# 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.

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Re: Tensors

anyone?

#### Phrak

Re: Tensors

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.

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#### latentcorpse

Re: Tensors

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

Re: Tensors

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.

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"Isotropic rank 3 pseudotensor help"

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