Module 1 : A Crash Course in Vectors
Lecture 5 : Curl of a Vector - Stoke's Theorem
  Contribution to the line integral from the two sides AB and CD are computed as follows.
On AB : $\int\vec F\cdot\vec {dl} = \int\vec F\cdot\hat\jmath dy=\int F_ydy$
  ON CD : $\int\vec F\cdot\vec {dl} = \int\vec F\cdot(-\hat\jmath) dy=\int F_ydy$
  Using Taylor expansion (retaining only the first order term), we can write
 
\begin{displaymath}F_y\mid_{CD} = F_y\mid_{AB}+\frac{\partial F_y}{\partial z}\Delta z\end{displaymath}
  Thus the line integral from the pair of sides AB and CD is
 
\begin{displaymath}-\int \frac{\partial F_y}{\partial z}\Delta z dy\approx - \frac{\partial F_y}{\partial z}\Delta z \Delta y\end{displaymath}
  In a similar way one can calculate the contributions from the sides BC and DA and show it to be
 
\begin{displaymath}\int \frac{\partial F_z}{\partial y}\Delta y dz\approx \frac{\partial F_z}{\partial y}\Delta y \Delta z\end{displaymath}
   
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