How should I interpret this notation of maps (diff geometry)

Question

I'm taking a course in general relativity (differential geometry).

Assume a smooth manifold $M$, where $x$ is a chart map. The following map (tangent vector) is defined: $$\frac{\delta }{\delta x^i}|_{p\in M}:C^\infty(M) \to \mathbb R$$ $$\frac{\delta }{\delta x^i}|_{p\in M}(f):=\delta_i{(f\circ x^{-1})}(x(p))$$

However, then the symbol is generalized using $TM=\{v_p | p\in M,v_p:C^\infty(M)\to \mathbb R \}$ as follows: $$\frac{\delta }{\delta x^i}:M\to TM$$ $$p \mapsto \frac{\delta}{\delta x^i}|_p$$

This means that $\frac{\delta }{\delta x^i}$ is a map that takes a point on the manifold, and outputs a tangent vector.

Here is my question:

However, the lecturer then continues and writes down $\frac{\delta }{\delta x^i}(f)$, where $f$ is a function on the manifold. This notation seems to not square with the definitions. Because now it is suddenly taking a function, whereas it was defined as taking a point on the manifold. Of course if we think loosely about this, the notation is reasonable, since $\frac{\delta }{\delta x^i}(f)$ outputs a map $M \to \mathbb R$, and I understand the meaning of this.

But from a strict mathematical point of view, is this notation meaningful? Are we simply allowed to insert an object into a map $x$ that is not the input of the map $x$ itself, but of the map that the map $x$ outputs?

Answer 1

I think your confusion lies when the evaluation is suppressed. Recall that in smooth manifold theory we identify tangent vectors with derivations at points in the manifold i.e linear maps from the tangent space to $\mathbb{R}$. So yes, $\partial^i: p \mapsto \partial^i(p) \in T_pM$, but now as $\partial^i(p)$ is a derivation at $p$, it acts on $C^{\infty}$ functions defined in neighborhoods of $p$. The notation $\partial^i(p)$ on a manifold is just notation, since the $\partial^i$ were created superficially to denote directional vectors and thus their action on smooth function should give the directional derivative. Using a chart $(U, \phi)$ we can give a well-defined definition of its value on smooth functions at $p$ which reflects what the notation should be doing:

$$\frac{\partial}{\partial x^i} \Bigr|_p f := \frac{\partial}{\partial r^i}\Bigr|_{\phi(p)} (f \circ \phi^{-1})$$

where $r^1,...,r^n$ denote the standard coordinates on $\mathbb{R}^n$.