Calculo - Volume 3

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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 Curvas 43
2.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Curvas Parametrizadas . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Parametrizações . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.1 Cônicas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.2 Curvas Planas Clássicas . . . . . . . . . . . . . . . . . . . 55
2.4 Curvas no Espaço . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4.1 Hélice Circular Reta . . . . . . . . . . . . . . . . . . . . . 62
2.5 Eliminação do Parâmetro . . . . . . . . . . . . . . . . . . . . . . 64
iii
iv CONTEÚDO
2.6 Continuidade e Diferenciabilidade . . . . . . . . . . . . . . . . . 66
2.6.1 Continuidade . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.6.2 Diferenciabilidade . . . . . . . . . . . . . . . . . . . . . . 68
2.6.3 Reta Tangente . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.6.4 Aplicação . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.7 Comprimento de Arco . . . . . . . . . . . . . . . . . . . . . . . . 87
2.8 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3 Bolas, Conjuntos Abertos e Fechados 97
3.1 Bolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2 Conjuntos Abertos . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3 Fronteira de um Conjuntos . . . . . . . . . . . . . . . . . . . . . . 99
3.4 Conjuntos Fechados . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4 Campos de Vetores 103
4.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.2 Campos Gradientes . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 Campos Rotacionais . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.4 Divergência . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.5 Campos Conservativos . . . . . . . . . . . . . . . . . . . . . . . . 116
4.5.1 Determinação do Potencial . . . . . . . . . . . . . . . . . 118
4.6 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5 Integrais 125
5.1 Integrais sobre Trajetórias . . . . . . . . . . . . . . . . . . . . . . 125
5.2 Integrais de Linha de Campos de Vetores . . . . . . . . . . . . . 128
5.3 Integrais de Linha e Reparametrizações . . . . . . . . . . . . . . 133
5.4 Aplicação . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.5 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.6 Teorema de Green . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6.1 Extensão do Teorema de Green . . . . . . . . . . . . . . . 153
5.6.2 Caracterização dos Campos Conservativos no Plano . . 158
5.7 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6 Superfícies 165
6.1 Introdução . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.2 Superfícies Parametrizadas . . . . . . . . . . . . . . . . . . . . . 166
6.3 Exemplos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.3.1 Parametrização de Gráficos . . . . . . . . . . . . . . . . . 167
6.3.2 Superfícies de Revolução . . . . . . . . . . . . . . . . . . 169
6.3.3 Esferas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.3.4 Cilindro . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
CONTEÚDO v
6.4 Superfícies Regulares . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.5 Área de uma Superfície . . . . . . . . . . . . . . . . . . . . . . . . 181
6.6 Aplicações . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7 Integrais sobre Superfícies 187
7.1 Integrais de Funções com Valores Reais . . . . . . . . . . . . . . 187
7.1.1 Aplicações . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.2 Integrais de Campos de Vetores . . . . . . . . . . . . . . . . . . . 190
7.2.1 Definição da Integral . . . . . . . . . . . . . . . . . . . . . 190
7.2.2 Interpretação Geométrica da Integral . . . . . . . . . . . 196
7.3 Teorema de Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.3.1 Aplicação . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.3.2 Interpretação do Teorema de Stokes . . . . . . . . . . . . 206
7.3.3 Caracterização dos Campos Conservativos no Espaço . . 207
7.4 Teorema de Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.4.1 Interpretação do Teorema de Gauss . . . . . . . . . . . . 212
7.4.2 Aplicação . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.4.3 Interpretação da Divergência . . . . . . . . . . . . . . . . 215
7.5 Exercícios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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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