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Demo:    • a type 2,0 tensor that is initially symmet...  

Derivation sheet: https://viadean.notion.site/Proving-S...

Summary: A tensor $T^{ab}$ is defined as symmetric if its components remain identical when its indices are swapped ($T^{ab} = T^{ba}$), and the sources establish that this symmetry is an intrinsic characteristic and a fundamental property that is preserved during coordinate transformations. To prove this invariance, one must apply the specific transformation rule for type (2,0) contravariant tensors, which involves the product of partial derivatives of the new coordinates with respect to the old ones. This mathematical framework demonstrates that even as the coordinate system rotates and individual components change, the symmetrical relationship between elements remains constant, ensuring that the transformed tensor $T^{\prime a^{\prime} b^{\prime}}$ is equal to $T^{\prime b^{\prime} a^{\prime}}$. Ultimately, this confirms that symmetry is a permanent quality of the tensor itself rather than a temporary feature of a specific representation or coordinate choice.