📢Defining the Transformation Law for a Type (0,2) Tensor Through Scalar Invariance

The components of a type ( 0,2 ) tensor, are defined by how they transform under a change of coordinates. The proof demonstrates that if the expression is a scalar, meaning it remains unchanged during a coordinate transformation, then the components must transform in a specific way. This transformation rule, derived from the invariance of the scalar and the known transformation laws for vectors, is the defining characteristic of a type ( 0,2 ) tensor. Essentially, the behavior of the scalar product dictates the behavior of the components, proving that them are indeed the components of such a tensor.

Proof of a Tensor's Invariance Property | Proof and Derivationviadean on Notion