Module 1 : A Crash Course in Vectors
Lecture 5 : Curl of a Vector - Stoke's Theorem
(ii)
 For the cylindrical cup, we have two surfaces : the curved face of the cylinder on which $\hat n = \hat\rho$ and the top
circular face on which $\hat n=\hat k$. The contribution from the top circular cap is $\pi R^2$, as before because the two caps only differ in their $z$ values (the z-component of the curl is independent of $z$). The surface integral from the curved surface is (the area element is )
 
\begin{displaymath}\int_0^{2\pi}Rd\theta\int_0^h dz (-R\sin\theta\cos\theta + z\cos\theta)\end{displaymath}
  For both the terms of the above integral, the angle integration gives zero. Thus the net surface integral is $\pi R^2$, as expected.

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