A beautiful Mind

Jerri Jordan

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)=?

Delphi91

Jerri Jordan wrote: »

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)=?

Interesting, in doing a little "googling" to find a possible answer, I came across the following 7 year old post:http://www.physicsforums.com/showthread.php?t=25174 Look familiar???


Anyway, I digress. Have a look here: http://www.oocities.org/halgravity/papers/bmath.pdf

Jerri Jordan

haha caught lovely!

equivariant

Jerri Jordan wrote: »

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)=?

As far as I can make out from this statement - the problem is to compute the dimension of H^1(R3|X). Here H^1 stands for "the first cohomology group" and R3|X is presumably menat to be the complement of some subspace X of R^3 (the notation is not clear). To fine an explanation of this, you could start with http://en.wikipedia.org/wiki/Curl_(mathematics)#Identities (and especially look at the section on differential forms)

Anyway, I digress. Have a look here: http://www.oocities.org/halgravity/papers/bmath.pdf

That paper seems to be gobbledygook - it was certainly not written by a competent mathematician (in my opinion, at least)

LeixlipRed

Think the clue is when he calls Russel Crowe, Rustle Crowe

Delphi91

LeixlipRed wrote: »

Think the clue is when he calls Russel Crowe, Rustle Crowe

Hehehe, never spotted that!

Michael Collins

equivariant wrote: »

That paper seems to be gobbledygook - it was certainly not written by a competent mathematician (in my opinion, at least)

Exactly, didn't spend much time on it but it did seem fairly nonsensical (to me).

By the way I think the actual question is:

[latex] \displaystyle \Large V = \{F:\rm I\!R^{3}\setminus X \to \rm I\!R^{3}| \nabla \times F = 0\} [/latex]

[latex] \displaystyle W = \{F = \nabla g\} [/latex]

[latex] \displaystyle \rm{dim}\left(V/W\right) = 8 [/latex]

Where the first equation says that V is the set of functions F, where F maps from the 3-dimensional reals with the exception of a subset X, to the 3-dimensional reals, such that the curl of F is zero i.e. any closed line integral outside the space X is zero.

The second equation says W is the set such that there exists a potential function g for F, i.e. the set such that F is a conservative field.

And finally you're told that the dimension of the quotient space between the two is 8, and this is where I don't know any more.

There seems to be a good PDF on it here: http://www.math.harvard.edu/~huizenga/Math21aF10/LECTURE35WS.PDF