Find a Conservative Vector Field from a Non-Conservative One

[dshadowwalker]

Homework Statement

My problem is, I have a scalar field and I take the gradient of this field. It is known that the gradient of a scalar function is a conservative vector field; but I need to run a procedure in this field that will modify the vector field; the modified vector field could be a non-conservative vector field, creating some problems to continue passing through another processing step. I want to find a conservative vector field that is as close as possible to this possible non-conservative vector field.
2. The attempt at a solution

A vector field is considered conservative if its curl is 0 and if it is simply-connected; considering that the field that I'm working is simply-connected, I need to make the curl of the modified vector field equals 0 (ZERO). I was studying the Helmholtz decomposition (http://en.wikipedia.org/wiki/Helmholtz_decomposition) to decompose the modified vector field into 2 components (for more explanations see the link), and I was wondering that if I just use the component that has curl=0 and generate a new vector field, this vector field would look similar.

Another way that I was thinking is about to obtain a scalar field that represents the non-conservative vector field, making easy to calculate the gradient of this scalar field to find a conservative vector field.

Which way I should take to achieve the result I want? In pratical, I have a non-conservative vector field and I just want to turn it into a conservative one.

Thank you.

 

[Muphrid]

I don't think you can take a scalar field to represent the non-conservative field--by definition, this should not be possible. It would have to be a vector field.

But I think the Helmholtz decomposition is the right track; here's a slightly modified version. Let's say there's a vector field [itex]F[/itex] such that it has divergence [itex]\nabla \cdot F = \rho[/itex] and curl [itex]\nabla \times F = J[/itex]. In general, then, the free space Green's function in 3D allows us to compute that

[tex]F(r) = \int \frac{-\rho(r') (r-r') + J(r') \times (r-r')}{4\pi|r - r'|^3} \; dV'[/tex]

I think that's right; it ought to be within some plus or minus signs. The first term describes exactly the currless field, the second term is the divergenceless field. This way, you can just take the field you have, calculate its divergence and curl, and subtract out the part from the curl.

 

[dshadowwalker]

Yes, it is not possible to take a scalar field from a non-conservative field; what I was trying to say is that it could be easier to find the scalar field that represents the closest conservative vector field than find directly the conservative field.

But about you said, then I will need just to calculate the curl of the non-conservative field and subtract it ?