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I would like to really understand what the mathematical as well as Physical meaning of coordinate invariance is. I have pretended to know what this means, but upon thinking a little harder today, I am now convinced I don't know what it means. In any case, as far as I know calculations still are always done in some coordinates, in some space.

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    $\begingroup$ Coordinate invariance does not imply coordinate free. $\endgroup$ – user99917 Jan 13 '16 at 4:11
  • $\begingroup$ I was really wonder what does it mean. For example, consider the wave equation for $\psi$ in a coordinate called $S$. By Galilean transformation, in another coordinate $S^{\prime}$ it hasn't its form: $\psi$ doesn't satisfy the wave equation. However, one can define a new $\phi$ in $S^{\prime}$ does satisfy the wave equation but this $\phi$ depends on $\psi$ so the wave equation is not invariant by Galilean transformation. $\endgroup$ – heaven-of-intensity Jan 13 '16 at 8:08
  • $\begingroup$ Means that if you change coordinates, your equations still predict the same thing. For example, if your equations say that gravity is attractive in one coordinate but repulsive in the other, then clearly your equations are not coordinate invariant. $\endgroup$ – Horus Jan 14 '16 at 12:31
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Coordinate invariance in physics simply means that laws of physics are and should be independent of the coordinate system used. It does not imply coordinate free. This is because the laws of physics are written in the form of tensorial equations, and tensors are independent of the coordinate system used.

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Perhaps you could explain what exactly led you to question your understanding... The physical meaning of coordinate invariance is pretty simple. It's just that the laws of physics cannot depend upon your choice of coordinates as long as the reference frame you're working in is inertial. So if you were to perform a coordinate transformation from one inertial frame to another, the mathematical form of your equations of motion will not change and are said to be coordinate invariant.

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