- #1
binbagsss
- 1,326
- 12
I just have a quick question on which order around the numerator and denominator should be in the jacobian matrix that multiplies the expression.
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.
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.