Manifold notes in more informal way

Question

When defining the properties of scalar functions that live in manifold $M$ in a less formal way, the following is said:

"We no longer refer to a covering by coordinate patches. Instead we conceive of the manifold as a set whose points may be described by many different coordinate systems, say ($x^0, x^1,...,x^{D-1}$) and ($x'^0,x'^1,...,x'^{D-1}$). Any 2 sets of coordinates are related by a set of $C^\infty$ functions, e.g $x^\mu(x^\nu)$ with non-singular Jacobian $\partial x'^{\mu}/\partial x^{\nu}$."

1) Why can we conceive of the manifold as a set whose points may be described by many different coordinate systems instead? What does this practically mean?

2) "Any 2 sets of coordinates are related by a set of $C^\infty$ functions, e.g $x^\mu(x^\nu)$ with non-singular Jacobian $\partial x'^{\mu}/\partial x^{\nu}$."

I don't get the example of $x^\mu(x^\nu)$ and how does it relate to what have been said just before it?

Answer 1

Usually the abstract definition of a manifold starts with the assumption that $M$ is a metrizable topological space, and continues with a description of the covering by coordinate patches.

Each coordinate patch has the form of an open subset $U \subset M$ together with a coordinate homeomorphism $\phi : U \to V \subset \mathbb{R}^D$ where $V \subset \mathbb{R}^D$ is an open subset. The "coordinate system" that is associated to this data is simply the coordinate system $(x^0,\ldots,x^{D-1})$ on $\mathbb{R}^D$ restricted to $V$ and pulled back via the function $\phi$ to the open set $U$.

One of the properties in the definition is that there exist a set of coordinate patches of this form, denoted $\{\phi_i : U_i \to V_i\}_{i \in I}$, such that $$\cup_{i \in I} U_i = M $$ It is quite possible, in fact it is almost inevitable, that there exist different coordinate patches $U_i,U_j$ such that $U_i \cap U_j \ne \emptyset$. So a point $p \in U_i \cap U_j$ will be described by two "different coordinate systems", namely the coordinates obtained by pulling back $\phi_i$, and the coordinates obtained by pulling back $\phi_j$. The "coordinate change map" (which you refer to using the notation $x^\mu(x^\nu)$) is then simply the map $$\phi_i(U_i \cap U_j) \xrightarrow{\phi_j \phi_i^{-1}} \phi_j(U_i \cap U_j) $$ and it is this map which is required to have nonsingular Jacobian.

For a simple example in 1-dimension where the Jacobian is just the ordinary derivative from first semester Calculus, think of two coordinate patches on the unit circle in the $xy$ plane, the first being the open upper semicircle using the $x$ coordinate, and the second being the open right semicircle using the $y$ coordinate. These two coordinate patches overlap in the open first quadrant of the circle, giving one coordinate $0<x<1$ and another coordinate $1>y>0$ which are related by the change of coordinate mapping $$y(x) = \sqrt{1-x^2} $$ whose Jacobian $\frac{d}{dx} \sqrt{1-x^2}$ is nonsingular on the interval $0<x<1$.