Let $r=(x^2+y^2)^{1/2}$ and consider the vector field $\vec F = r^A(-y\vec i + x\vec j)$, where $r\ne 0$ and $A$ is a constant. $\vec F$ has no $z$-component and is independent of $z$.

(a) Find $\mbox{curl }(r^A(-y\vec i+x\vec j))$, and show that it can be written in the form
$\mbox{curl }\vec F =$ $r^a \vec k$, where $a =$ , for any constant $A$.

(b) Using your answer to part (a), find the direction of the curl of the vector fields with each of the following values of $A$ (enter your answer as a unit vector in the direction of the curl):
$A = -7$: direction =
$A = 4$: direction =

(c) For each values of $A$ in part (b), what (if anything) does your answer to part (b) tell you about the sign of the circulation around a small circle oriented counterclockwise when viewed from above, and centered at $(1,1,1)$?
If $A = -7$, the circulation is ? positive negative we don't have enough information to say
If $A = 4$, the circulation is ? positive negative we don't have enough information to say .

(Be sure you can say how your answer in part (c) would change if the question were about a small circle centered at $(0,0,0)$.)