The differential $df_x$ of a map $f$ between two Euclidean spaces is the linear approximation which satisfies $f(x + v) = df_x(v) + o(|v|)$. The meaning of the word "approxiamtion" is clear here. I'm struggling to get what is the meaning of the differential of a map between two smooth manifolds as an approximation. In general there is no concept of a distance between two points in a manifold or a length of a vector in a tangent space of a manifold. Then how could we say about approximation precisely? In what sense does the tangent space at a point of a manifold locally approximate the manifold? In what sense does the differential of a map between smooth manifolds approximate the map?
[Math] Meaning of the differential of a map between manifolds as an approximation
differential-geometrymanifolds
Best Answer
Given $f: M \to N$, we use charts $(U, \phi)$ and $(V, \psi)$ for $U,V$ respectively such that $\phi(U) \cap V \not = \emptyset$. If you wish to do calculus in the traditional sense then we use $\tilde{f}: \phi(U) \subset \mathbb{R}^n \to \phi(V) \subset \mathbb{R}^m$ defined by,
$$ \tilde{f} = \psi \circ f \circ \phi^{-1}$$
If you like, you could just write that $\tilde{f}$ is just $f$, how? Well, $\psi(\textbf{x})$ is just the coordinate representation of $\textbf{x}$ in $U$ and if we let $\psi = (y^1,...,y^m)$ then $\psi(f(\textbf{x})) = (y^1(\textbf{x}),...,y^m(\textbf{x}))$ is just the coordinate rep. of $f$ in $V$. Hence, if the use of a chart is understood then we just write,
$$f(x^1,...,x^n) = (y^1,...,y^m)$$
The moral is, in order to make sense of smoothness, we use charts. The idea behind smooth manifold theory is that if you always assume that you manifold is embedded in some ambient space then you're probably deducing things about your space as a result of being in another i.e you've lost focus of the primary object, your manifold.
To counteract this though, you consider your manifold as an abstract space with just a differentiable structure and a few other things, but you never mention a metric or anything of that kind. You now run into a problem if you want to do traditional calculus, but if you make the right identifications there may be a chance. The above is an illustration of some of these identifications.