Editing Diffeomorphism (section)

===Surface deformations===
In [[mechanics]], a stress-induced transformation is called a [[deformation (mechanics)|deformation]] and may be described by a diffeomorphism.
A diffeomorphism <math>f:U\to V</math> between two [[Surface (topology)|surface]]s <math>U</math> and <math>V</math> has a Jacobian matrix <math>Df</math> that is an [[invertible matrix]]. In fact, it is required that for <math>p</math> in <math>U</math>, there is a [[neighborhood (topology)|neighborhood]] of <math>p</math> in which the Jacobian <math>Df</math> stays [[Invertible matrix|non-singular]]. Suppose that in a chart of the surface, <math>f(x,y) = (u,v).</math>

The [[total differential]] of ''u'' is
:<math>du = \frac{\partial u}{\partial x} dx +  \frac{\partial u}{\partial y} dy</math>, and similarly for ''v''.
Then the image <math> (du, dv) = (dx, dy) Df </math> is a [[linear transformation]], fixing the origin, and expressible as the action of a complex number of a particular type. When (''dx'',&thinsp;''dy'') is also interpreted as that type of complex number, the action is of complex multiplication in the appropriate complex number plane. As such, there is a type of angle ([[angle|Euclidean]], [[hyperbolic angle|hyperbolic]], or [[slope]]) that is preserved in such a multiplication. Due to ''Df'' being invertible, the type of complex number is uniform over the surface. Consequently, a surface deformation or diffeomorphism of surfaces has the '''conformal property''' of preserving (the appropriate type of) angles.