Module 1 : A Crash Course in Vectors
Lecture 5 : Curl of a Vector - Stoke's Theorem
Curl of a Vector - Stoke's Theorem
We have seen that the line integral of a vector field $\int\vec F\cdot\vec{dl}$ is essentially a sum of the component of $\vec F$ along the curve. If the line integral is taken over a closed path, we represent it as $\oint\vec F\cdot\vec{dl}$. If the vector field is consevative, i.e., if there exists a scalar function $V$ such that one can write $\vec F$ as $\nabla V$, the contour integral is zero. In other cases, it is, in general, non-zero.
  Consider a contour ${\cal C}$ enclosing a surface $S$. We may split the contour into a large number of elementary surface areas defined by a mesh of closed contours.
  Since adjacent contours are traversed in opposite directions, the only non-vanishing contribution to the integral comes from the boundary of the contour ${\cal C}$. If the surface area enclosed by the $i-th$ cell is $\Delta S_i$, then
 
\begin{eqnarray*} \oint_{\cal{C}}&=& \sum_i\int_{{\cal C}_i} \vec F\cdot\vec{dl}... ...rac{\int_{{\cal C}_i} \vec F\cdot\vec{dl}}{\Delta S_i}\Delta S_i \end{eqnarray*}
   
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