# Definition Suppose $U$ and $V$ are overlapping [[Neighborhood|neighborhoods]] of a [[Manifold]] (i.e. $U \cap V \neq \emptyset$). Their intersection is open by definition. Suppose we have the [[Coordinate Chart|charts]] $(U, f)$ and $(V, g)$. Note that $f(U) \subset \mathbb{R}^n$ may be completely distinct from $g(V) \subset \mathbb{R}^n$. Thus, there is some equation relating these coordinate systems. Pick a point in the [[Domain, Co-domain, Image|image]] of the overlap under $f$ and call it $x^i$ as shown in the figures below. Since $f$ is a [[Types of Maps|bijection]], it has an inverse $f^{-1}$ that maps to a unique point in the overlap, $f^{-1}(x^i) = S$. Now take a look at the image of $S$ under $g(V)$, call it $y^i$, i.e. $g(S) = g(f^{-1}(x^j)) = y^i,\, i, j = 1 \ldots n$. In this way, we obtain a *coordinate transformation* $y^i = y^i(x^j)$ Or in a more verbose manner: $\begin{align} y^1 &= y^1(x^1, x^2, \ldots x^n) \\ y^2 &= y^2(x^1, x^2, \ldots x^n)\\ &\vdots \\ y^n &= y^n(x^1, x^2, \ldots x^n)\\ \end{align}$ ![[Pasted image 20210123181322.png]] ![[Pasted image 20210123181337.png]]