Can we prove "Lumped element model" mathematically? Maybe this question is both in the fields of engineering and physics.
As it seems the electrical quantities like Resistance, Capacitance, Inductance, and so on are quantities we assign to distributed bodies(like solid cylinders, solid cubes, etc). But later on by assuming something named "Lumped element model of electrical components" to be correct, we easily localize these quantities to some single points and then easily draw schematic models of electrical systems with zero diameter lines as wires and pointy entities as resistors or capacitors and then it turns out the calculations are always correct.
My question is:
Is there any mathematical proof to this so called model? I mean how can we assume that for example the resistance of a solid disk is located at a point on (say) its center?
P.S. I think the proof should be in a manner like how we prove the forces exerted on a not rotating rigid body could be considered as just exerted on a point particle at the body's center of mass which has a mass equal to the body's total mass.
 A: The lumped elements approximation of electrical circuits uses the quasi-stationary approximation for the solution of Maxwell's equations. This means that the speed of electromagnetic field propagation c can be neglected (can be assumed to be infinite). Roughly this means that the dimensions $l$ of the circuit are much smaller than the vacuum wave length $l≪\lambda = c/f$ at the considered frequencies $f$.
A mathematical proof based on retarded potentials solutions of Maxwell's equations can be found, e.g., in chapter 4 of the textbook Ramo, Whinnery, van Duzer, "Fields and Waves in Communication Electronics, John Wiley & Sons Inc., 1994     
A: Whether a particular model is applicable in a physical situation cannot be proven mathematically. This can only be decided by comparing the predictions of the model with the results of experiments. As the saying goes "The proof of the pudding is in the eating."
If Maxwell's equations said the model was not appropriate while practical use says that it is, the result would not be that we abandon the model but that we look for some hidden factor which we failed to take into account when applying Maxwell's Equations. 
We do not use the lumped element circuit model because we are confident that it is a good approximation to Maxwell's Equations, we use it because it has been found to work to the level of accuracy to which it is commonly used. Applying Maxwell's equations only confirms what we already knew, and enables us to predict what the limitations of the model will be. (The model itself cannot make such predictions about its own limitations.)
What you are calling a mathematical proof is only a confirmation that Maxwell's Equations are consistent with our observation that the Lumped Element Model works.
