Transformation

Question Why to transform?

Answer The transform equation is in simpler form, and subsequent analysis of solving the equations is easy.

Definition 7 (Change of coordinates/a non-singular transformation)   The transformation $(\xi, \eta) = (\xi(x, y), \eta(x,y)$ is called a change of coordinates (or a non-singular transformation) if the Jacobian of transformation $J = \dfrac{\partial(\xi, \eta)}{\partial(x, y)} := \xi_{x} \eta_{y} - \xi_{y} \eta_{x}$ does not vanish at any point $(x,y)$.

Does the type of PDE change with transformation?