Differential
Let
be a differentiable map. To each , we associate a linear map called the differential of at and is defined as follows: For all , and be a differentiable curve such that , , then
Regular Surface
A subset
is a regular surface if
- for each
, there exists a neighbourhood of and a smooth homeomorphism such that is an open set. - For each
, the differential is injective. Such mapping
is called a local coordinate system or a chart of .