例 5.1.1 （例3）已知$\frac{\mathrm{d}x}{\mathrm{d}z}=-\frac{3z}{x}$，而$\frac{{\mathrm{d}}^{2}x}{\mathrm{d}{z}^2}\ne -\frac{3}{x}$！实际上应该写作$\frac{\mathrm{d}x}{\mathrm{d}z}=-\frac{3z}{x(z)}$，故有$$\begin{array}{}\text{(1)}& \frac{{\mathrm{d}}^{2}x}{\mathrm{d}{z}^2}=-\frac{3}{x}+\frac{3z}{x^2}\frac{\mathrm{d}x}{\mathrm{d}z}=-\frac{3}{x}-\frac{9z^2}{x^2}\end{array}$$

例 5.1.2 （例4）$(x,y)\mapsto (u,v)\mapsto z$，则有$\frac{\partial z}{\partial x}=\frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+\frac{\partial z}{\partial v}\frac{\partial v}{\partial x}$，而$\frac{\partial u}{\partial x}\ne {\left(\frac{\partial x}{\partial u}\right)}^{-1}$！应当使用矩阵求逆计算，即$$\begin{array}{}\text{(2)}& \left(\begin{array}{cc}\frac{\partial u}{\partial x}& \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\end{array}\right)={\left(\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\right)}^{-1}\end{array}$$

例 5.1.3 （例5）所有结果都要用$f,g,h$来表示。本题实际上需要找到$(x,y)\mapsto (z,t)$，因此充分条件为$det\frac{\partial (g,h)}{\partial (z,t)}\ne 0$。时刻注意：隐函数定理需要验证的是对因变量的Jacobi矩阵可逆！