~ 1 ~ Time-dependent Complex Vector Fields & the Drifting Vessel John Gill January 2014 Abstract: Elementary comments on time-dependent vector fields and their contours. How does a floating vessel drift in such a vector field? Where to place the vessel so its path ends at a certain point? Evaluating work done by an object in a force field above the pond as it tracks the vessel. A continued fraction is interpreted as a TDVF. Definition Zeno contour: Let , , () () kn kn g z z z η ϕ = + where z S ∈ and , () kn g z S ∈ for a convex set S in the complex plane. Require , lim 0 kn n η →∞ = , where (usually) 1,2,..., k n = . Set 1, 1, () () n n G z g z = , ( ) , , 1, () () kn kn k n G z g G z - = and , () () n nn G z G z = with () lim () n n Gz G z →∞ = , when that limit exists. The Zeno contour is a graph of this iteration. The word Zeno denotes the infinite number of actions required in a finite time period if , kn η describes a partition of the time interval [0,1]. Normally, () () z fz z ϕ = - for a vector field () = F fz , and (,) (,) ϕ = - zt f zt z for a time-dependent vector field , in which case , , () (,) η ϕ = + ⋅ k kn kn n g z z z . Begin with , 1 η = kn n and , 1 () (,) ϕ ≡ + k kn n g z z z n with (,) ϕ zt continuous on a domain [0,1] × S , and , () kn z S g z S ∈ ⇒ ∈ . (A Zeno contour forms by iteration ( ) 1, , , , : ( ,) k n k n kn n kn kn n z z fz z η + ϒ = + - ) We have 3 1 2 , 1, 2, 1, 1 1 1 1 () (,) ( ( ), ) ( ( ), ) ( ( ), ) ϕ ϕ ϕ ϕ - = + + + + + n nn n n n n n n n n G z z z G z G z G z n n n n . Now, imagine a function ( ) [ ] ( ) ( ) 1 1, 0 , , 0,1 and , lim ( ), , with , defined ψ τ τ ψ ϕ ψ τ τ - →∞ ∈ ≡ ∫ k mk mn n m k z z G z z d n : 1 0 1 1 1 2 1 3 1 () , , , , (,) ψ ψ ψ ψ ψ τ τ - = + + + + ≈ ∫ n n G z z z z z z z d n n n n n n n n And for τ irrational, (,) lim (, ) τ τ ψ τ ψ τ → = r r z z for rational τ r .
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Time-Dependent Complex Vector Fields & the Drifting Vessel
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~ 1 ~
Time-dependent Complex Vector Fields & the Drifting Vessel
John Gill January 2014
Abstract: Elementary comments on time-dependent vector fields and their contours. How does a floating vessel
drift in such a vector field? Where to place the vessel so its path ends at a certain point? Evaluating work done by
an object in a force field above the pond as it tracks the vessel. A continued fraction is interpreted as a TDVF.
Definition Zeno contour: Let , ,
( ) ( )k n k n
g z z zη ϕ= + where z S∈ and ,
( )k n
g z S∈ for a convex
set S in the complex plane. Require ,lim 0
k nn
η→∞
= , where (usually) 1,2,...,k n= . Set
1, 1,( ) ( )
n nG z g z= , ( ), , 1,
( ) ( )k n k n k n
G z g G z−= and ,
( ) ( )n n n
G z G z= with ( ) lim ( )nn
G z G z→∞
= , when
that limit exists. The Zeno contour is a graph of this iteration. The word Zeno denotes the
infinite number of actions required in a finite time period if ,k nη describes a partition of the time
interval [0,1]. Normally, ( ) ( )z f z zϕ = − for a vector field ( )=F f z , and ( , ) ( , )ϕ = −z t f z t z
for a time-dependent vector field , in which case , ,
( ) ( , )η ϕ= + ⋅ kk n k n n
g z z z .
Begin with ,
1η =
k nn
and ,
1( ) ( , )ϕ≡ + k
k n ng z z z
n with ( , )ϕ z t continuous on a domain [0,1]×S , and
,( )
k nz S g z S∈ ⇒ ∈ . (A Zeno contour forms by iteration ( )1, , , ,