Kirchhoff's Voltage Law in a General Electromagnetic Field Recently, Prof. Walter Lewin and YouTuber ElectroBOOM started a discussion about KVL, after Dr. Lewin claimed that KVL did not hold in the presence of an magneto-dynamic field. I would argue that Dr. Lewin is incorrect in his interpretation, and KVL does hold under scrutiny. 
Lewin's argument follows from the given definitions. The electric potential, $\phi$, is defined as the line integral of the electric field, $\vec{E}$, 
$$\phi = \int_C \vec{E} \cdot d\vec{l}$$
And in the general case, e.g. when a changing magnetic field is present, $\vec{E}$ is non-conservative, so $\phi$ is path-dependent. That is, the potential difference between two points in space is not unique, and therefore "voltage" is not well-defined. This is a direct result from vector calculus.
Now, as an extension, from the Faraday-Maxwell law, 
$$ \oint_C \vec{E} \cdot d\vec{l} + \int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{s} = 0$$
Where $C = \partial S$ is the boundary of surface $S$. This can be interpreted as an "induced EMF" term, the flux of the time derivative of $\vec{B}$, contributing to the potential around a given loop, $C$. Thus, when the induced EMF is accounted for, the net potential around a given loop is zero. Yet, the first term, the closed line integral of $\vec{E}$, is still ambiguous. 
We can provide the circuit-analysis technique with the following definitions. We assume that radiation may be neglected, and that components are lumped. Quasi-static conditions are assumed. We rewrite the Faraday-Maxwell law as a sum of $n+1$ discrete paths which are piece-wise continuous, such that
$$ \sum_{k=0}^n C_k = C$$
Where $C$ is defined as above. From the definition of $\phi$, we substitute the first term in Faraday's law, using
$$ \sum_{k=0}^n \oint_{C_k} \vec{E} \cdot d\vec{l} = \sum_{k=0}^n \phi_k $$
And we replace the symbol $\phi$ by $V$ to put the sum in circuit theory symbols...
$$ \sum_{k=0}^n V_k $$
Faraday's law then becomes
$$ \int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{s} + \sum_{k=0}^n V_k = 0 $$
We now make the substitution
$$ V_i = \int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{s} $$
Where $V_i$ is called the induced EMF for its ability to induce current in a conductor (hence being an electromotive force). This yields
$$ V_i + \sum_{k=0}^n V_k = 0 $$
Which is Kirchhoff's voltage law as seen in circuit theory. This definition appears to hinge on $V_k$ being well-defined, but in fact it does not require a path-independent $V_k$, simply one that is defined along the circuit in question, as we will see. 
To prove his point, Dr. Lewin places a loop of wire with two resistors in series around a solenoid, and switches the solenoid on, while probing the same position with two oscilloscope probes. The two probes detect different voltages when the switch is thrown, and because the same point in a circuit shouldn't measure two different voltages, KVL has failed, and the potential is shown to be path-dependent. 
This is, however, an incorrect conclusion. In general, the integration path for the Faraday-Maxwell law is arbitrary, but the circuit path is fixed. The only ambiguity in the potential around the loop is due to the nature of induced EMF, which depends on the area of the loop. If the integration path is perturbed, the potential between two points will change, but the induced EMF term changes as well, directly opposite. By accounting specifically for the path taken by the circuit (from a macroscopic perspective, e.g. with one-dimensional wires and resistors), the potential difference around the loop is still zero. 
The probing of the circuit, as ElectroBOOM pointed out, is responsible for the perceived failure of KVL. While Dr. Lewin correctly demonstrated the path-dependence of electric potential, he neglected the straight-forward behavior of this dependence. By careful choice of integration path, including probing loops, there is no discrepancy in the total potential measured around a loop. 
While it is possible to consider $\vec{E}$ in terms of a scalar and vector potential,
$$ \vec{E} = -\nabla \Phi - \frac{\partial \vec{A}}{\partial t} $$
And to observe that the conservative and rotational components of $\vec{E}$ allow direct application of KVL to the scalar potential, this is unnecessary, and further, $\Phi$ is only an accurate measure of the electric potential in the electrostatic case. So while using potentials is mathematically convenient for general classical electrodynamics, they are not required to resolve path dependence of $\phi$, and the failure of $\vec{E} = \nabla \phi$. 
While Kirchhoff's voltage law is not an electrodynamic law, it is a circuit theory law, and it is upheld in the example which Lewin provides, contrary to his results. The definition of KVL given above is necessary to analyze circuits including such as elements as transformers, below the limit where transmission line analysis techniques are in common use (typically when component lengths are $>>\lambda$ for a signal propagating in the circuit).
Have I missed anything in my derivation, or misinterpreted a result? I've been mulling this over for a few days now, and I originally defended Dr. Lewin's position, however the logic against him appears quite sound. 
 A: 
$$ V_i + \sum_{k=0}^n V_k = 0 $$
  Which is Kirchhoff's voltage law as seen in circuit theory.

This is not the usual form of KVL that we use in circuit theory. For example, 


*

*K. S. Suresh Kumar in Electric Circuits and Networks states the law as "The algebraic sum of voltages in any closed path in a circuit is zero."

*Dorf and Svoboda, Introduction to Electric Circuits, gives, "The algebraic sum of the voltages around any closed loop in a circuit is identically zero for all time."

*C. L. Wadhwa, Network Analysis, has "The sum of voltage rises and drops in a closed loop at any instant of time are equal"


These are all essentially just the $\sum_k V_k$ term from your statement of the law.
By including the $V_i$ term to account for changing magnetic flux encircled by the circuit, you have indeed formed a version of the law that accounts for changing magnetic flux encircled by the circuit. But that doesn't mean that the most common forms of KVL account for it.

The definition of KVL given above is necessary to analyze circuits including such as elements as transformers

This isn't really true. Usually we simply model transformers or inductors as devices that  directly affect only the voltages between the nodes they're directly connected to. For example we model an ideal inductor connected between nodes a and b with the relation
$$ V_{ab} = L\frac{{\rm d}I}{{\rm d}t}.$$
With this relation the inductor voltage is simply one of the $V_k$ in our KVL equation, and we don't need to introduce $V_i$.
What we can't model without the $V_i$ term, and don't normally model under KVL, is the effect of flux through the loops enclosed by the circuit elements and the wires that connect them.
For example, in this simple circuit

the usual KVL can't account for EMF produced by changing magnetic flux in the loop formed by the switch, cell, and bulb and the wires connecting them.
(image source here)
A: Prof. Walter Lewin is correct.
Kirchhoff's Voltage Law holds only when the voltage source is inside and part of the current circuit. Even in a transformer circuit the secondary coil of the transformer acts as the connected voltage source of the components circuit connected to the secondary. Dr. Lewin's explanation https://www.youtube.com/watch?v=LzT_YZ0xCFY shows clearly that the voltage source (i.e. electromagnet) is not connected and part of the target load circuit.
There is no voltage source in the circuit to begin with and apply Kirchhoff's Voltage Law. Kirchhoff's Voltage Law is not wrong but simply not applicable in this case.
Antenna theory is more suitable for describing the case where different impedance elements are EM irradiated with a single transient signal and according to their impedance develop a voltage drop transient reception signal.
That what is not well understood is that it is not the case of two parallel connected different resistive paths (elements) in a voltage source circuit  that must have the same voltage drop value in each path but instead the case of an inductive loop with non uniform distributed resistance along  the loop. Therefore, the different resistivity segments of the loop are not in parallel but in series.
Excellent analysis of the Lewin paradox, videos:
https://www.youtube.com/watch?v=xMePTKuAixE
https://www.youtube.com/watch?v=OmlnGei1xo8
Also experimental new evidence of the path dependence (video): https://www.youtube.com/watch?v=lehe18VoeNM
Final conclusion, as demonstrated by Prof. Walter Lewin, KVL cannot handle induced emf's from non-conservative fields like from a time varying external magnetic field and integrate them as being normal voltage sources (i.e. conservative emf) in an electrical component circuit. KVL law demands the emf source to be connected to the circuit so that the energy flow from the source is directed inside the path of the specific circuit each time and is independent of other paths outside the circuit.
A: 
The electric potential, $\phi$, is defined as the line integral of the electric field, $\vec{E}$, 
  $$\phi = \int_C \vec{E} \cdot d\vec{l}$$
  ...so $\phi$ is path-dependent.
Have I missed anything in my derivation, or misinterpreted a result?

Think about this for a while: if electric potential was, in general, defined as integral of total electric field, it would always be path-dependent. Because there are always some solenoidal electric fields somewhere and we could achieve any value just by extending and twisting the path appropriately. Yet, in all practical uses of electric potential, it is always assumed that potential has unique value in each point of space. How could that work so well? Perhaps potential is defined differently?
The general definition of electric potential is this: a single-valued function of position $\Phi$ that obeys
$$
\mathbf E  = - \nabla \Phi - \partial_t \mathbf A
$$
where $\mathbf A$ is single-valued function of position as well, which gives magnetic field via curl operation. So, in general, electric potential cannot be expressed solely as integral of total electric field. This is a surprisingly common misconception.
The integral of total electric field is used as definition of voltage only in electrostatics where total field is the same as its electrostatic component. There, $\partial_t \mathbf A$ vanishes, so it works there, but in general, the integral formula is ambiguous and does not give correct voltage.
In practical "engineering" use of the potential concept, including cases where total field is not electrostatic, one uses the Coulomb gauge, where the potential is a function of instantaneous positions of all charges of the system. Voltage, when written/said with no qualification (unlike in "induced voltage" or "electromotive voltage" (non-English for emf)) means simply difference of values of this Coulomb potential. Thus sum of voltages along any closed path is zero. The KVL law, when understood as statement about this voltage, is thus always true. Although it is also true that KVL may not always be very useful, especially if several of those voltages cannot be reliably measured by a voltmeter (like in Lewin's examples).
A: This is to amplify on what @The_Photon wrote. Prof. Lewin demonstrates in a simple experiment that voltage cannot be defined uniquely when there is a changing magnetic field: connecting two voltmeters to the same pair of points results in two completely different measurements. Lewin blames KVL (Kirchhoff Voltage Law -loop) for the discrepancy but I think the problem is one step before we can even apply KVL, namely by having a circuit in which the magnetic field is spread across the whole circuit and we do not have lumped elements for which the KVL and KIL are applicable. What makes it inapplicable is not just simply that the circuit consists of a pair of resistors and a solenoid with varying current in between but that we are allowed to probe inside that circuit with conducting leads that are affected by the time varying magnetic field. If instead of $\textbf{B}$ we would be using the vector potential $\textbf{A}$ we would notice that even when $\textbf{B}=0$, the vp is not zero, and there is no question that voltmeters are in the time varying magnetic field defined by $\textbf{A}$: $\textbf{curlE}= -\frac{\partial{\textbf{curlA}}}{\partial t}$, and therefore for an arbitrary loop $\mathcal{L}$ 
$\oint_{\mathcal{L}} \textbf{E}\cdot d\ell = -\frac{\partial}{\partial t}\oint_{\mathcal{L}} \textbf{A}\cdot d\ell $. 
So, even though the $\textbf{B}$ field is esentially zero outside the solenoid everywhere we measure the voltage, neither the vector potential $\textbf{A}$ nor its time derivative $\dot {\textbf{A}}$ is zero there and our voltmeters, therefore, through their leads, are also affected.
