Given a curve ##\gamma: I \to M## where ##I\subset \mathbb{R}## and ##M## is a manifold, the tangent vector to the curve at ##\gamma(0) = p \in M## is defined in some modern differential geomtery texts to be the differential operator(adsbygoogle = window.adsbygoogle || []).push({});

$$V_{\gamma(0)}= \gamma_* \left(\frac{d}{dt}\right)_{t=0}.$$

However, quite often when I read differential geometry texts I encounter expressions like

$$V_{\gamma(0)}= \left(\frac{d}{dt}\right)_{t=0} \gamma(t).$$

What is the meaning of the latter expression, and what is the relation between the two?

The only relation I can think of is that in a coordinate chart ##(U,\phi)## on ##M## and for ##f \in C(M)## we have

$$V_{\gamma(0)}f= \gamma_* \left(\frac{d}{dt}\right)_{t=0}f = \left(\frac{d}{dt}\right)_{t=0} f\circ \gamma(t) = \left(\frac{d}{dt}\right)_{t=0} f\circ \phi^{-1} \circ (\phi \circ \gamma(t)) = \frac{dx^\mu}{dt}(\gamma(0)) \left(\frac{\partial}{\partial x^\mu}\right)_{x^\mu(\gamma(0))} f\circ \phi^{-1}$$

where I have defined ##\phi\circ \gamma(t) = x^\mu(\gamma(t))##. Is the latter expression only meant to be a shorthand for

$$V_{\gamma(0)} = \frac{dx^\mu}{dt}(\gamma(0)) \left(\phi^{-1}_*\frac{\partial}{\partial x^\mu}\right)_{\gamma(0)}?$$

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# Tangent vector to curve - notational confusion.

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