🧄Metric Determinant and Cross Product in Scaled Coordinates

The problem demonstrated how coordinate scaling affects the geometry of space, starting with the transformation $y^3=2 x^3$. This scaling leads to a diagonal metric tensor where only the $g_{33}$ component is altered, becoming 1 / 4, resulting in a metric determinant of $g=1 / 4$. The key implication is how this value scales the vector calculus operations: the Levi-Civita density $\eta^{a b c}$, crucial for the cross product, is scaled by $1 / \sqrt{g}=2$. Consequently, the contravariant components of the cross product, $(v \times w)^a=\eta^{a b c} v_b w_c$, are simply twice the magnitude of the standard Cartesian cross product involving the covariant components of the vectors, illustrating the general principle that all tensor operations in non-Cartesian coordinates must incorporate factors derived from the metric determinant.

Metric Determinant and Cross Product in Scaled Coordinates | Proof and Derivationviadean on Notion