# Curl of a gradient and the anti Curl

1. May 7, 2017

### LaplacianHarmonic

1. The problem statement, all variables and given/known data
Is there a vector field D that produces The position vector <x,y,z> if we take the curl of vector field D?

2. Relevant equations
Curl of gradient f = 0

Curl of Vector D = <x,y,z>

3. The attempt at a solution

Curl of vector D
Where vector D=<A,B,C>

Cy - Bz = x
Az - Cx = y
Bx - Ay = z

I can't solve what component functions A, B, C are.

HELP

2. May 7, 2017

### Fred Wright

You are asked to determine if a vector exists s.t. $$\vec \nabla \times \vec D = \vec r$$
I suggest you expand the cross product in it's components. For example, the x component would be:$$\partial_y D_z -\partial_z D_y=x$$
Clearly $D_z$ must be of the form $a_zyx$, where $a_z$ is a constant. Do the same for the other two components.

Last edited: May 7, 2017
3. May 7, 2017

4. May 7, 2017

### LaplacianHarmonic

Divergence of a curl is zero

5. May 7, 2017

Yes. So....?

6. May 8, 2017

### LaplacianHarmonic

So... divergence of a curl measures how much the vector diverges outward after measuring how much that vector was curling. Thus, it is always zero.

7. May 8, 2017

### LCKurtz

But what does that observation have to do with your problem?