What Test to Show that Inverse Curl Exists?

[greswd]

Main Question or Discussion Point

What test can we perform on a vector field to determine if there exist vector field(s) that describe its inverse curl?

 

[jedishrfu]

What do you mean by inverse curl? Are you saying you have the curl and now want to find the field or fields for it?

Are you looking for a physical test or a mathematical test?

Physically, you could place a small rotor in the flow and see what happens.

Mathematically you should know already ie what is the curl of a conservative field.

 

[greswd]

What do you mean by inverse curl? Are you saying you have the curl and now want to find the field or fields for it?

Are you looking for a physical test or a mathematical test?

Physically, you could place a small rotor in the flow and see what happens.

Mathematically you should know already ie what is the curl of a conservative field.

The curl operator is not injective, hence there is no unique solution for the inverse curl. However, I just want to know the method by which we determine whether an inverse curl vector field does exist, given an existing vector field.

Mathematically speaking.

 

[greswd]

Mathematically you should know already ie what is the curl of a conservative field.

the curl of a conservative vector field is zero.

 

[micromass]

The curl operator is not injective, hence there is no unique solution for the inverse curl. However, I just want to know the method by which we determine whether an inverse curl vector field does exist, given an existing vector field.

Mathematically speaking.

What is an inverse curl vector field? You mean you are given a vector field and you want to find out whether this is the curl of another vector field?

 

[greswd]

What is an inverse curl vector field? You mean you are given a vector field and you want to find out whether this is the curl of another vector field?

yes. whether it could be the curl of another vector field

 

[micromass]

yes. whether it could be the curl of another vector field

This is an important question that sadly does not have an easy answer. The answer depends crucially on the domain of the vector field. The idea is that the divergence of the curl is $0$. This is a necessary condition for your answer to be positive. So if the divergence of your vector field is not zero, then it cannot be the curl of some field. The question is whether the converse holds.

The good news is that Poincare's lemma gives important conditions on when this is true. This says that whenever the domain (ie where your vector field is defined) is contractible/star shaped/convex, then it is true. So if your vector field is defined everywhere on $\mathbb{R}^3$ or on a convex domain $[0,1]^3$, then checking whether the divergence is zero is enough to conclude it is the curl of a vector field.

This is cool, but Poincare's lemma sometimes fails when the domain is not so nice. If your domain is not nice, then the situation becomes more complicated and this is where De Rham's cohomology comes in.