1.) Prove that the infinitesimal volume element d3x is a scalar
2.) Let Aijk be a totally antisymmetric tensor. Prove that it transforms as a scalar.
The Attempt at a Solution[/B]
1.) Rkh = ∂x'h/∂xk
By coordinate transformation, x'h = Rkh xk
dx'h = (∂x'h/∂xk) (∂xk/∂x'j) (∂x'i/∂xj) dxj
dx'h = δih Rji dxj
dx'h = Rjh dxj
This shows that the differential doesn't affect the transformation hence by performing the differential three times it would not affect the transformation, that is, it is a scalar. Can anyone verify if this is correct?
A'mnl = ( T'mnl - T'lnm )
= RimRjnRklTmnl - RklRjnRimTlnm
= RimRjnRkl(Tmnl - Tlnm)
What should I do next here?