Module 1 : A Crash Course in Vectors
Lecture 5 : Curl of a Vector - Stoke's Theorem
  We define the quantity
 
\begin{displaymath}\lim_{\Delta S_i\rightarrow 0}\frac{\int_{{\cal C}_i} \vec F\cdot\vec{dl}}{\Delta S_i}\hat n_i\end{displaymath}
as the curl of the vector $\vec F$ at a point P which lies on the surface $\Delta S_i$. Since the area $\Delta S_i$ is infinitisimal it is a point relationship. The direction of $\hat n_i$ is, as usual, along the outward normal to the area element $\Delta S_i$. For instance, the x-component of the curl is given by
 
\begin{displaymath}({\rm curl})_x=\lim_{\Delta y, \Delta z\rightarrow 0}\frac{\int_{{\cal C}_i} \vec F\cdot\vec{dl}}{\Delta y\Delta z}\hat n_i\end{displaymath}
  Thus
 
\framebox{\parbox[b]{2in}{$\displaystyle{\oint_{{\cal C}}\vec F\cdot\vec{dl}=\int_S{\rm curl}\vec F\cdot{dS}}$}}
  This is Stoke's Theorem which relates the surface integral of a curl of a vector to the line integral of the vector itself. The direction of $\vec{dl}$ and $\vec{dS}$ are fixed by the right hand rule, i.e. when the fingers of the right hand are curled to point in the diurection of $\vec{dl}$, the thumb points in the direction of $\vec{dS}$.
   
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