📢Impact of Coordinate Scaling on Metric and Cross Product

The problem demonstrated how coordinate scaling affects the geometry of space, starting with the transformation. This scaling leads to a diagonal metric tensor where only the one component is altered, 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, crucial for the cross product, is scaled. Consequently, the contravariant components of the cross product, 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