How to define the differential on a manifold ? How to prove a function is differentiable over a manifold ?
I've seen some definitions but can't get to a proper understanding of them.
To what i've understand , we could prove a function is differentiable over a manifold if it's a local restriction of a differentiable function over a vectorial space for example , but as i don't have any proposition or theorem to support on , that's complicated.
What would be a good way to define it ? Or , source for a good definition ,
Given two smooth manifolds $(M,\mathcal{A}_M)$ and $(N,\mathcal{A}_N)$ of dimensions $m$ and $n$ respectively, a function $f:M\to N$ is defined to be of class $C^r$ ($r \geq 0$ any integer, usually we consider $C^{\infty}$), if for every chart $(U,\alpha)\in\mathcal{A}_M$ and $(V,\beta)\in\mathcal{A}_N$ with $f(U)\subset V$, we have that the composite map \begin{align} \beta\circ f\circ \alpha^{-1}:\alpha[U]\subset \Bbb{R}^m\to \Bbb{R}^n \end{align}
is of class $C^r$ in the usual sense that one learns in multivariable calculus.
If this is your first time studying smooth manifolds, this can all be very abstract, so I highly recommend you watch this lecture series by Frederic Schuller on General Relativity. Particularly relevant for your question are lectures 4 and 5 (though if you have time I'd suggest you watch atleast lectures $1$-$6$). Following this, you should of course supplement it with any standard book on the subject (for example the first few chapters of John Lee's smooth manifolds book).