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I'm new to tensors and what I've learned about them is that they are objects that are 'invariant' under transformations, is this true for all ranks of tensors? If I got $n$ functions $A^i$ where $i = 1,2,3....,n$ in the $S$ coordinate for example and would like to transform these functions to $\bar A^i$ to the $\bar S$ coordinate, by the law of transformation:

$\bar A^i = \frac{\partial{\bar x^i}}{\partial{x^j}} A^j$ [1st rank tensor transformation]

But according to my understanding, if tensors are invariant the tensor $A^i$ before the transformation should equal $\bar A^i$ , is this correct? Do I even make sense in anything? Thanks

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