We have notions of derivative for a continuous and differentiable vector fields. The operations like curl,divergence etc. have well defined precise notions for these fields.
We know electrostatic and magneto static fields aren't actually well behaved. They blow up at the sources, have discontinuities and yet we use the same mathematical formulations for them as we would have done for continuous and differentiable vector field.
Why is this done ? Why are laws of electromagnetism(maxwell's equations) expressed in the so called differential forms when clearly that mathematical theory is not perfectly consistent with the electromagnetic field. Why not use a new mathematical structure ?
Is there a resource which can help me overcome these issues without handwaving at particular instances when the methods seem to give wrong results?
Also one of the major concerns is that, given a charge distributions, the maxwell equations in differential form, will always give a nicely behaved continuous and differentiable vector field solution. But the integral form (alone, not satisfying the differential form) can give a discontinuous solution as well. Leading to two different answers for the same configuration of charges. hence there is an inconsistency. Like there is an discontinuous solution for the boundary condition of 2D surface, the perpendicular component of the electric field is discontinuous. ( May be it is just an approximation) and actually the field is continuous but due to not being able to solve the differential equation we give such an approximation, but this isn't mentioned in the textbooks.