# Inverse of the curl

by ikaal
Tags: curl, inverse
 P: 1 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.
 P: 748 It's called vector potential
 Mentor P: 11,901 There is no unique solution for A. You can always add a vector field of zero curl to one solution and get another solution.
PF Patron
Thanks
Emeritus
P: 38,395

## Inverse of the curl

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".
 PF Patron HW Helper Sci Advisor Thanks P: 11,935 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?
P: 83
 Quote by 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.
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.

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