# Confusion about the notation of directional derivative

by Oliver G   Last Updated August 10, 2019 03:20 AM

From An Introduction to Manifolds by Tu:

$$(1)$$ Let $$D_v = \sum v^i\frac{\partial}{\partial x^i}|_p$$ where $$v = [v^1, \dots, v^n]$$ is a vector in $$\Bbb R^n$$ and $$p = (p^1, \dots, p^n)$$ a point in $$\Bbb R^n.$$ Then $$D_v$$ is a map that sends a function to a number $$D_vf$$.

Let $$C^{\infty}_p$$ be the set of all germs of $$f$$ at $$p$$, where a germ is an equivalence class of pairs $$(f, U)$$ where $$U$$ is a neighborhood of $$p$$ in $$\Bbb R^n$$ and $$(f,U)$$ is similar to $$(g, V)$$ if and only if there's an open subset $$W \subset U \cap V$$ containing $$p$$ such that $$f=g$$ when restricted to $$W$$. Each $$f$$ is a $$C^{\infty}$$ function.

Tu then writes:

For each tangent vector at a point $$p$$ in $$\Bbb R^n$$, the directional derivative at $$p$$ gives a map of real vector spaces $$(2)\space D_v: C^{\infty}_p \rightarrow \Bbb R.$$

$$D_v$$ is $$\Bbb R$$-linear and satisfies the Leibniz rule $$D_v(fg) = (D_vf)g(p) + f(p)D_vg.$$

To me, this looks like Tu's giving two different definitions in $$(1)$$ and $$(2)$$. The first is a function from a space of functions, and the second is a function from the set of all germs at $$p$$.

But if he's using the second definition, how is $$D_v(f)$$ defined? $$D_v$$ should map an equivalence class $$[(f,U)]$$ to the real numbers, but I don't understand what the operation is on equivalence classes or how to show this equality on equivalence classes.

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