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Specifically, I went through eqn.s 93.29-93.38.

However the sign of the Levi-Civita Symbol is bugging me:

It says that in 4D Euclidean space,

[tex]\epsilon^{1234}=+1[/tex] in Cartesian coordinates

implies

[tex]\epsilon^{\rho \chi \psi \phi}=-1[/tex]

in spherical coordinates

below eqn 93.17, by computing the Jacobian of coordinate change.

But from my calculation, the Jacobian is positive, simply

[tex]\rho^3 sin^2 \chi sin\psi[/tex]

In the simplest case, e.g. 2D or 3D, it seems to me that the Levi-Civita symbol doesn't change sign either under this coordinate transformation.

Can anyone explain why the sign of the Levi-Civita symbol is changed?

Thanks!