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## Homework Statement

1.) Prove that the infinitesimal volume element d

^{3}x is a scalar

2.) Let A

^{ijk}be a totally antisymmetric tensor. Prove that it transforms as a scalar.

## Homework Equations

## The Attempt at a Solution

[/B]1.) R

^{kh}= ∂x'

^{h}/∂x

^{k}

By coordinate transformation, x'

^{h}= R

^{kh}x

^{k}

dx'

^{h}= (∂x'

^{h}/∂x

^{k}) (∂x

^{k}/∂x'

^{j}) (∂x'

^{i}/∂x

^{j}) dx

^{j}

dx'

^{h}= δ

^{ih}R

^{ji}dx

^{j}

dx'

^{h}= R

^{jh}dx

^{j}

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?

2.)

A'

^{mnl}= ( T'

^{mnl}- T'

^{lnm})

= R

^{im}R

^{jn}R

^{kl}T

^{mnl}- R

^{kl}R

^{jn}R

^{im}T

^{lnm}

= R

^{im}R

^{jn}R

^{kl}(T

^{mnl}- T

^{lnm})

What should I do next here?