<PERSON>'s law for a current loop being deformed
I'm working through <PERSON>'s Classical Electrodynamics text (3rd version, chapter 5.15) about <PERSON>'s law:
<PERSON>'s law is pretty familiar:
$\int_c E \cdot dl = -\frac{d}{dt}(\int_s B \cdot n da)$
which states that the contour integral of the electric field around a bounded surface is equal to the negative time rate of change of the flux bounded by that surface. No problem. Here we have a total derivative and <PERSON> addresses 2 cases.
"The flux though the circuit may change because (a) the flux changes with time at a point, or (b) the translation of the circuit changes the location of the boundary"
In either of these cases, it is easy to show that:
$\frac{d}{dt}(\int_s B \cdot n da) = \int_s(\frac{\partial B}{\partial t} \cdot n da) + \int_c(B \times v)\cdot dl$
by using $\frac{d}{dt} = \frac{\partial}{\partial t} + v \cdot \nabla$ and then applying a "product rule", $\nabla \cdot B = 0$ and stokes theorem. 903
i.e.:
$\frac{d}{dt}\int_s B \cdot n da = \frac{\partial}{\partial t}\int_s B \cdot n da + (v \cdot \nabla) \int_s B \cdot n da$
In the case where the surface is constant, we can exchange the order of differentiation and integration:
$\frac{d}{dt}\int_s B \cdot n da = \int_s \frac{\partial}{\partial t} B \cdot n da + \int_s (v \cdot \nabla) B \cdot n da$
I still have no problem with this. Now we can work on the second term on the right (apply a product rule, drop terms with $\nabla \cdot B$ and then apply stokes theorem) to get the $\int_c(B \times v)\cdot dl$ term.
which, when applied to <PERSON>'s law gives the nice result:
$\int_c [ E' - (v \times B)] \cdot dl = - \int_s \frac{\partial B}{\partial t} \cdot n da$
The problem arises when you allow the shape of the bounded surface to change. (Consider a circular wire loop in a magnetic field that you then hit with a hammer causing a dent in one side). How does changing the surface influence the result that we derived above? It seems to me that this would prevent us from being able to exchange the order of differentiation and integration which seems like it leaves us in a pretty tight spot ... 903