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# Let $\mathbf{F}=2 z \mathbf{i}+2 y \mathbf{k}$, and let $\partial S$ be the intersection of the cylinder $x^2+y^2=a y$ with the hemisphere $z=\sqrt{a^2-x^2-y^2}$, $a>0$. Assuming distances in meters and force in newtons, find the work done by the force $\mathbf{F}$ in moving an object around $\partial S$ in the counterclockwise direction as viewed from above.

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Using the Stokes Theorem we have that:

\begin{aligned} \oint_{\delta S}{\bf F}\cdot{\bf T}ds&=\iint_S(\nabla\times{\bf F})\cdot{\bf n}dS \end{aligned}

Let's find the curl of the given vector field. Since ${\bf F}=(2z,0,2y)$ then:

\begin{aligned} \nabla\times{\bf F}&=\left|\begin{array}{lcl} {\bf i} & {\bf j} & {\bf k}\\ \dfrac{\partial }{\partial x} & \dfrac{\partial }{\partial y} & \dfrac{\partial }{\partial z}\\ 2z & 0 & 2y \end{array}\right|\\ &=(2,2,0) \end{aligned}

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