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(4) i. Copyright by Mauricio A. Vilches Todos os direitos reservados Proibida a reprodução parcial ou total.

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(6) Conteúdo 1 Introdução 1.1 Espaços Euclidianos . . . . . . . . . . . . . . . . . . . 1.2 O Espaço Euclidiano Tridimensional . . . . . . . . . . 1.3 Sistemas de Coordenadas . . . . . . . . . . . . . . . . 1.4 Produto Escalar . . . . . . . . . . . . . . . . . . . . . . 1.5 Norma Euclidiana de um Vetor . . . . . . . . . . . . . 1.6 Ângulos Diretores . . . . . . . . . . . . . . . . . . . . . 1.7 Produto Vetorial . . . . . . . . . . . . . . . . . . . . . . 1.8 Distância entre Pontos . . . . . . . . . . . . . . . . . . 1.9 Retas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Paralelismo e Perpendicularismo . . . . . . . . 1.9.2 Forma Simétrica da Equação da Reta . . . . . . 1.9.3 Distância de um Pontos a uma Reta . . . . . . 1.10 Planos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Ângulo entre Planos . . . . . . . . . . . . . . . 1.10.2 Paralelismo e Perpendicularismo entre Planos 1.10.3 Distância de um Pontos a um Plano . . . . . . 1.11 Generalizações . . . . . . . . . . . . . . . . . . . . . . . 1.12 Superfícies . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.1 Superfícies Quadricas . . . . . . . . . . . . . . 1.13 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Curvas 2.1 Introdução . . . . . . . . . . . . 2.2 Curvas Parametrizadas . . . . . 2.3 Parametrizações . . . . . . . . . 2.3.1 Cônicas . . . . . . . . . . 2.3.2 Curvas Planas Clássicas 2.4 Curvas no Espaço . . . . . . . . 2.4.1 Hélice Circular Reta . . 2.5 Eliminação do Parâmetro . . .. iii. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. 1 1 1 2 4 4 6 8 11 11 13 14 14 15 16 18 19 19 21 21 37. . . . . . . . .. 43 43 46 51 51 55 61 62 64.

(7) CONTEÚDO. iv 2.6. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 97 . 97 . 98 . 99 . 101. 4 Campos de Vetores 4.1 Introdução . . . . . . . . . . . . . . 4.2 Campos Gradientes . . . . . . . . . 4.3 Campos Rotacionais . . . . . . . . 4.4 Divergência . . . . . . . . . . . . . 4.5 Campos Conservativos . . . . . . . 4.5.1 Determinação do Potencial 4.6 Exercícios . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 103 103 109 113 115 116 118 122. 5 Integrais 5.1 Integrais sobre Trajetórias . . . . . . . . . . . . . . . . . . . . 5.2 Integrais de Linha de Campos de Vetores . . . . . . . . . . . 5.3 Integrais de Linha e Reparametrizações . . . . . . . . . . . . 5.4 Aplicação . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Teorema de Green . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Extensão do Teorema de Green . . . . . . . . . . . . . 5.6.2 Caracterização dos Campos Conservativos no Plano 5.7 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . .. 125 125 128 133 142 144 147 153 158 163. . . . . . . .. 165 165 166 167 167 169 171 171. 2.7 2.8. Continuidade e Diferenciabilidade 2.6.1 Continuidade . . . . . . . . 2.6.2 Diferenciabilidade . . . . . 2.6.3 Reta Tangente . . . . . . . . 2.6.4 Aplicação . . . . . . . . . . Comprimento de Arco . . . . . . . Exercícios . . . . . . . . . . . . . . .. 3 Bolas, Conjuntos Abertos e Fechados 3.1 Bolas . . . . . . . . . . . . . . . . 3.2 Conjuntos Abertos . . . . . . . . 3.3 Fronteira de um Conjuntos . . . . 3.4 Conjuntos Fechados . . . . . . . .. 6 Superfícies 6.1 Introdução . . . . . . . . . . . . . . 6.2 Superfícies Parametrizadas . . . . 6.3 Exemplos . . . . . . . . . . . . . . . 6.3.1 Parametrização de Gráficos 6.3.2 Superfícies de Revolução . 6.3.3 Esferas . . . . . . . . . . . . 6.3.4 Cilindro . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 66 66 68 78 84 87 91.

(8) CONTEÚDO 6.4 6.5 6.6. v. Superfícies Regulares . . . . . . . . . . . . . . . . . . . . . . . . . 175 Área de uma Superfície . . . . . . . . . . . . . . . . . . . . . . . . 181 Aplicações . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184. 7 Integrais sobre Superfícies 7.1 Integrais de Funções com Valores Reais . . . . . . . . . . . . 7.1.1 Aplicações . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Integrais de Campos de Vetores . . . . . . . . . . . . . . . . . 7.2.1 Definição da Integral . . . . . . . . . . . . . . . . . . . 7.2.2 Interpretação Geométrica da Integral . . . . . . . . . 7.3 Teorema de Stokes . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Aplicação . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Interpretação do Teorema de Stokes . . . . . . . . . . 7.3.3 Caracterização dos Campos Conservativos no Espaço 7.4 Teorema de Gauss . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Interpretação do Teorema de Gauss . . . . . . . . . . 7.4.2 Aplicação . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Interpretação da Divergência . . . . . . . . . . . . . . 7.5 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 187 187 187 190 190 196 199 205 206 207 209 212 213 215 216. 8 Apéndice 221 8.1 Teorema de Green . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 8.2 Teorema de Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . 224 8.3 Teorema de Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 9 Bibliografia. 231.

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