Isn't the following addition wrong on manifold as done in Frankel book?

Question

In ch-$4$ when calculating expression of Lie derivative using Hadamard's Lemma before $(4.4)$ Frankel's do following manipulation: $$\lim_{t\rightarrow0}\frac{\textbf{Y}_{\phi_tx}(f)-\textbf{Y}_x(f)}{t}$$ $$=\textbf{X}_x\{\textbf{Y}(f)\}$$ various symbol stands for

• $\textbf{Y}$, $\textbf{X}$ are vector field on manifold $M^n$
• $\phi_t$ is the flow associated with $\textbf{X}$
• $x$ is the point where we're taking the Lie derivative
• $f$ is a test function

My problem is how the subtraction is done of $\textbf{Y}$ at two different points. Intuitively the result make sense thats how vector differentiation is done in $\mathrm{R}^n$ but as far as I know we can add vectors only on the Tangent space of a point they're defined by the operator $\frac{d}{d\lambda}$. So does the flow $\phi_t$ here acts as the curve and $t$ can be called the parameter analogous to $\lambda$?

Answer 1

Remember that $\mathbf Y_x$ is the vector (not vector field) from $\mathbf Y$ at the point $x$. A vector eats a function and spits out a real number. As both $\mathbf Y_{\phi_t x}(f)$ and $\mathbf Y_x(f)$ are numbers, you can subtract them without issue.

Answer 2

The tangent space can be defined in multiple ways (see https://en.wikipedia.org/wiki/Tangent_space). I have not read Frankels book, but I assume from your post that he defines a tangent vector as a linear map of smooth functions $f$ on the manifold to $\mathbb{R}$, i.e. $\mathbf{Y}_x: C_{\infty}(M^n) \rightarrow \mathbb{R}$, so $\mathbf{Y}_x(f), \mathbf{Y}_{\phi_tx}(f)$ are just real numbers.
The equivalence with the tangent-space as defined by equivalence classes of curves is also explained in the wiki article. The identification of the tangent-space with tangential arrows attached to a (sub)manifold is more apparent with the curve definition.