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Задать вопрос экспертам

Консультация № 184605

29.11.2011, 01:51

59.93 руб.

0 1 1

Образование

Здравствуйте! Прошу помощи в следующем вопросе:

Найти функцию V(x,y,z) по её полному дифференциалу (dV=P*dx+Q*dy+R*dz) с помощью вычисления криволинейного интеграла

P=(y-1)*z² ;Q=(x+1)*z² ;R=2*(x+1)*(y-1)*z

Заранее благодарен за помощь!

Обсуждение

давно

Гордиенко Андрей Владимирович

Мастер-Эксперт
17387
18345

29.11.2011, 09:55

общий

это ответ

Здравствуйте, G-buck!

Проверим, действительно ли задан полный дифференциал:
[$8706$]P/[$8706$]y = z², [$8706$]Q/[$8706$]x = z²;
[$8706$]Q/[$8706$]z = 2z(x + 1), [$8706$]R/[$8706$]y = 2z(x + 1);
[$8706$]R/[$8706$]x = 2z(y - 1), [$8706$]P/[$8706$]z = 2z(y - 1).
Следовательно, задан полный дифференциал.

Найдём первообразную функцию U, воспользовавшись формулой
V(x, y, z) = _x0[$8747$]^xPdx + _y0[$8747$]^yQdy + _z0[$8747$]^zRdz
и приняв за точку (x₀; y₀; z₀) начало координат. Получим
V(x; y; z) = ₀[$8747$]^x(y - 1)z²dx + ₀[$8747$]^y(x + 1)z²dy + ₀[$8747$]^z2(x + 1)(y - 1)zdz = (y - 1)z² [$183$]x|₀^x + (x + 1)z² [$183$] y|₀^y + (x + 1)(y - 1) [$183$] z²|₀^z =
= x(y - 1)z² + (x + 1)yz² + (x + 1)(y - 1)z².

Ответ: V = x(y - 1)z² + (x + 1)yz² + (x + 1)(y - 1)z².

С уважением.

5

Об авторе:
Facta loquuntur.

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