Editing Homotopy principle (section)

=== Monotone functions ===
Perhaps the simplest partial differential relation is for the derivative to not vanish: <math>f'(x) \neq 0.</math> Properly, this is an ''ordinary'' differential relation, as this is a function in one variable. 

A holonomic solution to this relation is a function whose derivative is nowhere vanishing, i.e. a strictly monotone differentiable function, either increasing or decreasing. The space of such functions consists of two disjoint [[convex set]]s: the increasing ones and the decreasing ones, and has the homotopy type of two points.

A non-holonomic solution to this relation would consist in the data of two functions, a differentiable function f(x), and a [[continuous function]] g(x), with g(x) nowhere vanishing.  A holonomic solution gives rise to a non-holonomic solution by taking g(x) = f'(x).  The space of non-holonomic solutions again consists of two disjoint convex sets, according as g(x) is positive or negative.  

Thus the inclusion of holonomic into non-holonomic solutions satisfies the h-principle. 

[[File:Winding Number Around Point.svg|thumb|The [[Whitney–Graustein theorem]] shows that immersions of the circle in the plane satisfy an h-principle, expressed by [[turning number]].]]
This trivial example has nontrivial generalizations:
extending this to immersions of a circle into itself classifies them by order (or [[winding number]]), by lifting the map to the [[universal covering space]] and applying the above analysis to the resulting monotone map – the [[linear map]] corresponds to multiplying angle: <math>\theta \mapsto n\theta</math> (<math>z \mapsto z^n</math> in complex numbers). Note that here there are no immersions of order 0, as those would need to turn back on themselves. Extending this to circles immersed in the plane – the immersion condition is precisely the condition that the derivative does not vanish – the [[Whitney–Graustein theorem]] classified these by [[turning number]] by considering the homotopy class of the [[Gauss map]] and showing that this satisfies an h-principle; here again order 0 is more complicated.

Smale's classification of immersions of spheres as the homotopy groups of [[Stiefel manifold]]s, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes of maps of [[frame bundle]]s are much further-reaching generalizations, and much more involved, but similar in principle – immersion requires the derivative to have rank ''k,'' which requires the partial derivatives in each direction to not vanish and to be linearly independent, and the resulting analog of the Gauss map is a map to the Stiefel manifold, or more generally between frame bundles.