JSFHJSFHCap\303\255tulo 23-Calculando integrais de linha O cap\303\255tulo trata de integrais de linha. Parece valer a pena usar o Maple para configurar e avaliar integrais de linha em uma curva parametrizada. Seguiremos um procedimento passo a passo e, em seguida, procederemos a novos procedimentos que fazem tudo em um \303\272nico comando.
1-Um exemplo passo a passo
Primeiro, precisamos definir um campo vetorial que queremos integrar, o caminho parametrizado sobre o qual estamos integrando e os limites do par\303\242metro. No nosso exemplo, usaremos o familiar campo vetorial circular e integraremos mais da metade de uma elipse.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
O processo de integra\303\247\303\243o da integral de linha sobre a curva parametrizada pode ser dividido em 6 etapas:
1) Plote um gr\303\241fico do campo vetorial e da curva parametrizada.
2) Configure a integral de linha, com o integrando sendo o produto escalar do campo vetorial e a derivada do caminho parametrizado.
3) Substitua a parametriza\303\247\303\243o do caminho no campo para tornar o campo uma fun\303\247\303\243o com valor vetorial do par\303\242metro.
4) Pegue a derivada do caminho em rela\303\247\303\243o ao par\303\242metro.
5) Avalie o produto escalar e simplifique. Isso reduz o problema a uma integral por um intervalo.
6) Avalie a integral.
Estamos prontos para percorrer as etapas, uma de cada vez. \303\200 medida que trabalhamos no exemplo com o Maple, voc\303\252 tamb\303\251m deve executar cada passo manualmente.
O primeiro passo \303\251 examinar um gr\303\241fico do campo vetorial e a curva parametrizada.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O pr\303\263ximo passo \303\251 configurar a integral da linha. 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O terceiro passo \303\251 substituir x e y e o campo vetorial pelas parametriza\303\247\303\265es de x e y no ponto apropriado do caminho. 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O quarto passo \303\251 obter a derivada do caminho parametrizado. 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 quinto passo \303\251 avaliar o produto escalar no integrando e simplificar. 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Agora, avaliamos a integral e avaliamos o resultado numericamente, se necess\303\241rio.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
2-Uma abordagem automatizada
Por conveni\303\252ncia, bloqueamos o c\303\263digo em dois procedimentos que podemos usar, um para plotagem e outro para configurar a integral e avaliar. Isso nos permite modificar o problema com uma quantidade m\303\255nima de trabalho.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
Assim, podemos obter os resultados inserindo o campo de vetor e o caminho e executando os dois comandos.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
3-Integrais de linha em 3-D
Se o campo vetorial estiver em tr\303\252s dimens\303\265es, precisamos modificar levemente o comando lineinegral. Nesse caso, n\303\243o tra\303\247aremos a curva.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
Usando este comando, refazemos o exemplo 4 do livro.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
Exerc\303\255cios
1) Avalie a integral da reta onde F = [ln (y), ln (x)] e C \303\251 a curva dada parametricamente por [2t, t ^ 2] com t = 2..4.
>
2) Avalie a integral da linha em que F = [2y, -sin (y)] e C \303\251 o c\303\255rculo unit\303\241rio tra\303\247ado no sentido anti-hor\303\241rio a partir do ponto (1, 0).
>
3) Avalie a integral da reta em que F = e ^ xi + e ^ yj e C \303\251 a parte da elipse x ^ 2 + 4y ^ 2 = 4 do ponto (0, 1) ao ponto (2, 0) em no sentido hor\303\241rio.
>
4) Avalie a integral da reta em que F = -yi + xj + 5k e C \303\251 a h\303\251lice [cos (t), sin (t) t] para t = 0..4Pi.
>
>