ICE: NASH’S PROBLEM Math 21a In this ICE, you solve Nash’s problem, he gave to a multivariable calculus class. Remember Nash saying in the movie ”A beautiful mind”: ”This problem here will take some of you many months to solve, for others amoung you it might take a life time”. You will solve it here in 10 minutes ... NASH’s PROBLEM. Find a subset X of R 3 with the property that if V is the set of vector fields F on R 3 \ X which satisfy curl(F ) = 0 and W is the set of vector fields F which are conservative: F = ∇f . Then, the space V/W should be 8 dimensional. Remark. The meaning of the last sentence means that there should be 8 vectorfields F i which are not gradient fields and which have vanishing curl outside X . Furthermore, you should not be able to write any of the 8 vectorfields as a sum of multiples of the other 7 vector fields. Here is a two dimensional version of the problem: 2D VERSION OF NASH’s PROBLEM. If X = {0} and V is the set of vector fields F on R 2 \ X which satisfy curl(F ) = 0 and W is the set of vector fields F which are gradient fields, then dim(V/W ) = 1. The vector field F is F (x, y)=(-y/(x 2 + y 2 ), x/(x 2 + y 2 )) has vanishing curl in R 2 \ X but the line integral around the origin is 2π. It is not a gradient field. Now: What set X would you have to take to get dim(V/W ) = 8? The SOLUTION OF THE 3D VERSION OF NASH’s PROBLEM can be obtained directly from the solution of the 2D version. How?

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In this ICE, you solve Nash's problem

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ICE: NASH’S PROBLEM Math 21a

In this ICE, you solve Nash’s problem, he gave to a multivariable calculus class. RememberNash saying in the movie ”A beautiful mind”:

”This problem here will take some of you many monthsto solve, for others amoung you it might take a life time”.

You will solve it here in 10 minutes ...

NASH’s PROBLEM. Find a subset X of R3 with the property that if V is the set of vectorfields F on R3 \ X which satisfy curl(F ) = 0 and W is the set of vector fields F which areconservative: F = ∇f . Then, the space V/W should be 8 dimensional.

Remark. The meaning of the last sentence means that there should be 8 vectorfields Fi whichare not gradient fields and which have vanishing curl outside X. Furthermore, you should notbe able to write any of the 8 vectorfields as a sum of multiples of the other 7 vector fields.

Here is a two dimensional version of the problem:

2D VERSION OF NASH’s PROBLEM.

If X = {0} and V is the set of vector fields F on R2 \X which satisfy curl(F ) = 0 and W isthe set of vector fields F which are gradient fields, then dim(V/W ) = 1.

The vector field F is F (x, y) = (−y/(x2 + y2), x/(x2 + y2)) has vanishing curl in R2 \ X butthe line integral around the origin is 2π. It is not a gradient field.Now: What set X would you have to take to get dim(V/W ) = 8?

The SOLUTION OF THE 3D VERSION OF NASH’s PROBLEM can be obtained directlyfrom the solution of the 2D version. How?