🎬how a quantity's value changes with a change in the coordinate system by visualizing the difference

This 3D demo is that a physical quantity's measurement can change depending on your chosen coordinate system, revealing if it's a scalar or a scalar density. The Physical Density ( $\rho$ ) of the sphere is a true scalar (a rank-0 tensor), and its value remains constant because it represents the mass per physical volume, which doesn't change. The Coordinate Density ( $\tilde{\rho}$ ), however, is a scalar density (a rank- 0 tensor density with weight 1). Its value changes as you stretch or compress the grid because it represents the mass per coordinate volume. This change is directly proportional to the Jacobian Determinant (J), which quantifies the scaling of the coordinate system. The demo visually proves this relationship: $\overline{ \rho }= \rho \cdot J$.

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