Suppose you have an infinite cylinder and are considering a field $\mathbf{D}$ caused by physical elements within the cylinder such that it satisfies $\int \mathbf{D}\cdot d\mathbf{a} = Q_{free}$. Where $Q_{free}$ refers to the so-called "free charges".
In mathematical terms, why does it follow that this field has to be radial (without resorting to physical arguments)?
I can see that because the cylinder is infinite the only thing which is needed to specify the field is the distance $s$ from the central axis. But I don't see why the solution can't have some non-radial components as well...
Is there any mathematical argument about symmetries that guarantees a specific form of a solution?
As an example of exploiting symmetries and knowing a priori the form of the solution is Laplace's equation $\nabla^2 V(x,y,z)=0$. If one sees that regardless of translations in $z$ the environment is always the same, then one can easily conclude the potential $V$ has no dependence on $z$.