ss-Integrales-linea_book - Integrales de Linea Calculo II 1 Integrales de Linea Campo Vectorial Conservativo Definicin Un campo vectorial F se dice que

ss-Integrales-linea_book - Integrales de Linea Calculo II 1...

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Integrales de Linea Calculo II 1. Integrales de Linea Campo Vectorial Conservativo Definición. Un campo vectorial ¯ F se dice que es conservativo si existe una función diferenciable f tal que ¯ F = f . La función f es la función potencial de ¯ F Ejemplo. Hallar la función potencial de ¯ F ( x, y, z ) = 4 xe z , cos y, 2 x 2 e z Teorema Fundamental para Integrales Sea C una curva suave por trozos contenida en una región abierta R parametrica. ¯ r ( t ) = ( x ( t ) , y ( t )) , a t b , si f es continuamente diferenciable entonces C f r ) d ¯ r = f r ( b )) - f r ( a )) Demostración: Ejemplo Ejemplo. Hallar el trabajo realizado por el campo de fuerzas ¯ F ( x, y ) = 1 2 xy, 1 4 x 2 sobre una partícula que se mueve de (0 , 0) a (1 , 1) a lo largo de cada una de las trayectorias 1. C 1 : y = x 2. C 2 : x = y 2 3. C 3 : y = x 3 1
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Teorema. C ¯ F ¯ dr es indepediente de la trayectoria en D si y solo si C ¯ F ¯ dr para cualquier curva cerrada C .
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