A Impossible Curl of a Vector Field

1. Mar 21, 2017

laplacianZero

Let's assume the vector field is NOT a gradient field.

Are there any restrictions on what the curl of this vector field can be?

If so, how can I determine a given curl of a vector field can NEVER be a particular vector function?

2. Mar 21, 2017

Staff: Mentor

Can give us a context here or some example that you're looking at?

3. Mar 21, 2017

laplacianZero

No example in particular... but I guess I can come up with one.

Here

Curl of vector field F = <2x, 3yz, -xz^2>

Is this possible??

4. Mar 21, 2017

zwierz

sure. if a vector field v is a curl of some another vector field then $\mathrm{div}\,v=0$ Locally the inverse is also true

5. Mar 21, 2017

laplacianZero

Well, is the above post #3 a possibility?

6. Mar 21, 2017

laplacianZero

????

7. Mar 30, 2017

laplacianZero

Nvm. I got it.

8. Mar 31, 2017

jostpuur

You can obtain some results concerning that question by examining the Fourier transforms. This approach suffers from the obvious shortcoming that not all functions have Fourier transforms, but anyway, it could be that Fourier transforms still give something.