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

In summary: However, if you add a vector field of zero curl to one solution, you get another solution, which has the same vector field as the original but with the curl field zero. That is, \mathbf{A}(\mathbf{x}) = \mathbf{B}(\mathbf{x}) +\mathbf{0}\, \mathbf{C}(\mathbf{x}).
  • #1
ikaal
1
0
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.
 
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  • #2
It's called vector potential
 
  • #3
There is no unique solution for A. You can always add a vector field of zero curl to one solution and get another solution.
 
  • #4
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".
 
  • #5
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?
 
  • #6
ikaal said:
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:

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

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

1. What is the inverse of a curl operator?

The inverse of a curl operator is a mathematical operation that takes a vector field as input and outputs a vector field that represents the magnitude and direction of the rotation of the original vector field at every point in space.

2. How is the inverse of a curl operator represented?

The inverse of a curl operator is typically represented by the notation ∇ x^-1, where ∇ is the del or nabla operator, and x^-1 indicates the inverse operation. It can also be represented as the cross product between the nabla operator and the inverse of the original vector field.

3. What is the relationship between the inverse of a curl operator and the original vector field?

The inverse of a curl operator is a function of the original vector field, meaning that it depends on the values of the original field at each point in space. It represents the inverse operation to the curl operator, which is a measure of the rotational behavior of the vector field.

4. How is the inverse of a curl operator used in scientific research?

The inverse of a curl operator is used in various fields of science, including physics, engineering, and fluid dynamics. It is a useful tool for studying and analyzing vector fields, particularly those that exhibit rotational behavior. It is also used in the development of mathematical models and simulations.

5. What are some real-life applications of the inverse of a curl operator?

The inverse of a curl operator has many practical applications, such as in weather forecasting, fluid flow analysis, and electromagnetic field modeling. It is also used in computer graphics for generating realistic animations of fluid and smoke movements. Additionally, it is utilized in the study of ocean currents and atmospheric circulations.

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