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I'm new to tensors and am trying to understand how a change to the reference coordinate system affects the tensor. For example, if the coordinate system is cartesian and the tensor is this:

$$ \begin{bmatrix} \partial_xu_x & \partial_yu_x & \partial_zu_x\\ \partial_xu_y & \partial_yu_y & \partial_zu_y\\ \partial_xu_z & \partial_yu_z & \partial_zu_z \end{bmatrix} $$

And I, for example, 'reverse' the y-axis, or swap the x- and z- axes, how is this realised in the tensor?

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Let $T$ be your tensor. You can express your change of coordinates with a matrix $M$, i.e. the coordinates of point in the new system $x'$ are given by $x' = M x$ where $x$ are the coordinates in the current system. The tensor in the new system can be written as $T' = M T M^T$. You can find plenty of material on this if you search for 'tensor under transformation'.

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