definition of map from open subset of Euclidean \(C^\infty\) manifold into subset of Euclidean \(C^\infty\) manifold differentiable at point
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean \(C^\infty\) manifold.
- The reader knows a definition of map.
Target Context
- The reader will have a definition of map from open subset of Euclidean \(C^\infty\) manifold into subset of Euclidean \(C^\infty\) manifold differentiable at point.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( \mathbb{R}^{d_1}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\( \mathbb{R}^{d_2}\): \(= \text{ the Euclidean } C^\infty \text{ manifold }\)
\( U\): \(\in \{\text{ the open subsets of } \mathbb{R}^{d_1}\}\)
\( S\): \(\in \{\text{ the subsets of } \mathbb{R}^{d_2}\}\)
\( p\): \(\in U\)
\(*f\): \(: U \to S\)
//
Conditions:
\(f \text{ 's all the } 1 \text{-st partial derivatives exist at } p\)
//
2: Note
\(f\) is required to be from an open subset of \(\mathbb{R}^{d_1}\) in order for the derivatives of \(f\) to exist at \(p\).
There is a definition for the case that the domain is an arbitrary subset of \(\mathbb{R}^{d_1}\).
\(S\) does not need to be open on \(\mathbb{R}^{d_2}\), because taking the derivatives does not require \(S\) to be open.
