Having investigated, in such a thorough manner, the geometry of curves, and in particular (or, rather, in general!) generic curves, it is natural to enquire about the geometry of surfaces. Will the same methods yield similar results? In short, the answer is yes. However, the classification of functions of several variables is much more complicated than the single variable case, where a few elementary ideas and explicit changes of coordinates sufficed to produce our list. This is the charm of functions of one variable, the charm that originally encouraged us to write this book. But this simplicity is misleading, and in this chapter we will introduce the remaining basic technique of real singularity theory, sometimes called the Mather yoga of vector fields.

We shall concentrate our efforts on the classification problem, and not deal with the question of versal unfoldings. We have taken this approach largely because we did not want this chapter to become absurdly long, but also because the key new ideas are exhibited in a simple setting. Also many of the other books available are rather short of good explicit calculations and the technical tools needed to make classifications.