# Proof of euclidean distance

HallsofIvy
$$\mathrm{d}s^2=\mathrm{d}x_i\mathrm{d}x^i$$
It finally dawned on me that Gernuk did NOT write $ds^2= dx^idx^i$, he wrote $ds^2= dx_idx^i$. If the space is NOT Euclidean then $dx_i= g_{ij}dx^j$ so what he really wrote was that $ds^2= g_{ij}dx^idx^j$ the general formula for a metric tensor.