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.

🎬Narrated Video

https://youtube.com/shorts/T6Apm9V4Uow?feature=share

🏗️Structural clarification of Poof and Derivation

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%% Proof and Derivation

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%% Proof and Derivation

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🗒️Downloadable Files - Recursive updates

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