Graduate Impossible Curl of a Vector Field

[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?

 

[jedishrfu]

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

 

[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??

 

[zwierz]

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

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

 

[laplacianZero]

Well, is the above post #3 a possibility?

 

[laplacianZero]

?

 

[laplacianZero]

Nvm. I got it.

 

[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.