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The issue with the normal differentiation is it varies with coordinate system change.

Covariant differentiation fixes this as it is in tensor form and so is invariant under coordinate transformations.

'If a tensor is zero in one coordinate system, it is zero in all coordinate systems.'This is probably a stupid question, but following this, by 'invariant' do we mean you get the same value/matrix components? What's the significance of the form being unchanged?

Would you get different matrix components but scaled suitable i.e. according to the coordinate tranistion laws which are such that the computed tensor will have the same physical meaning in all coordinate systems , and a tensor that is zero, will remain zero, and a non-zero tensor will remain non-zero.

I'm not sure I understand the concepts...Are these thoughts correct?

Thanks in advance.