Module 1 : A Crash Course in Vectors
Lecture 5 : Curl of a Vector - Stoke's Theorem
  Example 22 :
A vector field is given by $ \vec F = -y\hat\imath + z\hat\jmath + x^2\hat k$. Calculate the line integral of the field along a circular path of radius $R$ in the x-y plane with its centre at the origin. Verify Stoke's theorem by considering the circle to define (i) the plane of the circle and (ii) a cylinder of height $z=h$.
  Solution :
  The curl of $\vec F$ may be calculated as
 
\begin{displaymath}\nabla\times\vec F = -\hat\imath + 2x\hat\jmath + \hat k\end{displaymath}
  Because of symmetry, we use cylindrical (polar) coordinates. The transformations are $ x= \rho\cos\theta,\ \ y= \rho\sin\theta, \ \ z=z$. The unit vectors are
 
\begin{eqnarray*} \hat\imath &=& \hat\rho\cos\theta -\hat\theta\sin\theta\\ \ha... ...=& \hat\rho\sin\theta -\hat\theta\cos\theta\\ \hat k &=& \hat k \end{eqnarray*}

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