There are problems in classical electromagnetism where they ask you to find the electrical displacement given some geometry (like a sphere or a cylinder) and the dielectric constant ##\epsilon_r##. The solution to these problems typically employs symmetry arguments along with Gauss' laws for the D field. However, it got me thinking: Is this only valid in the case when the dielectric is linear? What justifies symmetry arguments for the D field in general? I have a vague intuition about the validity of this, but these arguments aren't as obvious as the case for the E field. I wouldn't like to say much more because I want a general argument that explains why exploiting symmetry works for the D field. You could take as an example the case of an infinite cylindrical capacitor with a dielectric in between. (Maybe its linear, maybe not) Why should the D field be radial in this case? I agree with you it has to look the same at whichever point separated a distance ##s## from the cylinder since the cylinder is infinite, but how would you counter someone who says that it's not perfectly radial but has some inclination as well? Thanks!