Given is a curve [tex]\gamma[/tex] from [tex]\mathbb{R} \rightarrow M[/tex] for some manifold M. The tangent to [tex]\gamma[/tex] at [tex]c[/tex] is defined as(adsbygoogle = window.adsbygoogle || []).push({});

[tex](\gamma_*c)g = \frac{dg \circ {\gamma}}{du}(c)[/tex]

Now, the curve is to be reparameterized so that [tex]\tau = \gamma \circ f[/tex], with f defining the reparametrization. (f' > 0 everywhere)

The book I'm reading claims that [tex]\tau_* = f' \cdot \gamma_* \circ f[/tex], however I do not quite see how this result is derived.

Using the chain rule, I get

[tex]

(\tau_*c)g = \frac{dg \circ \gamma \circ f}{du}(c) =\frac{dg \circ \gamma \circ f}{df} \cdot \frac{df}{du}(c)

[/tex]

The second part of the rhs is obviously f', but how is the first part equal to [tex]\gamma_* \circ f[/tex]?

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# Tangent to reparameterized curve

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