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This is quoted from Feynman's Lectures:

We would like to see what happens when the loop shrinks down to a point, so that surface boundary disappears - the surface becomes closed. Now, if the vector $\mathbf{C}$ is everywhere finite, the line integral around the loop must go to zero as we shrink the loop. According to Stokes' theorem, the surface integral of $(\nabla \times \mathbf{C})_n$ must also vanish.

Mathematically, from Stokes' theorem, it can be inferred that

$$\int_{\text{any closed surface}} (\nabla \times \mathbf{C}) \cdot n \cdot da = 0.$$ But physically, what is going on that is making the circulation zero in the closed surface?

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    $\begingroup$ Your equation is a mathematical theorem. It seems to make little sense to try to explain why it is true from experience. Aren't you looking for examples of its use in physics, instead? $\endgroup$
    – Ján Lalinský
    Commented Apr 5, 2015 at 13:05
  • $\begingroup$ @Ján Lalinský: Yes, sir. Actually I want the reason. $\endgroup$
    – user36790
    Commented Apr 5, 2015 at 13:09

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Stokes' theorem needs no physical reason to be true. However, there is a nice intuitive description of the two-dimensional case. Tesselate the surface with little (infinitesimal) oriented squares and consider the integral as the sum of the curl on all these little squares:

enter image description here

The inner sides of the squares have no contribution to this sum at all, because they are always canceled by the adjacent squares whose sides are oriented in the opposite direction. Therefore, only the contribution of the border of the surface remains, and since a closed surface has no border, the integral over closed surfaces vanishes.

Image and explanation adapted from the Wikipedia page linked in the first sentence.

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  • $\begingroup$ This is really intuitive! Even Feynman used the same approach to deduce Stokes' theorem. Sir, btw, can you please tell how a loop, when shrinked, gets transformed to a closed surface? $\endgroup$
    – user36790
    Commented Apr 5, 2015 at 16:18
  • $\begingroup$ @user36790: A loop doesn't become a closed surface. I think what Feynman describes there is that if you have a surface whose boundary is a loop, and you shrink the loop to a point, and the surface has no longer a boundary (since the point you shrank the loop to can be considered part of the surface), and is hence closed. $\endgroup$
    – ACuriousMind ♦
    Commented Apr 5, 2015 at 16:28

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