# 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.

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