# Curl of a gradient and the anti Curl

LaplacianHarmonic

## Homework Statement

Is there a vector field D that produces The position vector <x,y,z> if we take the curl of vector field D?

## Homework Equations

Curl of gradient f = 0

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

## 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[/B]

## Answers and Replies

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:
Homework Helper
Gold Member

## Homework Statement

Is there a vector field D that produces The position vector <x,y,z> if we take the curl of vector field D?
Think about what you know about the divergence of a curl...

LaplacianHarmonic
Divergence of a curl is zero

Homework Helper
Gold Member
Yes. So....?

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.