Stokes and skyrmion tensors and their application to structured light

We replace the familiar Stokes vector by a tensor. This allows us to introduce, for example, polar-coordinate components of the Stokes vector. From the tensor, we can derive the skyrmion field for mapping the polarization in structured light beams. These ideas have wider application in optics and in electromagnetic theory. We illustrate this with an example from non-paraxial optics and for the Poynting vector.