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

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

(b) For what values of $A$ is $\mbox{div }F$ positive, negative or zero?
$\mbox{div }F$ is positive for $A$ ? < <= = >= >
$\mbox{div }F$ is negative for $A$ ? < <= = >= >
$\mbox{div }F = 0$ for $A =$
(For the first two, select the appropriate comparison and fill in a value for the right-hand side of the expression; for the last, give the value or values at which the divergence is zero, as a comma-separated list.)

(c) Suppose that $A = -6$. What is the sign of the flux out of a small sphere centered at $(1,1,1)$? ? positive negative
If instead $A = 4$, what is the sign of the flux out of a small sphere centered at $(1,1,1)$? ? positive negative

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