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Консультация № 182586
22.03.2011, 08:50
56.19 руб.
0
1
1
Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Даны векторное поле F=Xi+Yj+Zk (F, i, j, k со знаком вектора) и плоскость р: Ax+By+Cz+D=0, которая совместно с координатными плоскостями образует пирамиду V. Пусть [$963$] - основание пирамиды, принадлежащее плоскости р; [$955$] - контур, ограничивающий [$963$]; вектор n - нормаль к [$963$], направленная вне пирамиды V. Требуется вычислить.
1) поток векторного поля F через поверхность [$963$] в направлении нормали n;
2) циркуляцию векторного поля F по замкнутому контуру [$955$] непосредственно и применив теорему Стокса к контуру [$955$] и ограниченной им поверхности [$963$] с нормалью n;
3) поток векторного поля F через полную поверхность пирамиды V в направлении внешней нормали к ее поверхности непосредственно и применив теорему Остроградского. Сделать чертеж.
F= (3x+4y+2z)j; p:x+y+2z-4=0
Даны векторное поле F=Xi+Yj+Zk (F, i, j, k со знаком вектора) и плоскость р: Ax+By+Cz+D=0, которая совместно с координатными плоскостями образует пирамиду V. Пусть [$963$] - основание пирамиды, принадлежащее плоскости р; [$955$] - контур, ограничивающий [$963$]; вектор n - нормаль к [$963$], направленная вне пирамиды V. Требуется вычислить.
1) поток векторного поля F через поверхность [$963$] в направлении нормали n;
2) циркуляцию векторного поля F по замкнутому контуру [$955$] непосредственно и применив теорему Стокса к контуру [$955$] и ограниченной им поверхности [$963$] с нормалью n;
3) поток векторного поля F через полную поверхность пирамиды V в направлении внешней нормали к ее поверхности непосредственно и применив теорему Остроградского. Сделать чертеж.
F= (3x+4y+2z)j; p:x+y+2z-4=0
Обсуждение
22.03.2011, 15:50
общий
это ответ
Здравствуйте, vera-nika!
1) Поток векторного поля F через поверхность σ равен поверхностному интегралу
так как F[sub]x[/sub] = F[sub]z[/sub] = 0. Из уравнения плоскости y = 4-x-2z, откуда
2) Контур λ является суммой трех отрезков: AB: {x+y=4, z=0}, BC: {y+2z=4, x=0}, CA: {x+2z=4, y=0}. Циркуляция векторного поля F по контуру λ равна линейному интегралу
Здесь опять же F[sub]x[/sub] = F[sub]z[/sub] = 0 и интеграл по отрезку CA так же равен 0, так как на этом отрезке dy = 0.
По формуле Стокса циркуляция равна
где
С учетом этого циркуляция равна
3) Поток векторного поля через полную поверхность пирамиды V равен
По формуле Остроградского этот же поток равен
где
Соответственно,
1) Поток векторного поля F через поверхность σ равен поверхностному интегралу
так как F[sub]x[/sub] = F[sub]z[/sub] = 0. Из уравнения плоскости y = 4-x-2z, откуда
2) Контур λ является суммой трех отрезков: AB: {x+y=4, z=0}, BC: {y+2z=4, x=0}, CA: {x+2z=4, y=0}. Циркуляция векторного поля F по контуру λ равна линейному интегралу
Здесь опять же F[sub]x[/sub] = F[sub]z[/sub] = 0 и интеграл по отрезку CA так же равен 0, так как на этом отрезке dy = 0.
По формуле Стокса циркуляция равна
где
С учетом этого циркуляция равна
3) Поток векторного поля через полную поверхность пирамиды V равен
По формуле Остроградского этот же поток равен
где
Соответственно,
5
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