📝 题目
例 3 设 $x\left( t\right) ,y\left( t\right)$ 可微, $r = \sqrt{{x}^{2} + {y}^{2}},\theta = \arctan \frac{y}{x}$ ,求 $\mathrm{d}r,\mathrm{\;d}\theta$ .
💡 答案与解析
解 $\mathrm{d}r = \frac{x\mathrm{\;d}x + y\mathrm{\;d}y}{\sqrt{{x}^{2} + {y}^{2}}},\mathrm{\;d}\theta = \frac{1}{{\left( 1 + \frac{y}{x}\right) }^{2}} \cdot \frac{x\mathrm{\;d}y - y\mathrm{\;d}x}{{x}^{2}} = \frac{x\mathrm{\;d}y - y\mathrm{\;d}x}{{x}^{2} + {y}^{2}}$ .