When deriving Faraday's law of induction, one needs to calculate the time derivative of the flux Φ=∫S(t)→B⋅→ndS, 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 ddt∫S(t)→B⋅→ndS=∫S(t)∂→B∂t⋅→ndS+∮∂S(t)→B⋅→v×d→l,
where I assume →v(t,→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 →v⋅→∇→B plus some kind of Stoke's theorem. Is that possible?
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