🎬how the metric tensor changes with the geometry of a coordinate system

The 3D visualization demonstrates that the metric tensor acts as a dynamic blueprint for a coordinate system's geometry. While the diagonal components of the tensor stay constant at one (since the basis vectors have unit length), the off-diagonal components become non-zero when the grid is skewed, with their values directly encoding the angles between the non-orthogonal axes. This shows how the tensor precisely captures the distortions of the space, allowing for calculations of distances and angles even in a non-Cartesian system.

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