Vector Calculus in A Beautiful Mind
Jason Parsley
Wake Forest University
29 November 2010
Wednesday Warmup, 1/20
Vector Calclus in A Beautiful Mind
John Nash
John Nash, older
Crowe, as Nash
A Beautiful Mind
John Nash (Russell Crowe) is teaching vector calculus, um, in
his own way. After discarding the book, he poses the following
problem:
Jason Parsley
A Beautiful Mind
Wednesday Warmup, 1/20
What did Crowe/Nash say?
Problem (Crowe’s Problem)
V ={F : (R3 \ X ) → R3
W ={F = ∇g}
so ∇ × F = 0}
{‘Gradients’}
dim(V \ W ) = ?
Here X is a subset of R3 ; ∇ × F is the curl of F.
Fact: The curl of ∇g is zero, for any gradient.
Crowe claims this it will take
“for some of you, many months to solve”
“for others among you, it will take you the term of your
natural lives”
Jason Parsley
A Beautiful Mind
Wednesday Warmup, 1/20
Solving Crowe’s Problem
Problem (Crowe’s Problem)
V ={F : (R3 \ X ) → R3
W ={F = ∇g}
so ∇ × F = 0}
{‘Gradients’}
dim(V \ W ) =?
Hollywood was being overly dramatic. This is a great problem,
and relates to my research. But, it’s not that hard to solve!
My Math 733 class solved this in a few lectures last spring!
It relates to how many different nontrivial loops you can draw on
the subset.
Jason Parsley
A Beautiful Mind
Wednesday Warmup, 1/20
Solving Crowe’s Problem
Problem (Crowe’s Problem)
V ={F : (R3 \ X ) → R3
W ={F = ∇g}
so ∇ × F = 0}
{‘Gradients’}
dim(V \ W ) =?
Answer:
dim(V \ W ) = # different nontrivial loops on X
If X is simply-connected, (e.g., a solid ball or a solid cube), then
there aren’t any nontrivial loops, and dim(V \ W ) = 0.
Let X be R3 − z-axis. Is this simply-connected? no.
There’s one nontrivial loop, which goes around the z-axis. The
vector field
−y
x
F= 2
i+ 2
j
2
x +y
x + y2
has ∇ × F = 0, but is not
gradient,
as Mind
we showed in class.
Jasona
Parsley
A Beautiful