Isotropic rank 3 pseudotensor help

[latentcorpse]

Can anybody show me how any isotropic rank 3 pseudotensor can be written as

a_{ijk}=\lambda \epsilon_{ijk}

for the isotropic rank 2 tensor case [i.e. a_{ij}=\lamda \delta_{ij} ], my notes prove it by considering an example i.e. a rotation by \frac{\pi}{2} radians about the z axis.

 

[latentcorpse]

anyone?

 

[Phrak]

No help yet? It's not clear to me how you're using indices. Are you writing vectors with lower indices?

Secondly, are you defining pseudotensors as tensors that change sign under a reflection of the coordinates such as (x,y,z)-->(-x,y,z)? There are two definitions of a pseudotensor.

Edit: Actually, it should be obvious from your question that you are using the first definition.

So the claim is that the only pseudotensors, whos elements remain unchanged under a general rotation, are equal to the totally antisymmetric tensor multiplied by some scaling factor.

 

[latentcorpse]

yes.
i guess then i meant that such a pseudotensor is proprotional to the levi civita rank 3 pseudotensor.

how does one actually show that though?

 

[Phrak]

They're not giving you much to go on, are they?

The easy problem (1) is to show that the elements of \lambda \epsilon_{ijk} are unchanged under othonormal transformations. The hard part is (2) showing that the elements of all rank 3 pseudotensors that remain unchanged are of the form \lambda \epsilon_{ijk}.

The only difference between tensors and pseudovectors in this proof is that you have to eventually account for an optional reflection of coordinates.

I would tackle the first problem first:


Tensors transform as the product of vectors.

An orthornormal transformation in three dimensions can be obtained as the product of the transformations that rotate vectors about the X, Y and Z axis.