A fluid of constant density rotates around the $z$-axis with velocity $vec{F} = 2omega(-yhat{i} + xhat{j})$, where $omega$ is a positive constant called the angular velocity of the rotation. Calculate the circulation of the field around a circle $C$ of radius $R$ bounding a disk $S$ on the $xy$-plane (Hint: Use the Stokes' Theorem).
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Penny R.
Let $f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2} .$ Show that the clockwise circulation of the field $\mathbf{F}=\nabla f$ around the circle $x^{2}+y^{2}=a^{2}$ in the $x y$ -plane is zero. a. by taking $\mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi,$ and integrating $\mathbf{F} \cdot d \mathbf{r}$ over the circle. b. by applying Stokes' Theorem.
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