Question

(1 point) Horizontal cross-sections of the vector fields F(x, y, z) and 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 2 (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 div(F) and div(G) positive, negative, or zero at the origin? Be sure you can explain your answer. At the origin, div(F) is Negative At the origin, div(G) is Zero (b) Are F and G curl free (irrotational) or not at the origin? Be sure you can explain your answer. At the origin, F is Irrotational At the origin, Ğ is Irrotational (c) Is there a closed surface around the origin such that F has nonzero flux through it? Be sure you can explain your answer by finding an example or a counterexample. Yes v (Click on a graph to enlarge it.) (d) Is there a closed surface around the origin such that G has nonzero circulation around it? Be sure you can explain your answer by finding an example or a counterexample. Yes

Answer 1

SOLUTION fields Given that the of the vector horizontal cross-sections ² (asy, z) and ã cas y, z) are given & is independent of " . → Each vector field has zero Z. component Rotational "o single ove Diverge div > tve So here , div (²) is negative at the origin div( f ive five dir (6) ▼ five div (6) is negative at the origin are sink type 2 > here in dive(f) (or) aiv (6) both so, curl (f) to .. curl (ã) to Fluse is given By By = f 7.ar So here, an w ê (By vector So flux, - S zo in ay plane) means, there are no closed surface so that flunt is non Bero.

The flux is given by a dn Å (By Wector in sy plane) so the flux is soko so, here there are no closed Suoface so that the flur is non gero