Applicability of the concept of voltage in electrodynamic circuits In electrostatics, we have $$\nabla \times \vec{E} = 0$$. Hence, we can define a scalar potential $V$, where $$\vec{E} = -\nabla V$$. We know from Faraday's law that $$\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$. In situations involving constant currents(D.C.) the rate of change of the magnetic field is zero, so we can continue to use the concept of scalar potential, also known as voltage in circuit theory, for analysis of DC circuits. However,in practice, in the literature, we find the concept of voltage being used to study alternating current circuits as well. Further, it is also used in the study of high frequency alternating current circuits used for generation of electromagnetic waves. In such situations $$ \oint \vec{E}.\vec{dl} \neq 0$$. So strictly speaking, we cannot use the concept of voltage to study alternating current circuits, especially in the high frequency applications. Yet, in practice we find that it continues to be used in the literature. So is it really justified to use this concept outside of its realm of validity, and why or in what way?
 A: Yes, it is possible to use the concept of voltage and related tools in AC circuits. The fact that the contour integral of $\vec E$ isn't zero isn't a problem. It would only be a problem if you first, "gradient" equation were right in this context. However, it's not. The correct equation needed for more general electromagnetic setups is
$$ \vec E = -\nabla \Phi  - \frac{\partial \vec A}{\partial t} $$
The identity for $\vec E$ implied by this definition isn't simply ${\rm curl}\,\vec E=0$ but the full Faraday's law Maxwell's equation you wrote down
$$ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} $$
because $\vec B = \nabla\times \vec A$. The equations above are fully consistent with one another and compatible with the fact that $\Phi$, the vector potential, is a well-defined field even in general situations involving time-dependent fields and magnetic fields. 
There is an ambiguity in the choice of $\Phi,\vec A$, the gauge invariance, but useful conventions may be assumed to fix this ambiguity in the case of AC circuits. When it's done, the non-magnetic parts of the circuits work just like before or in DC circuits. The voltage may be calculated as the differences of $\Phi$ just like in the DC case. 
For coils, one has to appreciate that the integrated electric field isn't the only relevant contribution. Instead, one finds the electromotive force, EMF, a modified sibling of "ordinary" voltage which has to be added to coils (and batteries) for the total sums of voltages over closed loops to vanish.
One may also study harmonic time-dependence in which all quantities depend on time as $\cos\omega t$ which is usually complexified to $\exp(i\omega t)$.
A: In the framework of EE theory there is widely accepted assumption of "lumped circuit". 
Lumped circuit is the circuit where all the elements are "infinitesimal" (size less), and the whole circuit is "infinitesimal" too (zero area). The above means that even in presence of time varying magnetic field, the magnetic flux through circuit elements and through the area of the circuit itself is assumed to be zero.
Just like any other physical model, the lumped model has its limitations too. The rule of thumb is that lumped model breaks when the wavelengths involved are comparable to circuit's dimensions (or shorter). However, even in these high frequency cases, the lumped model may still be applied if the high frequency effects are accounted for by adding to the circuit additional components (transmission lines, parasitic inductances and capacitances, etc...).
When the lumped model may not be applied (due to very high frequencies involved), the simple notations of EE theory may not be used anymore. In these cases one must build a physical model of the circuit and solve Maxwell's equations. However, to my best knowledge, these cases are treated by physics doctors in a specialized facilities, not by electrical engineers.
A: 
we cannot use the concept of voltage to study alternating current circuits, especially in the high frequency applications.

This is a common misconception, usually on part of physics teachers and students, electrical engineers seem to get this right more often.
What is going on here is that voltage, in its basic meaning used in AC/DC circuits and Kirchhoff's voltage law, is difference of the Coulomb potential, or equivalently, line integral of the Coulomb electric field from one point to another. The Coulomb field is a function of all charges magnitudes and  positions' and is conservative; any line integral of it over closed curve is zero. Thus the integral from one point to another does not depend on path, only on the endpoints. All this is true regardless of what total electric field looks like; voltage only cares about the Coulomb field.
When you see electric field written as
$$
\mathbf E = -\nabla \Phi -\partial_t \mathbf A
$$
this does not by itself fix $\Phi$, but in practice, in working with AC/DC circuits where radiation is negligible, the most natural way to fix $\Phi$ is to fix it to the electrostatic definition
$$
\Phi(\mathbf x, t) = \int K\frac{\rho(\mathbf x',t)}{|\mathbf x - \mathbf x'|}d^3\mathbf x'.
$$
Then, $-\nabla \Phi$ is the Coulomb field, and $-\partial_t \mathbf A$ is the rest of the field; in DC circuits this contribution is zero, but in AC circuits, it is not, and we call it induced electric field.
