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divergence-free vector fields in the plane and on torus
Posted:
Sep 16, 1991 6:28 AM
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DIVERGENCE-FREE VECTOR FIELDS IN R^2 AND ON TORUS
by
Wlodzimierz Holsztynski
Notation:
R - the field of reals
Z - the ring of (rational) integers
Pi - the smallest positive real for which sin(pi)=0
T = (R / 2*Pi*Z) ^ 2 - the torus
q : R^2 --> T - the canonical quotient hom. (of ab. gp's)
Div(V) = D1(V_1) + D2(V_2) is the divergence of vector
field V(x,y) = (V_1(x,y), V_2(x,y)) in R^2 or
on torus (D1 & D2 are the partial derivatives).
Q = R^2 \ { (0,0) }
S = { (x,y) in R^2 | x^2 + y^2 = 1 } - the unit circle
u : Q --> S - the normalizing map;
u(x,y) = (x,y)/(x^2 + y^2)^(1/2)
A notion named N introduced by a definition, appears
in that definition as *N* (so that it's easy to see
what is defined).
o - the symbol of composition of functions
------------------------------------
For homotopy classes of nowhere vanishing vector fields
on torus it is well known that:
[T, Q] = [T, S] = H^1(T) = Z^2
Definition. W : R^2 --> Q is *diperiodic* if there exists
V : T --> S such that V o q = u o W.
We may consider the homotopy equivalence within the diperiodic
vector fields in R^2. The homotopy classes of diperiodic vector
fields, [[R^2, Q]], are in a bijective correspondence (induced
by q and u) with [T, S] = Z^2.
We want to study the divergence-free vector fields, i.e. fields W
for which Div(W) = 0;
The following divergence-free diperiodic vector fields W : R^2 --> Q
represent all classes [[R^2, Q]] (one per class) ;
W(x,y) = exp(-k*x + n*y) * ( cos(n*x + k*y), sin(n*x + k*y) )
for every (x,y) in R^2, where k and n are arbitrary integers;
parameters (k,n) associate the above examples W with Z^2 = [T, S]
= [[R^2, Q]].
The diperiodic divergence-free vector fields are the best thing
next to divergence-free vector fields on torus. We think that:
CONJECTURE. All (everywhere non-vanishing) divergence-free vector
fields on torus are homotopically trivial.
I didn't just "guess" the above examples. If there is still
an interest in this topic I may "derive" my examples and provide
my motivation behind the conjecture.
Acknowledgment:
Kenton Yee has asked about "topologically non-trivial" (:-)
divergence-free vector fields on torus in an article on sci.math
(without requiring that they are nowhere vanishing). For this I am grateful
to him. However I didn't appreciate his "mathematical" style nor his
arrogant and egoistic attitude toward sci.math (because of that I didn't
feel like contributing to this otherwise interesting thread; only the
creation of sci.math.research somehow has caused me to change my mind).
Regards,
Wlodek
PS. I'd be grateful for related references and quotations
(since I have difficulties to reach any math library).
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