The algebraic manipulations for changing variables in PDE/ODE problems are often very simple once you know the transformation to use (at least at my level it's just applying the chain rule carefully). However, the geometric meaning of even a very simple non-linear transformation is often extremely difficult to visualise.
As I'm just starting out on a second course in PDE's (though the first was engineering-based) I thought it might be beneficial to ask for advice on what should be going through my mind when transforming a PDE. In particular, is there any way to 'understand' a transformation without relying on geometric intuition far better than my own. Currently I feel like a computer just using rules to transform an equation.