# A question on infinitesimal generator

Pages 120, 121 of W. Boothby's An Introduction to Differentiable Manifolds contain an explanation of infinitesimal generator, with the quotient:

$$\frac{1}{\Delta{t}}[f(\theta_{\Delta{t}}(p)) - {f}(p)]$$

Then the author assumes that $$\hat{f} (x^{1}, \ldots, x^{n})$$ is the local expression for $$f \in C^{\infty}(p)$$, and then he equals to the above quotient with

$$\frac{1}{\Delta{t}}[\hat{f}(h(\Delta{t}, x)) - \hat{f}(x)].$$

I am confused as to how these quotients may be equal.

I'd appreciate any useful explanation to help me out.

(I argue that $$\theta : \mathbb{R} \times M \to M$$ and $$f : M \to \mathbb{R}$$, so that $$f \circ \theta : \mathbb{R} \times M \to \mathbb{R}$$. On the other hand, $$h : \mathbb{R} \times \mathbb{R}^{n} \to \mathbb{R}^{n}$$ and $$\hat{f} : \mathbb{R}^{n} \to \mathbb{R}$$, which gives $$\hat{f} \circ h : \mathbb{R} \times \mathbb{R}^{n} \to \mathbb{R}$$.)

You put a chart $\phi:M\to\Bbb R^n$ in between, $$f∘θ=f∘\phi^{-1}∘\phi∘θ$$ and split in the middle, to get $$\hat f=f∘\phi^{-1},\\ h=\phi∘θ∘(id_{\Bbb R}, \phi^{-1})$$ with the described properties.