What is the 'Beautiful Mind' Equation and Does it Have a Solution?

[Loren Booda]

"Beautiful Mind" equation

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)=?

 

[M98Ranger]

The "Beautiful Mind" equation is a mathematical problem that was introduced by Prof. Nash in the movie A Beautiful Mind. It is a nontrivial equation that involves vector calculus and has a solution. The equation is written as V={F:R3|X-->R3 so (Del x F)=0}, which means the set of functions F in three-dimensional space that map from X to R3 such that the divergence of F is equal to zero. This equation is used to solve problems related to fluid mechanics and electromagnetism.

The second part of the equation, W={F=(Del g)}, introduces the concept of a gradient field and is related to the first part of the equation. The solution to this equation involves finding the dimension of the quotient space V/W, which is represented by dim(v/w)=?. This is a challenging problem that requires advanced mathematical skills to solve.

In the movie, Prof. Nash states that for some students, this equation may take the term of their natural lives to solve. This highlights the complexity and difficulty of the problem, emphasizing the brilliance and intelligence of Prof. Nash's mind.

In conclusion, the "Beautiful Mind" equation is a nontrivial mathematical problem that has a solution and is related to vector calculus and gradient fields. Its complexity and difficulty make it a perfect representation of Prof. Nash's genius mind.