数学试题讲解

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若二元函数

f (u, v)

对每个变量都具有二阶连续偏导数, 并且满足

u \frac{\partial f}{\partial u} + v \frac{\partial f}{\partial v} = 4 f (u, v)

, 并且满足

\frac{\partial^{2} f}{\partial u^{2}} + \frac{\partial^{2} f}{\partial v^{2}} = u^{2} + v^{2}

。
(1) 求证:

{\begin{cases} u^{2} \frac{\partial^{2} f}{\partial u^{2}} + 2 u v \frac{\partial^{2} f}{\partial u \partial v} + v^{2} \frac{\partial^{2} f}{\partial v^{2}} = 12 f (u, v) \\ v^{2} \frac{\partial^{2} f}{\partial u^{2}} - 2 u v \frac{\partial^{2} f}{\partial u \partial v} + u^{2} \frac{\partial^{2} f}{\partial v^{2}} = {(u^{2} + v^{2})}^{2} - 12 f (u, v) \end{cases}

(2) 记

g (x, y) = f (e^{λ x} \cos y, e^{λ x} \sin y)

, 其中

λ

是一个常数, 求解

div (grad g)

。