defining a tangent vector v as the equivalence class of of curves: [tex]v = [\sigma] = \left. \frac{df(\sigma)}{dt} \right|_{t=0}[/tex], i want to show that this definition is independent of the member of the equivalence class that i choose.(adsbygoogle = window.adsbygoogle || []).push({});

where [tex]\sigma[/tex] represents a function from the reals to the manifold and f is a coordinate function from the manifold to the reals.

so i am starting with [tex]\sigma_{1} (0) = \sigma_{2}(0)[/tex]since i am picking two curves that go through the same point. i then define the functions [tex]F = \phi o \sigma_{1}[/tex] and [tex] G = \phi o \sigma_{2}[/tex] and since these are both functions from R to R my goal is to now show that F'(0) = G'(0). i also know that f(0) = g(0) since [tex]\sigma_1(0) = \sigma_2(0)[/tex] but now i am stuck and can't figure out the next step. can someone give me a few hints in the right direction? thanks.

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# Tangent vectors as equivalence classes of curves

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