# Homework Help: Question From Beatiful Mind

1. Oct 23, 2005

### dglee

I was bored so im deciding that since im only at the level of undergraduate real analysis, i wonder if anybody could solve this problem. In the movie it said that it would take someone months or many yrs or even a life time. Wonder if this is true or not?

$$V = \{ F : \Re^3 \setminus X \rightarrow \Re^3 \left so\right \nabla \times F = 0 \}$$

$$W = \{ F= \nabla g\}$$

$$dim(V/W) = ?$$

Last edited: Oct 23, 2005
2. Oct 23, 2005

### James R

Does the problem as written make sense to anybody here? Mathematicians?

3. Oct 23, 2005

### HallsofIvy

As a rule, even mathematicians need to know what you are talking about before they can answer! I presume you are saying that V is the set of all functions defined on a given subset of R3 with curl 0- but I have no idea what g is.

4. Oct 23, 2005

### dglee

hmm well i copied that question straight from the movie.... that movie with Russel Crowe as John Nash. Where he teaches multivariable calculus and he put that question down on the board.

5. Oct 23, 2005

### James R

I think we'd need to know what X and g are, to start with.

I'm not even sure what the notation dim(V/W) means...

6. Oct 23, 2005

### Pyrrhus

It took me 5 seconds to solve this, the problem is meaningless because nothing is defined :rofl:

7. Oct 24, 2005

you mean that mathematics isn't being treated properly in a hollywood movie!? :tongue2:

8. Oct 24, 2005

### amcavoy

I believe this arose in physics, although that is the only thing I could find. (I, too, was interested in what this was after seeing the movie.)

9. Oct 25, 2005

### dglee

i hope somebody can answer this.... lol im really interested in what it means.

10. Oct 25, 2005

### Pyrrhus

"The problem Bayer finally chose (see photo) was a more complicated version of a classical physics problem: determining whether a static electric field (the F in lines 1 and 2) necessarily has a potential function (indicated by g). If the “electric field” is allowed to be infinite or simply nonexistent at certain points (collectively indicated by X), the question becomes physically unrealistic but mathematically very rich. The answer depends not only on the geometry of the set X but also on one’s assumptions about the field F—as the fictional Nash explains to Alicia rather brusquely when she offers her stab at a solution."

http://www.swarthmore.edu/bulletin/june02/bayer.html" [Broken]

Last edited by a moderator: May 2, 2017
11. Apr 26, 2006

Sorry about digging up an old thread, but I've never encountered such notation before. Is this an older notation? I'm not in physics so I don't understand the explaination given above.

12. Apr 26, 2006

### matt grime

the fancy R is just R as in the real numbers.

The question is, given V the space of functions {f} defined on V\X such that curl(f)=0, and W the subspace of functions that are grad(g) for some g, what is the dimension of the quotient space V/W? Or, more succinctly, what is the dimension of

$$H^2_{DR}(\mathbb{R}^3 - X)$$

for X some subset of R^3, or something. If X is the empty set then the answer is zero.