle suitable for a large class of problems in optimization. The main result is obtained through basic tools of convex analysis and duality theory. We establish a correct relation between the critical points of the primal and dual formulations and formally prove there is no duality gap between such formulations, in a local extremal context.

Submitted 18 June, 2019; originally announced June 2019.

Comments: 7 pages

MSC Class: 49N15

1. arXiv:1904.12379 [pdf, ps, other]

math.NA

On the generalized method of lines and its proximal explicit and hyper-finite difference approaches

Authors: Fabio Botelho

Abstract: This article firstly develops a proximal explicit approach for the generalized method of lines. In such a method, the domain of the PDE in question is discretized in lines and the equation solution is written on these lines as functions of the boundary conditions and domain shape. The main objective of introducing a proximal formulation is to minimize the solution error as a typical parameter… ▽ More

Submitted 7 May, 2019; v1 submitted 28 April, 2019; originally announced April 2019.

Comments: 18 pages, some typos corrected, a new result added

MSC Class: 65N40; 65N06

1.1. arXiv:1904.02286 [pdf, ps, other]

math.OC

A primal dual variational formulation and a multi-duality principle for a non-linear model of plates

Authors: Fabio Botelho

Submitted 18 April, 2019; v1 submitted 3 April, 2019; originally announced April 2019.

Comments: 20 pages, some typos corrected, a new result added

MSC Class: 49N15; 74P99

2. arXiv:1903.06014 [pdf, ps, other]

A duality principle for non-convex optimization in Rn

Fabio Botelho

Comments: 13 pages, some typos and errors corrected, in this version all proof details have been provided

Subjects: Optimization and Control (math.OC)

3. arXiv:1902.08811 [pdf, ps, other]

A note on optimization in Rn

Fabio Botelho

Comments: 10 pages

Subjects: Optimization and Control (math.OC)

4. arXiv:1902.04448 [pdf, ps, other]

Existence of solution for an optimal control problem associated to the Ginzburg-Landau system in superconductivity

Fabio Botelho, Eduardo Pandini Barros

Comments: 8 pages

Subjects: Optimization and Control (math.OC)

5. arXiv:1812.04097 [pdf, ps, other]

A variational formulation for relativistic mechanics based on Riemannian geometry and its application to the quantum mechanics context

Fabio Botelho

Comments: 15 pages, new results based on the Weinberg approach for relativistic mechanics

Subjects: Analysis of PDEs (math.AP)

6. arXiv:1809.09575 [pdf, ps, other]

On central fields in the calculus of variations

Fabio Botelho

Comments: 12 pages, typos corrected

Subjects: Optimization and Control (math.OC)

7. arXiv:1804.06283 [pdf, ps, other]

On General Duality Principles for Non-Convex Variational Optimization with Applications to the Ginzburg-Landau System in Superconductivity

Fabio Botelho

Comments: 32 pages, typos corrected, other results added

Subjects: Optimization and Control (math.OC)

8. arXiv:1712.04809 [pdf, ps, other]

A duality principle for a semi-linear model in micro-magnetism

Fabio Botelho

Comments: 7 pages

Subjects: Optimization and Control (math.OC)

9. arXiv:1712.03552 [pdf, ps, other]

On the numerical solution of non-linear first order ordinary differential equation systems

Fabio Botelho

Comments: 9 pages

Subjects: Numerical Analysis (math.NA)

10. arXiv:1712.02701 [pdf, ps, other]

A duality principle for non-linear elasticity

Fabio Botelho

Comments: 10 pages, more typos and errors corrected, some parts of the text have been re-written

Subjects: Optimization and Control (math.OC)

11. arXiv:1712.01595 [pdf, ps, other]

Global existence results and duality for non-linear models of plates and shells

Fabio Botelho

Comments: 28 pages, some parts of the text have been re-written, variational nature of the dual formulations retaken

Subjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)

This work is dedicated in memoriam to Professor J.J. Telega

12. arXiv:1712.01031 [pdf, ps, other]

On duality principles for non-convex variational models applied to a Ginzburg-Landau type equation

Fabio Botelho

Comments: 14 pages, more typos corrected

Subjects: Optimization and Control (math.OC)

Artigos completos publicados em periódicos

1. Fabio Botelho, A note on Riemannian geometry and the relativistic quantum mechanics context. CIÊNCIA E NATURA, v. 40, p. 58, 2018.

2. Fabio Botelho.,A Variational Formulation for the Relativistic Klein-Gordon Equation. CIÊNCIA E NATURA, v. 40, p. 57, 2018.

3. Fabio Botelho, On the Generalized Method of Lines Applied to Ginzburg-Landau Type Equations. International Journal of Applied and Computational Mathematics, v. 1, p. pp 1-15, 2016.

4. Lucas dos Santos Fernandez, Alexandre Molter, Fabio Silva Botelho, Simultaneous topology optimization and proportional actuators localization. SeMA Journal, v. 23, p. --, 2016.

5. Fabio Botelho, On the Lagrange multiplier theorem in Banach spaces. Matemática Aplicada e Computacional (Cessou em 1997. Cont. ISSN 1807-0302 Computational & Applied Mathematics), v. 32, p. 135-144, 2013.

6. Fabio Botelho. Existence of solution for the Ginzburg Landau system, a related optimal control problem and its computation by the generalized method of lines. Applied Mathematics and Computation, v. 218, p. 11976-11989, 2012.

7. Fabio Botelho, On duality principles for scalar and vectorial multi-well variational problems. Nonlinear Analysis, v. 75, p. 1904-1918, 2012.

8. Fabio Botelho, Dual Variational Formulations for a Non-linear Model of Plates. JOURNAL OF CONVEX ANALYSIS, v. 17, p. 131-158, 2010.