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Let [itex]\varphi[/itex] be a one-parameter group on a manifold M, and let [itex]f[/itex] be a differentiable function on M, the derivative of f with respect to [itex]\varphi[/itex] is the defined as the limit:

[tex]\lim_{t\to 0} \frac{\varphi^*_t[f]-f}{t}(x)=\lim_{t\to 0}\frac{f\circ \varphi_x(t)-f\circ \varphi_x(0)}{t}=D_{\varphi_x}f=X(x)f,[/tex]

where [itex]X(x)[/itex] is a tangent vector at x and the operator [itex]D_\varphi[/itex] is defined as [itex]D_\varphi f=\frac{df\circ \varphi}{dt}\bigg|_{t=0}[/itex]

I don't understand why [itex]D_{\varphi_x}f=X(x)f[/itex]. According to the chain rule, I would get [itex]D_{\varphi_x}f=d_x f \circ d_0 \varphi(x)=X(x)d_x f[/itex]

[tex]\lim_{t\to 0} \frac{\varphi^*_t[f]-f}{t}(x)=\lim_{t\to 0}\frac{f\circ \varphi_x(t)-f\circ \varphi_x(0)}{t}=D_{\varphi_x}f=X(x)f,[/tex]

where [itex]X(x)[/itex] is a tangent vector at x and the operator [itex]D_\varphi[/itex] is defined as [itex]D_\varphi f=\frac{df\circ \varphi}{dt}\bigg|_{t=0}[/itex]

I don't understand why [itex]D_{\varphi_x}f=X(x)f[/itex]. According to the chain rule, I would get [itex]D_{\varphi_x}f=d_x f \circ d_0 \varphi(x)=X(x)d_x f[/itex]

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