Can Solving H²_DR(ℝ³ - X) Take a Lifetime?

[dglee]

I was bored so I am deciding that since I am 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\}<br />

dim(V/W) = ?

 

[James R]

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

 

[HallsofIvy]

I was bored so I am deciding that since I am 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\}<br />
dim(V/W) = ?

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.

 

[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.

 

[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...

 

[Pyrrhus]

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

 

[Brad Barker]

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

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

 

[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.)

 

[dglee]

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

 

[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"

 

[Treadstone 71]

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 explanation given above.

 

[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.

It's a question about excision.