📝 题目
例 21 设 $F\left( {x,y,z}\right)$ 在 ${\mathbf{R}}^{3}$ 中有连续的一阶偏导数,并满足
$$ y\frac{\partial F}{\partial x} - x\frac{\partial F}{\partial y} + \frac{\partial F}{\partial z} \geq \alpha > 0\;\left( {\alpha \text{ 为常数 }}\right) . $$
求证: 当点(x, y, z)沿着曲线 $\Gamma : x = - \cos t,y = \sin t,z = t\left( {t \geq 0}\right)$ 趋向无穷时, $F\left( {x,y,z}\right) \rightarrow + \infty$ .
💡 答案与解析
证 考虑一元函数 $f\left( t\right) \overset{\text{ 定义 }}{ = }F\left( {-\cos t,\sin t,t}\right)$ ,则
$$ \frac{\mathrm{d}f}{\mathrm{\;d}t} = \frac{\partial F}{\partial x}\sin t + \frac{\partial F}{\partial y}\cos t + \frac{\partial F}{\partial z}. $$
当 $\left( {x,y,z}\right) \in \Gamma$ 时,有 $\frac{\mathrm{d}f}{\mathrm{\;d}t} = y\frac{\partial F}{\partial x} - x\frac{\partial F}{\partial y} + \frac{\partial F}{\partial z} \geq \alpha > 0$ . 因此,根据微分中值定理,有 $F\left( {x,y,z}\right) - F\left( {-1,0,0}\right) = f\left( t\right) - f\left( 0\right) = {f}^{\prime }\left( \xi \right) t \geq {\alpha t}$ ,即得 $F\left( {x,y,z}\right) \geq F\left( {-1,0,0}\right) + {\alpha t}$ . 于是 $F\left( {x,y,z}\right) \rightarrow + \infty \left( {t \rightarrow + \infty }\right)$ .