We define a curve $\Gamma_a$ parametrized by its arc length $s$ such that: \begin{equation} \Gamma = \{x_a(s), z(s)_a | s \in (0,L) \} \end{equation} were $(z,x)$ is the coordinate system under study. So $z$ is drawn as function of $x$: $z(x)$. The first derivative of $z$ can be written as: \begin{equation} \frac{\partial z_a}{\partial x_a} = \frac{\partial z_a}{\partial s}\frac{\partial s}{\partial x_a} = \frac{z'_a}{x'_a} \end{equation} The prime $(')$ here is defined with respect to $s$. The professor said that we can generalize and in better way this way of writing by introducing that $\tilde{z}$ in terms of x such that (because x_a is invertable):
\begin{equation} \tilde{z}_a(x) = z_a(x_a^{-1}(x)) \ \ \ \text{as} \ \ s = x_a^{-1}(x) \end{equation} then a better treatment with partial derivative is: \begin{equation} \frac{\partial \tilde{z}_a}{\partial x} (x) = \frac{1}{\frac{\partial x_a}{\partial s}} \frac{\partial z_a}{\partial s}(x_a^{-1}(x)) \end{equation} I didn't get how did he get $s=x_a^{-1}(x)$ and why do we need to do such step? because my step already works but the high mathematical level by using $\tilde{z}$ is still somehow weird?