🎬how the metric tensor in polar coordinates is used to compute the circumference of a circle

The metric tensor, $g_{a b}$, defines how distances are measured in any coordinate system, especially non-Cartesian ones. A metric tensor with non-zero off-diagonal elements, as seen in the skewed grid example, signifies non-orthogonal axes, while a diagonal matrix with non-unity elements, like the $g_{\phi \phi}=\rho^2$ term in polar coordinates, reveals how coordinate components change in length with position. The inverse metric tensor, $g^{a b}$, is essential for calculations in such systems, underscoring that Cartesian coordinates are a simplified, special case where the metric tensor is just the identity matrix.

The Metric Tensor Covariant Derivatives and Tensor Densities | Condensed Notesviadean on Notion