# What's the difference between deviatoric tensor and skew tensor?

I read two books about  deviatoric tensor and skew tensor, but they have diffrerent denote.

One says the deviatoric tensor is decomposed from the symmetric tensor; the other one state that  from any tensor.

So, I was confused!

### Deviatoric and Skew Tensors

There two different additive decompositions of a tensor: Symmetric & Skew, and Spherical & Deviatoric. The Skew and Deviatoric tensors are both traceless (The first principal invariant is zero in each case). A Skew tensor is antisymmetric and has only zero elements along the diagonal when represented by the components provided by the Cartesian coordinate system. The diagonal components of a deviatoric tensor are not necessarily zero; but in order for the tensor to still be traceless, the sum of these add to zero.

Symmetrical as well as shpherical tensors are also both symmetric in the sense of being indistinguishable from their transposes. In the case of spherical tensors, off diagonal elements (Cartesian Representation) all vanish. In regualr symmetric tensors that are not spherical, off diagonal elements do not necessarily vanish. 