Question about the method of characteristic coordinates to solve the wave equation

Question

I'm working on learning how to solve the 1-D wave equation using characteristic coordinates and I am getting stumped by one of the steps. Here is the part I am stuck on:

The textbook says that if $δ=x+ct$ and $φ=x-ct$ then $∂_x=∂_δ+∂_φ$ and $∂_t=c∂_δ-c∂_φ$.

I don't understand how the differentiation works here. Can someone explain the rules for how this works?

Answer 1

Use the chain rule: \begin{align*} \frac{\partial f}{\partial x}&=\frac{\partial f}{\partial\delta} \frac{\partial \delta}{\partial x}+\frac{\partial f}{\partial\varphi} \frac{\partial \varphi}{\partial x}=\frac{\partial f}{\partial\delta} +\frac{\partial f}{\partial\varphi} \\ \frac{\partial f}{\partial t}&=\frac{\partial f}{\partial\delta} \frac{\partial \delta}{\partial t}+\frac{\partial f}{\partial\varphi} \frac{\partial \varphi}{\partial t}=c\frac{\partial f}{\partial\delta} -c\frac{\partial f}{\partial\varphi} \end{align*}