# Thread: verify that a skew symmetric tensor transforms ?

1. ## verify that a skew symmetric tensor transforms ?

If $A_{ij}$ is a skew-symmetric tensor,verify that
$B_{ijk}=A_{ij,k}+A_{ki,j}+A_{jk,i}$ transforms as a tensor.

I have searched far and wide to find an example of verifying that an expression transforms as a tensor,without luck. I suppose that i show it takes the form of a covariant transformation $\bar{B}_{ijk}=\frac{\partial x^{l}}{\partial \bar{x}^{i}}\frac{\partial x^{m}}{\partial \bar{x}^{j}}\frac{\partial x^{n}}{\partial \bar{x}^{k}}B_{lmn}$ Where $B_{lmn}=B_{lm,n}+B_{nl,m}+B_{mn,l}$ ?

Where will i use the skew symmetric property for this question?

And can i even say $\bar{B}_{ijk} = \bar{A}_{ij,k}+\bar{A}_{ki,j}+\bar{A}_{jk,i}$ ? Or is this illegal?
I am an idiot and i need your help.

P.S Does anyone know where i can actually see examples worked out to do with tensors? I am doing self study,and have study material on differential geometry and relativity,where the lecturers decided it would be best not to include any worked out examples or any of those nasty illustrations that confuse us so. If you know of a good site please let me know.

2. I'm not sure, but I think you must check that you get "minus the expression" when you interchange two indices.
$
A_{ji,k}+A_{kj,i}+A_{ik,j} = -B_{jik}
$

You must just use the fact that $
A_{ij}
$
is a skew-symmetric tensor as an intermediate step.