4 - Regular values and smooth manifolds

Summary

‘Where there is no imagination there is no horror.’

(A Study in Scarlet)

In an attempt to describe the world in which we live it is natural to try to produce mathematical models of the objects about us, so that we can study their geometry, compare their form, possibly predict their growth. Of course we study the static world by modelling it on Euclidean space E³, which we usually identify with ℝ³ together with its usual distance function. (Indeed, we take this model so much for granted that it almost ceases to be a model at all.)

How are we to describe the objects that appear inside this space? One fairly natural method is to model them on solutions of equations f(x) = c for maps f : ℝ³ → ℝ^p and points c ∈ ℝ^p. The next natural question to ask is: what type of maps f should we work with? Various suggestions come to mind: polynomial, analytic (given by convergent power series), differentiable, continuous …. Each has its own advantages and disadvantages. From our point of view polynomial and analytic functions are too rigid – they hamper the imagination too much. Using merely continuous functions one forgoes the powerful techniques of differential calculus. Differentiable functions are about the right compromise – actually we require our functions to possess derivatives of all orders; such functions are called smooth or C^∞.