🎬the impact of the Jacobian on the distributions of both energy and momentum

The Jacobian is essential for conserving the number of particles when you change variables in a distribution function. Without it, the mathematical descriptions in the two different domains (energy and momentum) would be inconsistent, and the total number of particles would not be conserved. The demo visually proves this by showing that only when the Jacobian is included do the areas under both the energy and momentum curves match, confirming that they describe the same physical reality.

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