# Homework Help: Divergence and F=∇×A

1. Mar 8, 2010

### jrenman

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
div F = $$\nabla$$ . F

3. 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 $$\left\langle$$P,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!

2. Mar 9, 2010

### vela

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