Let be A an euclidean space and let be B a set whose elements form a coordinate system { $e_1, e_2, ... ,e_3$ }. We have to be clear that B elements are "Static" and any geometrical object represented on the last system is represented in any other system taking elements of the group of isometries in $R^n$ (Rotations, reflection and translations) and in that way any change of coordinates is only the action of the group on the basis vectors and define a differential calculus intrinsic to the space do not vary if we take the derivative on $x_0$ and $f(x_0)$ if we consider $f$ as the change of coordinates cause in fact we are not moving the frame of reference from one point to another, but in a non-static frame of reference like in polar coordinates for example where B = { $e_\theta, e_r$ } which vary from point to point I do not understand how differential calculus could be defined over a geometrical object in our euclidean space if we are taking different references fromaes at any two distinct points.
Could someone please explain me this ?
Let $M$ be a smooth Riemannian manifold (say, the plane). Let $p$ and $q$ be two points in $M$. You are wondering how we can do calculus on $M$ if the coordinate-induced basis of the tangent space $T_pM$ is not “the same” as the basis as the tangent space $T_qM$.
Suppose $p$ and $q$ are defined within the same neighborhood, so we can use the same coordinate system $\phi$ for both. The coordinate system, together with the Riemannian metric, defines a canonical tangent frame. When $\phi$ is the identity map on the plane, this gives the standard basis frame $\{e_1, e_2\}$. When $\phi$ is polar coordinates, you instead get $\{e_r, e_\theta\}$.
Now, the only reason you think the standard frame is “static” is that you are tacitly assuming the Levi-Civita connection on the plane. The connection is what allows you to say that the standard basis “stays the same” when viewed from either $p$ or $q$. What this really means is that the Christoffel symbols are trivial (that is, they are identically zero) for standard coordinates.
The reason you can do calculus with polar coordinates is that the Levi-Civita connection tells you how the polar frame changes as you move from $p$ to $q$. The Christoffel symbols are not zero for polar coordinates; they tell you exactly how the polar basis frame “twists” from tangent space to tangent space. So long as you keep track of this twisting, you can do calculus as usual, and the connection is the book-keeping device that helps you keep track.
(Note that the “twisting” of the polar frame reflects the curvilinear nature of the coordinates, not the curvilinear nature of the underlying manifold. The plane is not curved, after all: equipped with the standard Riemannian metric, it’s flat! The Levi-Civita connection allows you to account for — in mathier speak, mod out for — any curviness you introduce merely from the coordinate system. If you didn’t correct for the curviness introduced by the coordinates, you would distort your view of the underlying geometric object, like a stick looking bent when it’s submerged in water.)