📝 题目
5.3.3 对下列方程所确定的 $z = z\left( {x,y}\right)$ ,求二阶偏导数:
(1) ${xy} + {yz} + {zx} = 1$ ; (2) $\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx} = 1$ .
💡 答案与解析
5.3.3 (1) ${z}_{xx}^{\prime \prime } = \frac{2\left( {y + z}\right) }{{\left( x + y\right) }^{2}},{z}_{xy}^{\prime \prime } = \frac{2z}{{\left( x + y\right) }^{2}},{z}_{yy}^{\prime \prime } = \frac{2\left( {x + z}\right) }{{\left( x + y\right) }^{2}}$ ;
(2) ${z}_{xx}^{\prime \prime } = \frac{{2y}\left( {{yz} - 1}\right) }{{\left( 1 - xy\right) }^{2}},{z}_{xy}^{\prime \prime } = \frac{2z}{{\left( 1 - xy\right) }^{2}},{z}_{yy}^{\prime \prime } = \frac{{2z}\left( {{xz} - 1}\right) }{{\left( 1 - xy\right) }^{2}}$ .