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Консультация № 172473
22.09.2009, 12:55
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Добрый день, уважаемые! Помогите решить пример по дифференциальному исчислению функций нескольких переменных...
Дана функция z=f(x,y). Показать, что F(x,y,z,dz/dx;dz/dy;(d^2)z/dx^2;(d^2)z/dy^2;(d^2)z/dxdy)=0
z=cos y+(y-x)sin y; F=(x-y)(d^2)z/dxdy - dz/dy
Заранее благодарен...
Дана функция z=f(x,y). Показать, что F(x,y,z,dz/dx;dz/dy;(d^2)z/dx^2;(d^2)z/dy^2;(d^2)z/dxdy)=0
z=cos y+(y-x)sin y; F=(x-y)(d^2)z/dxdy - dz/dy
Заранее благодарен...
Обсуждение
Неизвестный
22.09.2009, 13:39
общий
это ответ
Здравствуйте, Попов Антон Андреевич.
Задание сводится к тому, чтобы проверить удовлетворяет указанная функция z(x, y) выражению F(...)=0 или нет.
Так как:
z = cos(y) + (y - x)*sin(y), то
dz/dx = z'x = [cos(y) + (y - x)*sin(y)]'x = - sin(y)
dz/dy = z'y = [cos(y) + (y - x)*sin(y)]'y = - sin(y) + 1*sin(y) + (y - x)*cos(y) = (y - x)*cos(y)
d2z/(dxdy) = (dz/dx)'y = (dz/dy)'x = (- sin(y))'y = - cos(y)
Тогда:
F = (x - y)*[d2z/(dxdy)] - (dz/dy) = (x - y)*(- cos(y)) - (y - x)*cos(y) = (y - x)*cos(y) - (y - x)*cos(y) [$8801$] 0
Следовательно функция z = z(x, y) удовлетворяет выражению F(...) = 0
Задание сводится к тому, чтобы проверить удовлетворяет указанная функция z(x, y) выражению F(...)=0 или нет.
Так как:
z = cos(y) + (y - x)*sin(y), то
dz/dx = z'x = [cos(y) + (y - x)*sin(y)]'x = - sin(y)
dz/dy = z'y = [cos(y) + (y - x)*sin(y)]'y = - sin(y) + 1*sin(y) + (y - x)*cos(y) = (y - x)*cos(y)
d2z/(dxdy) = (dz/dx)'y = (dz/dy)'x = (- sin(y))'y = - cos(y)
Тогда:
F = (x - y)*[d2z/(dxdy)] - (dz/dy) = (x - y)*(- cos(y)) - (y - x)*cos(y) = (y - x)*cos(y) - (y - x)*cos(y) [$8801$] 0
Следовательно функция z = z(x, y) удовлетворяет выражению F(...) = 0
Форма ответа
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