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I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment:

The effect of an infinitesimal coordinate transformation on any tensor $T$ is that the new tensor equals the old tensor at the same coordinate point, plus the Lie derivative of the tensor.

Is there a straightforward way to see this?

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    $\begingroup$ Consider vector fields as the generators of diffeomorphisms. The Lie derivative is the induced representation of the generators. $\endgroup$ – ACuriousMind Apr 7 '15 at 13:54
  • $\begingroup$ Note lie derivative is only defined for (n, 0) and (0, n) tensors. $\endgroup$ – zzz Jun 10 '15 at 0:01
  • $\begingroup$ Maybe say something about the context of the quote, im guessing by infinite small coordinate transformation he means infinite small step along the flow generated by a vector field $\endgroup$ – zzz Jun 10 '15 at 0:05

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