📝 题目
例 2 试求复合函数
$$ u = f\left( {x\left( {\xi ,\eta }\right) ,y\left( {\xi ,\eta }\right) }\right) $$
的二阶偏导数. 这里假设 $x\left( {\xi ,\eta }\right) ,y\left( {\xi ,\eta }\right)$ 和 $f\left( {x,y}\right)$ 都是二阶连续可微函数.
💡 答案与解析
解 利用链式法则逐次计算可得:
$$ \frac{\partial u}{\partial \xi } = \frac{\partial f}{\partial x}\frac{\partial x}{\partial \xi } + \frac{\partial f}{\partial y}\frac{\partial y}{\partial \xi }, $$
$$ \frac{\partial u}{\partial \eta } = \frac{\partial f}{\partial x}\frac{\partial x}{\partial \eta } + \frac{\partial f}{\partial y}\frac{\partial y}{\partial \eta }, $$
$$ \frac{{\partial }^{2}u}{\partial {\xi }^{2}} = \frac{\partial }{\partial \xi }\left( {\frac{\partial f}{\partial x}\frac{\partial x}{\partial \xi }}\right) + \frac{\partial }{\partial \xi }\left( {\frac{\partial f}{\partial y}\frac{\partial y}{\partial \xi }}\right) $$
$$ = \frac{\partial }{\partial \xi }\left( \frac{\partial f}{\partial x}\right) \frac{\partial x}{\partial \xi } + \frac{\partial f}{\partial x}\frac{{\partial }^{2}x}{\partial {\xi }^{2}} $$
$$ + \frac{\partial }{\partial \xi }\left( \frac{\partial f}{\partial y}\right) \frac{\partial y}{\partial \xi } + \frac{\partial f}{\partial y}\frac{{\partial }^{2}y}{\partial {\xi }^{2}} $$
$$ = \left( {\frac{{\partial }^{2}f}{\partial {x}^{2}}\frac{\partial x}{\partial \xi } + \frac{{\partial }^{2}f}{\partial y\partial x}\frac{\partial y}{\partial \xi }}\right) \frac{\partial x}{\partial \xi } + \frac{\partial f}{\partial x}\frac{{\partial }^{2}x}{\partial {\xi }^{2}} $$
$$ + \left( {\frac{{\partial }^{2}f}{\partial x\partial y}\frac{\partial x}{\partial \xi } + \frac{{\partial }^{2}f}{\partial {y}^{2}}\frac{\partial y}{\partial \xi }}\right) \frac{\partial y}{\partial \xi } + \frac{\partial f}{\partial y}\frac{{\partial }^{2}y}{\partial {\xi }^{2}} $$
$$ = \frac{{\partial }^{2}f}{\partial {x}^{2}}{\left( \frac{\partial x}{\partial \xi }\right) }^{2} + 2\frac{{\partial }^{2}f}{\partial x\partial y}\left( {\frac{\partial x}{\partial \xi }\frac{\partial y}{\partial \xi }}\right) $$
$$ + \frac{{\partial }^{2}f}{\partial {y}^{2}}{\left( \frac{\partial y}{\partial \xi }\right) }^{2} + \frac{\partial f}{\partial x}\frac{{\partial }^{2}x}{\partial {\xi }^{2}} + \frac{\partial f}{\partial y}\frac{{\partial }^{2}y}{\partial {\xi }^{2}}. $$
类似地可以求得
$$ \frac{{\partial }^{2}u}{\partial {\eta }^{2}} = \frac{{\partial }^{2}f}{\partial {x}^{2}}{\left( \frac{\partial x}{\partial \eta }\right) }^{2} + 2\frac{{\partial }^{2}f}{\partial x\partial y}\left( {\frac{\partial x}{\partial \eta }\frac{\partial y}{\partial \eta }}\right) $$
$$ + \frac{{\partial }^{2}f}{\partial {y}^{2}}{\left( \frac{\partial y}{\partial \eta }\right) }^{2} + \frac{\partial f}{\partial x}\frac{{\partial }^{2}x}{\partial {\eta }^{2}} + \frac{\partial f}{\partial y}\frac{{\partial }^{2}y}{\partial {\eta }^{2}}, $$
$$ \frac{{\partial }^{2}f}{\partial \xi \partial \eta } = \frac{{\partial }^{2}f}{\partial {x}^{2}}\frac{\partial x}{\partial \xi }\frac{\partial x}{\partial \eta } + \frac{{\partial }^{2}f}{\partial x\partial y}\left( {\frac{\partial x}{\partial \xi }\frac{\partial y}{\partial \eta } + \frac{\partial x}{\partial \eta }\frac{\partial y}{\partial \xi }}\right) $$
$$ + \frac{{\partial }^{2}f}{\partial {y}^{2}}\frac{\partial y}{\partial \xi }\frac{\partial y}{\partial \eta } + \frac{\partial f}{\partial x}\frac{{\partial }^{2}x}{\partial \xi \partial \eta } + \frac{\partial f}{\partial y}\frac{{\partial }^{2}y}{\partial \xi \partial \eta }. $$