= Solution
Use compatible orientations for the loop and spanning surface. The relevant <Maxwell equations> are
$$
\nabla\cdot\mathbf B=0,\qquad
\nabla\times\mathbf E=-\partial_t\mathbf B.
$$
The <magnetic flux transport for a moving circuit> separates the change of the field from the motion of its boundary. To see the boundary term, freeze $\mathbf B$ at time $t$. Orient the closed swept surface as $S(t+\delta t)-S(t)+S'$. Its side element is
$$
d\mathbf S'=-\delta t\,\mathbf v\times d\mathbf r,
$$
to first order. Hence its flux is $-\delta t\oint_C\mathbf B\cdot(\mathbf v\times d\mathbf r)$. By the <divergence theorem> and $\nabla\cdot\mathbf B=0$, the sum of the three fluxes is zero, giving the purely geometric change
$$
\delta\Phi_{\rm shape}
=\delta t\oint_C\mathbf B\cdot(\mathbf v\times d\mathbf r)
=-\delta t\oint_C(\mathbf v\times\mathbf B)\cdot d\mathbf r.
$$
Adding the temporal field change and applying <Stokes theorem> yields
$$
\frac{d\Phi}{dt}
=\int_{S(t)}\partial_t\mathbf B\cdot d\mathbf S
-\oint_C(\mathbf v\times\mathbf B)\cdot d\mathbf r
=-\oint_C(\mathbf E+\mathbf v\times\mathbf B)\cdot d\mathbf r.
$$
Thus
$$
\boxed{\frac{d\Phi}{dt}=-\mathcal E.}
$$
The boundary contribution is the <motional electromotive force>; the field contribution is <Faraday's law>.
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