2022-04-24

277: Directional Derivative

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A definition of directional derivative

Topics


About: \(C^\infty\) manifold
About: function

The table of contents of this article


Starting Context



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.


References


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