Solving for Divergence and F=∇×A

[jrenman]

Homework Statement

A vector field F for which div F = 0, is called incompressible (also called solenoidal). Consider the vector field F(x, y, z) = ⟨y, x + y, −z⟩.

(a) (1 point) Show that F is incompressible.

(b) (3 points) Find a vector field A such that F=\nabla×A.

Homework Equations

div F = \nabla . F

The Attempt at a Solution

I understand how to do part a and confirmed that div F = 0 and thus is incompressible.
But I'm not entirely sure how to find part b.

Is there a cross product operation in which A = some combination of F and \nabla?

Or would you assign A some arbitrary vectors \left\langleP,Q,R⟩ and take the cross of those with \nabla. Resulting in the vector
⟨y, x + y, −z⟩ = (dR/dy-dQ/dz)i-(dR/dx-dP/dz)j+(dQ/dx-dP/dy)k
and somehow solve for P,Q,R?

Or am I completely on the wrong track?

Any help or advice would be wonderful!

 

[vela]

You can find P, Q, and R by trial and error pretty quickly since F is relatively simple.