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How is one is to know that in Gauss's Law $$ \nabla \bullet \vec{E} = \dfrac{1}{\epsilon_0} \rho_f $$ that $$ \vec{E} = - \nabla \Phi $$ and is not $$ \vec{E} = -\nabla \Phi - \partial \vec{A} / \partial t $$?

Some people say that "this is a special case". Yes, exactly. But how in Maxwell's system of equations, where symbol "E" is shown in several different equations (Gauss's Law, Faraday's Law, Ampere's Law, Lorentz force equations) is one to know E is a special case and when E is not a special case?

Without explicit indication of when the special case is being invoked there is a mathematical problem.

Once a special case is invoked it must be stated explicitly so that it can be applied to

*all following equations*. Otherwise, you have no idea if special conditions are being invoked or not. So, as a complete mathematical/logical system, not as independent equations, isn't some indication (some notation) required to indicate special cases?

If I have a

*system*of equations that includes

$$ f = a x^2 $$

and

$$ f = a x^2 + b x^3 $$

where,

__in general__, b is not 0, isn't this a problem, because this says, in this

*system*of equations, that

$$ f = a x^2 \ne a x^2 + b x^3 = f $$

so then

$$ f \ne f $$

which is a contradiction for the general case b not equal to 0.

How is this handled in Maxwell's equations for E?