🧄Proof of a Tensor's Invariance Property

The components of a type ( 0,2 ) tensor, $T_{a b}$, are defined by how they transform under a change of coordinates. The proof demonstrates that if the expression $T_{a b} v^a w^b$ is a scalar (meaning it remains unchanged during a coordinate transformation), then the components $T_{a b}$ 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 whole (the scalar product) dictates the behavior of the parts ( $T_{a b}$ ), proving that $T_{a b}$ are indeed the components of such a tensor.

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