Hi everyone. I don't know whether this is an advanced or introductory topic but I I've always wondered how to prove symmetry arguments in electrostatics, magnetostatics etc mathematically. Suppose you have an infinite line charge and you need to calculate the electric field at some distance [itex]\rho[/itex] from its axis. Assume that I have absolutely no clue about symmetry and I write the electric field E as a sum of three components along the (ρ, [itex]\phi[/itex], z) unit vectors in cylindrical coordinates. Also I start with the assumption that each component of the field depends on the three cylindrical coordinate variables. I kinda get the physical intuition but I wonder if we can be more rigorous. How do I start ruling out and eliminating dependencies and get the answer? Do I need some knowledge of symmetry and group theory etc or could it be done by elementary methods? A few hints would be appreciated.