Using Curl to determine existence of Potential Function

[Ineedahero]

How does it work, exactly?

Assume I have a vector field function and I take the curl of it.

If I get a curl of zero, then does that guarantee that there is no potential function?
And if I get a curl of non-zero, does that guarantee that there is a potential function?

I googled this, but it wasn't clear.

Thanks.

 

[WannabeNewton]

You are mixing things up. As far as calculus on $\mathbb{R}^{3}$ is concerned, if $X$ is a smooth vector field defined on a simply connected region in $\mathbb{R}^{3}$ and $\nabla \times X = 0$ then there exists a smooth scalar field $\varphi$ such that $X= \nabla\varphi$ on that simply connected region. However if the vector field is defined on a non-simply connected region then this need not hold true.

If the curl of the vector field is non-zero, how can the vector field be the gradient of a scalar field?

More generally, we speak of exact and closed differential forms. The result on $\mathbb{R}^{3}$ can be generalized to smooth manifolds using the rather beautiful Poincare Lemma: http://en.wikipedia.org/wiki/Closed_and_exact_differential_forms#Poincar.C3.A9_lemma

EDIT: Note that we make use of the above in electromagnetism all the time because the electromagnetic field $F$ satisfies $dF = 0 \Rightarrow F = dA$ (at least locally) and we call $A$ the electromagnetic 4-potential.

 

[Ineedahero]

Thanks for the quick response!
So, I take it that the following statements are true (assuming the vector field is smooth and connected in R3 and so on):

If the curl is zero, then that GUARANTEES that there IS a potential function
If the curl is non-zero, then that GUARANTEES that there is NO potential function

 

[micromass]

If the curl is zero, then that GUARANTEES that there IS a potential function

No. But this is a very tricky subject. A lot depends on the domain of the vector field $X$. For example, if $X$ is defined on entire $\mathbb{R}^3$, then it's true: if $curl(X)=0$, then there is a potential function. But if $X$ is defined on $\mathbb{R}^3\setminus \{0\}$ (for example), then there are counterexamples.

The Poincare lemma states that if the domain of definition of $X$ is convex (or more generally: star shaped), then it is true that if the curl is zero, then there is a potential function. But on more general domains, this might fail.

The study of this question is done in De Rham cohomology. This is a formalism set up to study the situations when $curl(X)=0$ implies the existence of a potential function.

If the curl is non-zero, then that GUARANTEES that there is NO potential function

This is true.