Module 1 : A Crash Course in Vectors
Lecture 5 : Curl of a Vector - Stoke's Theorem
  The line integrals along the four sides are
\begin{eqnarray*} \oint\vec F\cdot\vec{dl} &=& \int_0^1 F_x\mid_{y=0}dx +\int_0^... ...int_1^0 x^2dx +\int_0^1 2dy\\ &=& \frac{1}{3}+0-\frac{1}{3}+2=2 \end{eqnarray*}
  Since the normal to the plane is along $\hat k$, we only need $z-$ component of $\nabla\times\vec F$ to calculate the surface integral. It can be checked that
 
\begin{displaymath}(\nabla\times\vec F)_z = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 2-0=2\end{displaymath}
  Thus
 
\begin{displaymath}\int_S(\nabla\times\vec F)\cdot\hat k dS= \int_0^1\int_0^1 2dxdy=2\end{displaymath}
  which agrees with the line integral calculated.

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