Solution
= Solution
The <Maxwell equations>[Maxwell equation] $\nabla\times\mathbf E=-\partial\mathbf B/\partial t$, integrated over a fixed spanning surface $S$, gives by <Stokes theorem>
$$
\oint_{\partial S}\mathbf E\cdot d\mathbf l
=-\int_S\frac{\partial\mathbf B}{\partial t}\cdot d\mathbf S
=-\frac{d}{dt}\int_S\mathbf B\cdot d\mathbf S.
$$
Thus the <electromotive force> and <magnetic flux> obey <Faraday's law>
$$
\boxed{\mathcal E=-\frac{d\Phi_B}{dt}}.
$$