🎬Visualize the area element (the Jacobian determinant) helps illustrate how the transformation stretc

Visualizing the area element, quantified by the Jacobian determinant ( $|\operatorname{det}(J)|$ ), is highly instructive as it clearly maps the local stretching or compression of space caused by a coordinate transformation. Since these transformations are static, a side-by-side comparison of the resulting area maps is more beneficial than an animation. The color intensity in the visualization directly represents the magnitude of the Jacobian, which is the factor by which the infinitesimal area in the transformed space ( $d u d v$ or $d t d s$ ) must be multiplied to yield the true Cartesian area (d.A). For the Hyperbolic system, the Jacobian factor is $2|v|$, showing a linear increase in stretching (brighter color) as you move radially away from the origin (increasing $|v|$ ), with the lines being most stretched along the $x^1$ and $x^2$ axes. Conversely, the Parabolic system has a Jacobian of $t^2+s^2$, which is zero only at the origin and grows quadratically in all directions as you move away, resulting in a rapid, symmetric outward stretching of the parabolic grid lines.

Volume Elements in Non-Cartesian Coordinates (Jacobian Method) | Proof and Derivationviadean on Notion