Polarization and energy ellipsoids for an introductory visualization of tensors

Abstract

In The Feynman Lectures on Physics is discussed an introduction to tensors, specifically a way of ``visualizing'' the polarization tensor, by means of the energy ellipsoid, which is drawn by the electric fields which produce the same polarization energy density in an anisotropic crystal. Here, we discuss an alternative way of visualizing the polarization tensor, by means of the polarization ellipsoid, which is based on the ideas of Lam\'{e}'s stress ellipsoid and is drawn by the polarization vectors produced by electric fields having the same magnitude. We compare both ellipsoids as a first introductory way of visualizing the polarization tensor.