- #1

jrenman

- 4

- 0

## 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**=[tex]\nabla[/tex]×

**A**.

## Homework Equations

div

**F**= [tex]\nabla[/tex] .

**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 [tex]\nabla[/tex]?

Or would you assign

**A**some arbitrary vectors [tex]\left\langle[/tex]P,Q,R⟩ and take the cross of those with [tex]\nabla[/tex]. 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!