Aplicaci\303\263n del Teorema de Gauss \303\263 Divergencia Alumno:Nicolas Palma Ruiz Profesor: Miguel Olivares B. Universidad Adolfo Iba\303\261ez
<Text-field style="Heading 1" layout="Heading 1">Problema.</Text-field> -Se nos pide encontrar el flujo que pasa a traves de una esfera de radio b ubicada en el campo gravitacional. -notar que el campo vectorial gravitacional tiene una singularidad en 0,0,0 la cual acotraremos por la esfera unitaria con b>1. -Se utilizar\303\241 el Teorema de Gauss o de Divergencia para calcular el pedido. -Se usar\303\241 b=4 para realizar graficos.
<Text-field style="Heading 1" layout="Heading 1">Grafica de esferas y campo vectorial</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJnBsb3RzRidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZA 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A continuacion mostramos la esfera en el campo vectorial. 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 Definimos el campo vectorial como F= 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 Por el Teorema de Gauss sabemos que 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 sabemos que el divergente de un campo se define como : divF=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 donde P,Q,R son las componentes de nuestro campo vectorial 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 Ya calculado el divergente de F lo cambiaremos a coordenadas cilindricas para calcular la integral triple, teniendo lo siguiente: 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 tenemos la siguiente region de integracion, para obtener las variaicones de 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRIkFGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRIkJGJ0YvRjIvJStleGVjdXRhYmxlR0ZFRkFGQUZlbkZB Finalmente la integral triple por el Teorema de Divergencia queda asi 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 por lo que el flujo que pasa atravez de esta superficie es 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