Ok for the purpose of this question let's stick to the flux integral: The general formula is ∫∫s (E-vector)*(dS-vector)=Flux where * stands for the dot-product. Now, I like it when my integrals make sense, and to do that I usually think of the Riemann Sum which might represent my integral. The problem I have though is this one: I can think of cases where this expression for the flux integral holds true, but I'm mainly thinking of very simple surfaces and vector fields. I just don't see how you can know this would be the expression of the flux integral considering the most complicated vector field you can think of on the most irregular surface you can imagine. Namely, I'm looking for the general argument that makes it necessary that the flux is "the double integral over some surface of E*dS". What is the rationale behind "Oh, sure it's simply the double integral of the dot product of the field and some differential of area". I'm specifically troubled by the double integral, what made people think the double integral was the operation needed? I can see how it is needed when I can think of a Riemann sum which represents the double integral where I have ΔxΔy or something on all terms, but I can't see this in a general way. I hope my question is clear, and thanks for helping me out. TL;DR: why is the flux the double integral of the dot product of the field and a differential of surface. I specifically want to know where the double integral came from.