A definition of directional derivative
Topics
About: \(C^\infty\) manifold
About: function
The table of contents of this article
Starting Context
- The reader knows a definition of \(C^\infty\) manifold.
- The reader knows a definition of germ of \(C^k\) functions at point.
- The reader knows a definition of curve.
Target Context
- The reader will have a definition of directional derivative.
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: Definition
For any \(C^\infty\) manifold, M, and any point, \(p \in M\), any map, \(D_v: C^1_p \rightarrow \mathbb{R}\) such that there is a \(C^\infty\) curve, c (t), \(p = c (t_0)\), such that \(D_v (f) = lim_{t \rightarrow t_0} \frac{f (c (t)) - f (p)}{t - t_0}\)
2: Note
The same directional derivative can be represented by many curves and can be especially by the straight line on any chart, because only the tangent of the curve at the point matters.
