"Beautiful Mind" equation

[Loren Booda]

In the movie A Beautiful Mind, Prof. Nash introduces his class at MIT to a problem (below) on the chalkboard and states "...for others among you it will take the term of your natural lives." What is this equation, is it nontrivial, and does it have a solution?

V={F:R3|X-->R3 so (Del x F)=0}

W={F=(Del g)}

dim(v/w)=?

[kuengb]

Interesting; I thought about that too when I saw the movie. I couldn't figure out the solution yet. It's certainly trivial when the set X in the first line is empty, because then V=W and dim(V/W)=0.

And when X is non-empty? Maybe you have to deal with the number of connected components of X.

edit: the same again in clear latex:
V=\{ F:\mathbb{R}^3 \backslash X \rightarrow \mathbb{R}^3 \:so\: curl F=0\}
W=\{F=\nabla g \}
dim(V/W)=?