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Calculo Diferencial e Integral II Mauricio A1 Vilches e Ma

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Prévia do material em texto

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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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