On the differential of a mapping

From on Warner’s book.

Smooth curve on manifold {M}:

A {C^\infty} mapping {\alpha : (a, b) \rightarrow M}. Let {t \in (a, b)}, we define the tangent vector of the curve {\alpha} at {t} is the vector

\displaystyle d\alpha\bigg(\dfrac{d}{dt}\bigg|_{t = 0}\bigg) \in T_{\alpha(0)}M.

we apply the formula

\displaystyle d\psi(v)(g) = v(g \circ \psi),

where {g} is an any function on {M}.

Put {\psi = \alpha(t)} and {v = \dfrac{d}{dt}\alpha(t)\big|_{t = 0}}, the above formula implies

\displaystyle d\alpha(\frac{d}{dt}\big|_{t=0})(f) = (\frac{d}{dt}\big|_{t=0})(f \circ \alpha) = \frac{d}{dt}(f \circ \alpha)\big|_{t=0}.

This is directional derivative.

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