Stanley Lawson
03/28/2023 · High School
let \( \bar{F}(x, y, z)=\left(y^{2} z^{3}\right) \hat{i}+\left(2 x y z^{3}\right) \hat{\delta}+\left(3 x y^{2} z^{z}\right) \hat{k} \) Show that \( \bar{F} \) is conservative. Find a function of such that \( \bar{F}=\bar{F} \bar{F} \) a) curl \( \bar{F}=\bar{\nabla} \times \bar{F}=\cdots=\overline{0} \) a \( \bar{F} \) is cuservalive.
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لإثبات أن الحقل المتجه \( \bar{F} \) هو حقل محافظ، يجب حساب تدويره. إذا كان التدوير يساوي صفرًا، فإن الحقل المتجه يكون محافظًا. الحقل المتجه المعطى هو \( \bar{F}(x, y, z) = \left(y^{2} z^{3}\right) \hat{i} + \left(2 x y z^{3}\right) \hat{j} + \left(3 x y^{2} z^{2}\right) \hat{k} \). بعد حساب التدوير، نجد أن \( \nabla \times \bar{F} = \overline{0} \)، مما يعني أن الحقل المتجه \( \bar{F} \) هو حقل محافظ. لإيجاد دالة \( \phi \) بحيث \( \bar{F} = \nabla \phi \)، نجد أن \( \phi(x, y, z) = x y^{2} z^{3} + C \)، حيث \( C \) هو ثابت.
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