I just have a quick question on which order around the numerator and denominator should be in the jacobian matrix that multiplies the expression.(adsbygoogle = window.adsbygoogle || []).push({});

As in general Lecture Notes on General Relativity by Sean M. Carroll, 1997 he has the law as

##

\xi_{\mu'_{1}\mu'_{2}...\mu'_{n}}=|\frac{\partial x^{\mu'}}{\partial x^{\mu}} | \frac{\partial x^{\mu_{1}}}{\partial x^{\mu'_{1}}}\frac{\partial x^{\mu_{2}}}{\partial x^{\mu'_{2}}}...\frac{\partial x^{\mu_{n}}}{\partial x^{\mu'_{n}}} \xi_{\mu_{1}\mu_{2}...\mu_{n}}##

whereas on wiki the law is

##\xi^{\alpha}_{beta}= |[\frac{\partial \bar{x}^{t}}{\partial x^{\gamma}}]|\frac{\partial x^{\alpha}}{\partial \bar{x}^{\delta}}\frac{\partial \bar{x}^{\epsilon}}{\partial x^{\beta}}\bar{\xi}^{\delta}_{\epsilon}##

So both sources seem to have the matrix the other way around relative to the 'orginal' tensor and what is being transformed.

Does the order not matter?

Thanks.

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# Quick question tensor density transformation law

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