Inverse of Curl Operator: A as a Function of B?

[ikaal]

I want to express A as a function of B in the following equation:

curl{A}=B

So I need the inverse of the curl operator, but I don't know if it exist.

Thanks.

 

[Curl]

It's called vector potential

 

[Redbelly98]

There is no unique solution for A. You can always add a vector field of zero curl to one solution and get another solution.

 

[HallsofIvy]

If A= f(x,y,z)i+ g(x,y,z)j+ h(x,y,z)k then curl A = (h_y- g_z)i+ (f_z- h_x)j+ (g_x- f_y)k.

If you are given that curl A= B= p(x,y,z)i+ q(x,y,z)j+ r(x, y, z)k then you must solve the system of equation h_y- g_z= p, f_z- h_x= q, g_x- f_y= r.

Since those are partial differential equations, the "constants of integration" will be functions of x, y, z. That is why, as RedBelly98 says, "You can always add a vector field of zero curl".

 

[arildno]

Note that this non-uniqueness is not something that is a strange facet by the vector potential only.

You know of it from before, as the so-called "constant of integration".
When doing partials, functions of the other variables will be "constants" with respect to that variable you have differentiated with respect to.


A far more important question, though, is, not uniqueness vs. non-uniqueness, but existence vs. non-existence of the vector potential(s)!

Do you know, given B, how to be certain that at least one "A" exists?

 

[Anthony]

I want to express A as a function of B in the following equation:

curl{A}=B

So I need the inverse of the curl operator, but I don't know if it exist.

Thanks.

Consider the vector field defined by:

\mathbf{A}(\mathbf{x}) = \int_0^1 \mathbf{B}(\lambda \mathbf{x}) \wedge (\lambda\mathbf{x})\, \mathrm{d}\lambda.

You might like to show that if \nabla\cdot\mathbf{B}=0, then \nabla \wedge \mathbf{A} = \mathbf{B}. Obviously this \mathbf{A} is not unique.