Also in the following line for equation2

We have "w=(âˆ‚tâˆ’câˆ‚x)u=utâˆ’cux"

Shouldn't it be "w=(âˆ‚tâˆ’câˆ‚x)u=wt - wux"

But why would the w come into the equation here? I think we are saying that \begin{equation}

w = (\partial_t - c\partial_x)u = \partial_tu - c\partial_xu \end{equation}

As in, we are expanding what is in the brackets so that it acts on u, and thus this is what w is equal to. Maybe I am misunderstanding?

Edit: I agree with the original post though, I think a u is missing.