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In the proof of Proposition 9.7 of Lee's Introduction to Smooth manifolds, there is a certain notation which is giving me trouble.

We have a smooth map $\theta :\mathbb R\times M\to M$ and smooth function $f:M\to R$. What is the meaning of $$\left.\frac{\partial}{\partial t}\right|_{(0,p)}f(\theta(t,p))$$ Is there a more generalized form of this notation, where instead of $\mathbb R$ we have some other manifold?

I am aware of a similar notation, which is as follows. Let $M$ be a smooth manifold and $(U,\phi)$ be a smooth chart on $M$. Say $\phi$ is expressed as $(x_1,\ldots, x_n)$ in local coordinates. Then $\left.\frac{\partial}{\partial x_i}\right|_p$ is defined by

$$\left.\frac{\partial}{\partial x_i}\right|_pf=\left.\frac{\partial (f\circ\phi^{-1})}{\partial x_i}\right|_{\phi(p)}$$

where the las term is nothing but the usual partial derivative of ordinary calculus.

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Composing the function $f$ with $\theta$ gives another smooth function $f \circ \theta \colon \mathbb{R} \times M \to \mathbb{R}$, so let's work in a chart: the local coordinates will be $(t,x_1,\ldots,x_n)$, where $(x_1,\ldots,x_n)$ are local coordinates on $M$. Then, we can differentiate with respect to $t$ as we normally would to get the function $\frac{\partial}{\partial t} f \circ \theta$. Once again, this is a function on the product, so we would like to evaluate it at a point of $\mathbb{R} \times M$. Specifically, we're evaluating it at the point $t = 0$ and $p \in M$.

Let's look at an example where our manifold has only one chart: let $M = \{ (x,y) \in \mathbb{R}^2 \colon x^2 + y^2 < 1 \}$ be the unit disk in $\mathbb{R}^2$ and let's say the smooth function $f \colon M \to \mathbb{R}$ is given by $(x,y) \mapsto x^2 + y^2$. If $\theta \colon \mathbb{R} \times M \to M$ is the smooth function given by $(t,x,y) \mapsto (\sin(2t)\cdot x, \cos(5t)\cdot y)$, then the composite $f \circ \theta$ is $(t,x,y) \mapsto \sin^2(2t)x^2 + \cos^2(5t) y^2$. Computing partial derivatives, we get that $$ \frac{\partial}{\partial t} f(\theta(t,x,y)) = \frac{\partial}{\partial t} \left(\sin^2(2t)x^2 + \cos^2(5t)y^2 \right)= 4\sin(2t)\cos(2t)x^2 -10\cos(5t)\sin(5t)y^2$. $$ And then we can evaluate at any point $(t,x,y)$. (In general, we pass to a chart of our manifold, where we can perform the same calculations as above using just multivariable calculus.)

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