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� � � � � ��� � ��� � �� �� � ��� ������ ��� � �� � �� �������� !�"� i Copyright by Mauricio A. Vilches Todos os direitos reservados Proibida a reprodução parcial ou total ii Conteúdo 1 Introdução 1 1.1 Espaços Euclidianos . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 O Espaço Euclidiano Tridimensional . . . . . . . . . . . . . . . . 1 1.3 Sistemas de Coordenadas . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Produto Escalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Norma Euclidiana de um Vetor . . . . . . . . . . . . . . . . . . . 4 1.6 Ângulos Diretores . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Produto Vetorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.8 Distância entre Pontos . . . . . . . . . . . . . . . . . . . . . . . . 11 1.9 Retas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.9.1 Paralelismo e Perpendicularismo . . . . . . . . . . . . . . 13 1.9.2 Forma Simétrica da Equação da Reta . . . . . . . . . . . . 14 1.9.3 Distância de um Pontos a uma Reta . . . . . . . . . . . . 14 1.10 Planos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.10.1 Ângulo entre Planos . . . . . . . . . . . . . . . . . . . . . 16 1.10.2 Paralelismo e Perpendicularismo entre Planos . . . . . . 18 1.10.3 Distância de um Pontos a um Plano . . . . . . . . . . . . 19 1.11 Generalizações . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.12 Superfícies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.12.1 Superfícies Quadricas . . . . . . . . . . . . . . . . . . . . 21 1.13 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2 Funções de Várias Variáveis 43 2.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2 Domínio e Imagem . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 Gráficos de Funções de Várias Variáveis . . . . . . . . . . . . . . 51 2.4 Conjuntos de Nível . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.5 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 iii iv CONTEÚDO 3 Limites e Contínuidade 67 3.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Limites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3 Continuidade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.4 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4 Derivadas Parciais 87 4.1 Definições e Exemplos . . . . . . . . . . . . . . . . . . . . . . . . 87 4.2 Generalizações . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3 Interpretação Geométricas das Derivadas Parciais . . . . . . . . 92 4.4 Derivadas Parciais com Taxa de Variação . . . . . . . . . . . . . 95 4.5 Diferenciabilidade . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.6 Aproximação Linear . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.7 Derivadas de Ordem Superior . . . . . . . . . . . . . . . . . . . . 110 4.8 Regra da Cadeia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.9 Derivada Direcional . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.10 Derivada Direcional como Taxa de Variação . . . . . . . . . . . . 128 4.11 Gradiente de uma Função . . . . . . . . . . . . . . . . . . . . . . 129 4.11.1 Observações Geométricas sobre Gradientes . . . . . . . . 133 4.12 Funções Implícitas . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.13 Gradiente e Conjuntos de Nível . . . . . . . . . . . . . . . . . . . 144 4.14 Gradiente e Curvas de Nível . . . . . . . . . . . . . . . . . . . . . 144 4.14.1 Ângulos entre Curvas . . . . . . . . . . . . . . . . . . . . 147 4.15 Gradiente e Supefícies de Nível . . . . . . . . . . . . . . . . . . . 149 4.15.1 Ângulos entre Supefícies . . . . . . . . . . . . . . . . . . . 155 4.16 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5 Máximos e Mínimos 169 5.1 Definições e Exemplos . . . . . . . . . . . . . . . . . . . . . . . . 169 5.2 Determinação dos Máximos e Mínimos Locais . . . . . . . . . . 175 5.3 Problemas de Otimização . . . . . . . . . . . . . . . . . . . . . . 186 5.3.1 Mínimos Quadrados . . . . . . . . . . . . . . . . . . . . . 190 5.4 Máximos e Mínimos Absolutos . . . . . . . . . . . . . . . . . . . 195 5.5 Método dos Multiplicadores de Lagrange . . . . . . . . . . . . . 199 5.6 Determinação dos Extremos Condicionados . . . . . . . . . . . . 200 5.7 Problemas de Otimização . . . . . . . . . . . . . . . . . . . . . . 206 5.8 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6 Integração Múltipla 221 6.1 Integração Dupla sobre Retângulos . . . . . . . . . . . . . . . . . 221 6.2 Significado Geométrico da Integral Dupla . . . . . . . . . . . . . 222 CONTEÚDO v 6.3 Integrais Iteradas . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 6.4 Teorema de Fubini . . . . . . . . . . . . . . . . . . . . . . . . . . 228 6.4.1 Extensão do Teorema de Fubini . . . . . . . . . . . . . . . 232 6.5 Integração Dupla sobre Regiões mais Gerais . . . . . . . . . . . 234 6.5.1 Regiões Elementares . . . . . . . . . . . . . . . . . . . . . 234 6.5.2 Extensão da Integral Dupla . . . . . . . . . . . . . . . . . 237 6.6 Integral Dupla e Volume de Sólidos . . . . . . . . . . . . . . . . 238 6.7 Mudança de Coordenadas . . . . . . . . . . . . . . . . . . . . . . 248 6.7.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.8 Mudança Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 6.9 Mudança Polar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.9.1 Regiões Limitadas por Círculos . . . . . . . . . . . . . . . 258 6.10 Outras Aplicações da Integral Dupla . . . . . . . . . . . . . . . . 268 6.10.1 Massa Total . . . . . . . . . . . . . . . . . . . . . . . . . . 268 6.10.2 Momento de Massa . . . . . . . . . . . . . . . . . . . . . . 268 6.10.3 Centro de Massa . . . . . . . . . . . . . . . . . . . . . . . 268 6.10.4 Momento de Inércia . . . . . . . . . . . . . . . . . . . . . 272 6.11 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 7 Integração Tripla 277 7.1 Integração Tripla sobre Paralelepípedos . . . . . . . . . . . . . . 277 7.2 Integração Tripla sobre Região mais Gerais . . . . . . . . . . . . 280 7.2.1 Regiões Elementares no Espaço . . . . . . . . . . . . . . . 280 7.2.2 Extensão da Integral Tripla . . . . . . . . . . . . . . . . . 283 7.3 Mudança de Coordenadas . . . . . . . . . . . . . . . . . . . . . . 287 7.3.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 7.4 Coordenadas Cilíndricas . . . . . . . . . . . . . . . . . . . . . . . 288 7.5 Coordenadas Esféricas . . . . . . . . . . . . . . . . . . . . . . . . 294 7.6 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 8 Apéndice 303 9 Respostas 313 10 Bibliografia 317 � � � ���� � � � � � ������� � ������ � �����! 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