Evaluate the circulation of G⃗ =xyi⃗ +zj⃗ +3yk⃗ around a square of side length 9, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis.

Given that[tex]\vec G(x,y,z)=xy\,\vec\imath+z\,\vec\jmath+3y\,\vec k[/tex]has a fairly simple curl,[tex]\nabla\times\vec G(x,y,z)=2\,\vec\imath-x\,\vec k[/tex]we can take advantage of Stokes' theorem by transforming the line integral of [tex]\vec G[/tex] along the boundary of the square (call it [tex]S[/tex]) to the integral of [tex]\nabla\times\vec G[/tex] over [tex]S[/tex] itself. Parameterize [tex]S[/tex] by[tex]\vec s(u,v)=u\,\vec\jmath+v\,\vec k[/tex]with [tex]-\dfrac92\le u\le\dfrac92[/tex] and [tex]-\dfrac92\le v\le\dfrac92[/tex]. Then take the normal vector to [tex]S[/tex] to be[tex]\vec s_u\times\vec s_v=\vec\imath[/tex]so that[tex]\displaystyle\int_{\partial S}\vec G\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec G)\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv[/tex][tex]=\displaystyle\int_{-9/2}^{9/2}\int_{-9/2}^{9/2}(2\,\vec\imath)\cdot(\vec\imath)\,\mathrm du\,\mathrm dv=\boxed{162}[/tex]