277: Directional Derivative

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A definition of directional derivative

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Topics

About: $C^{\mathrm{\infty }}$ manifold
About: function

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Definition
• 2: Note

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Starting Context

• The reader knows a definition of $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of germ of $C^k$ functions at point.
• The reader knows a definition of curve.

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Target Context

• The reader will have a definition of directional derivative.

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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.

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Main Body

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1: Definition

For any $C^{\mathrm{\infty }}$ manifold, M, and any point, $p\in M$, any map, $D_v:C_p^1\to \mathbb{R}$ such that there is a $C^{\mathrm{\infty }}$ curve, c (t), $p=c(t_0)$, such that $D_v(f)=li{m}_{t\to t_0}\frac{f(c(t))-f(p)}{t-t_0}$

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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.

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References

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