🎬a 2D type (1,1) tensor under rotation under both linear transformation and non-linear transformation

The defining characteristic of a true Type (1,1) Tensor Transformation is physical invariance: the resulting vector fields (the "output") must look the same regardless of the coordinate system chosen. The partial derivative ( $\partial_a v^b$) fails this test because its transformation rule is incomplete, lacking the necessary non-tensorial correction term (the second-order derivatives of the coordinate map). While this failure is hidden in simple linear transformations (like rotation), the Non-Linear Shear demo reveals the fundamental flaw: the transformed components of the partial derivative yield a physically incorrect result (divergence) because the coordinate grid's changing orientation is not properly accounted for. The covariant derivative ( $\nabla_a v^b$ ) is the necessary remedy, as it adds the Christoffel symbol correction (which exactly cancels the nontensorial term), ensuring that the result is always a true tensor and thus, physically invariant.

Tensor vs. Non-Tensor Transformation of Derivatives | Proof and Derivationviadean on Notion