For simplicity, we consider a 2-dimensional manifold (a surface) with coordinates axis \(x^1\),$x^2$. consider an infinitesimal shift from the point ($x^1$,$x^2$). $dx^1$ and $dx^2$ along $x^1$ and $x^2$, respectively. The formula for converting coordinate labels to distances was provided by the great mathematician Carl Friedrich Gauss as follows:
The famous equation is \((E=mc^2)\).
my code \(x^1\)
Note: $dx^1$ and $dx^2$ are coordinates differences and not distances.\\
In modern scientific texts, the standard form of this formula, expressed using Einstein’s summation convention, is as follows:
$a=g_{11}$ , $b=g_{12}$ , $c=g_{21}$ , $d=g_{22}$ ,
\begin{equation}
ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu} \hspace{1cm} \mu \in 1,2 \hspace{0.5cm} \nu \in 1,2
\end{equation}