Module 1 : A Crash Course in Vectors
Lecture 5 : Curl of a Vector - Stoke's Theorem
  Substituting the above, the field $\vec F$ and its curl are given by
\begin{eqnarray*} \vec F&=& \hat\rho(-\rho\sin\theta\cos\theta+z\sin\theta)\\ ... ...theta)\\ &+&\hat\theta(\sin\theta + 2\rho\cos^2\theta) +\hat k \end{eqnarray*}
  The line integral of $\vec F$ around the circular loop :
  Since the line element is $\vec{dl} = Rd\theta\hat\theta$,
 
\begin{displaymath}\oint \vec F\cdot\vec{dl} = \int_0^{2\pi}(R\sin^2\theta + z\cos\theta)Rd\theta \end{displaymath}
  On the circle $z=0$. The integral over $\sin^2\theta$ gives 1/2. Hence
 
\begin{displaymath}\oint\vec F\cdot\vec{dl} = \pi R^2\end{displaymath}

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