- #1

Andrea

I consider only Cartesian tensors in the following. The definition of

isotropic tensor function I know is

1) T = F ( G )

such that, for any rotation ( ' = transpose),

2) O F( G ) O' = F( O G O' )

But, if I change to component notation, it seem to me that any tensor

function is isotropic, which cannot obviously be. Denoting the

components in the new basis with ^*, I have

3a) T_ij^* = O_ir T_rs (O_sj)'

3b) G_ij^* = O_ir G_rs (O_sj)'

since T and G are tensors. Then, by 1),

4a) T_rs = F_rs ( G_mn )

4b) T_rs^* = F_rs^* ( G_mn^* )

Then, substituting 3a) and 3b) into 4b), I get

5) O_ir T_rs (O_sj)' = F_rs^* ( O_mk G_kl (O_ln)' )

Finally, substituting 4a) into 5), I have

6) O_ir F_rs ( G_mn ) (O_sj)' = F_rs^* ( O_mk G_kl (O_ln)' )

that is,

O F( G ) O' = F (O G O' )

So any tensor function would be isotropic. Clearly that's false, but I

don't see where the error is. Can you help me find it? Thanks,

Andrea