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Given a Lie groupGand a smooth pathγ:[-ε,ε]→Gcentered atg∈G(i.e.,γ(0)=g), and assuming I have a chartΦ:G→U⊂ℝ, how do I define the derivative [itex]\frac{d\gamma}{dt}\mid_{t=0}[/itex] ?^{n}

I already know that many books define the derivative ofmatrix Lie groupsin terms of an "infinitesimal change" between matrices, but I still have troubles accepting that definition because such an infinitesimal change involves the calculation of adifferencebetween matrices, while it is assumed that the only binary operation we can perform between elements of a matrix Lie group ismatrix multiplication.

The answer I am looking for should be valid for general Lie groups and it should be general enough to contain the definition of derivative of matrix Lie groups as a special case.

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# A Derivative of smooth paths in Lie groups

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