📢Proof of the Contravariant Nature 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. By substituting the known transformation law for these vectors under a coordinate change, the derivation shows that the components in the new coordinate system that are related to the original components by the specific tensor transformation law. This result, with its two partial derivative terms in the numerator, is the hallmark of a contravariant tensor and proves the desired property.

Proof of the Contravariant Nature of the Inverse Metric Tensor | Audiosviadean on Notion