electromagnetism - Rigorous derivation of Faraday's induction

When deriving Faraday's law of induction, one needs to calculate the time derivative of the flux $\Phi = \int_{S(t)} \vec{B} \cdot \vec{n} \;dS$, where $S(t)$ is the surface over which we define this flux. Now, Jackson and Wikipedia (see first equation) state that it is easy to prove that $$\frac{d}{dt}\int_{S(t)} \vec{B} \cdot \vec{n} \;dS = \int_{S(t)} \frac{\partial \vec{B}}{\partial t} \cdot \vec{n}\;dS + \oint_{\partial S(t)} \vec{B} \cdot \vec{v} \times d\vec{l},$$ where I assume $\vec{v}(t,\vec{x})$ is the vector field describing the velocity of the surface $S(t)$.

I fail to see how the second term is obvious and how I could get it from straightforward differentiation, no drawings.

My first idea would be to use that this is in fact a material derivative, and so the second term should come from $\vec{v} \cdot \vec{\nabla} \vec{B}$ plus some kind of Stoke's theorem. Is that possible?