🧄Transformation of the Inverse Metric Tensor

This is accomplished by starting with the fundamental definition of the inverse metric tensor in terms of the dual basis vectors, $g^{a b}=E^a \cdot E^b$. By substituting the known transformation law for these vectors under a coordinate change, the derivation shows that the components in the new coordinate system, $g^{l a b}$, are related to the original components by the specific tensor transformation law: $g^{\prime a b}=\frac{\partial x^a}{\partial x^{\prime c}} \frac{\partial x^b}{\partial x^{\prime a}} g^{c d}$. This result, with its two partial derivative terms in the numerator, is the hallmark of a contravariant tensor and proves the desired property.

Transformation of the Inverse Metric Tensor | Proof and Derivationviadean on Notion