Horizontal cross-sections of the vector fields $\vec{F}(x,y,z)$ and $\vec{G}(x,y,z)$ are given in the figure. Each vector field has zero z-component (i.e., all of its vectors are horizontal) and is independent of $z$ (i.e., is the same in every horizontal plane). You may assume that the graphs of these vector fields use the same scale. (a) Are $\mathrm{div}(\vec{F})$ and $\mathrm{div}(\vec{G})$ positive, negative, or zero at the origin? Be sure you can explain your answer. At the origin, $\mathrm{div}(\vec{F})$ is Choose Positive Negative Zero At the origin, $\mathrm{div}(\vec{G})$ is Choose Positive Negative Zero (b) Are $\vec{F}$ and $\vec{G}$ curl free (irrotational) or not at the origin? Be sure you can explain your answer. At the origin, $\vec{F}$ is Choose Irrotational Not irrotational At the origin, $\vec{G}$ is Choose Irrotational Not irrotational (c) Is there a closed surface around the origin such that $\vec{F}$ has nonzero flux through it? Be sure you can explain your answer by finding an example or a counterexample. Choose Yes No (d) Is there a closed surface around the origin such that $\vec{G}$ has nonzero circulation around it? Be sure you can explain your answer by finding an example or a counterexample. Choose Yes No

$\vec{F}$ $\vec{G}$ (Click on a graph to enlarge it.)

You can earn 50% partial credit for 3 - 5 correct answers.