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This report summarizes work done as part of the Calculus of Variations PFUG under Rice University's VIGRE program. VIGRE is a program ofVertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. APFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed around the study of a common problem. This modulecontinues work from last year's research into {}``kinetic energy\char`\"{} of unit-length vector fields on surfaces of rotation, focusing mainlyon minimizing energy given a surface and boundary conditions. We answer some of the questions posed by the previous summer's PFUG and extendtheir work onto new surfaces. This work was studied in the Rice University VIGRE 2010 Summer Undergraduate Internship Program.


It is highly recommended that the reader familiarize themselves with [2] before approaching this module. Both the problems and the techniques used to solve them will draw heavily from that work. Wewill first answer questions previously raised concerning the unit cylinder before moving on to the study of similar vector fields onthe torus.

The cylinder

First, we turn our attention to the unit cylinder (with radius and height 1) for futher investigation on [2] work.

Parametrization and equations

We borrow heavily from [2] . Our surface, the unit cylinder with radius r = 1 may be parameterized as Φ ( θ , t ) = ( cos θ , sin θ , t ) . Our vector field is parameterized as V ( ϕ ) = ( - sin θ cos ϕ , cos θ cos ϕ , sin ϕ ) : 0 θ 2 π , 0 t 1 and the corresponding energy equation on the cylinder is

E ( ϕ ) = 0 1 0 2 π cos ϕ 2 + ϕ θ 2 + ϕ t 2 d θ d t

which, of course, is a specific case of the general equation

As [2] shows, the minimizing ϕ must satisfy

Δ ϕ + sin 2 ϕ 2 = 0 , 0 ϕ 2 π .

θ -independence

We recall and restate Theorem 5 from [2] :

Let a surface S with radius r ( t ) be given. Suppose that the boundary conditions ϕ ( θ , 0 ) , ϕ ( θ , h ) of a vector field on S do not depend on θ : that is, ϕ ( θ , 0 ) = ϕ 0 , ϕ ( θ , h ) = ϕ h for all θ [ 0 , 2 π ] and constant ϕ 0 , ϕ h . The function ϕ ( θ , t ) which minimizes energy given constant boundary conditions ϕ ( θ , 0 ) = ϕ 0 , ϕ ( θ , h ) = ϕ h does not depend on θ . In other words, the vector field described by ϕ is constant along every ”horizontal slice" of the surface.

Existence and uniqueness

We now aim to show that such a unique energy-minimizing ϕ does in fact exist on this cylinder.

Theorem 1

Let ( - sin θ cos ϕ 0 , cos θ cos ϕ 0 , sin ϕ 0 ) and ( - sin θ cos ϕ 1 , cos θ cos ϕ 1 , sin ϕ 1 ) be “constant” boundary data on the bottom and top of the unit cylinder respectively. Then the minimizer exists.


Let E = inf { V t a n g e n t s m o o t h u n i t v e c t o r f i e l d } E ( V ) . Let C be the following the set of smooth functions:

C = { ϕ C ( [ 0 , 1 ] ) : ϕ ( 0 ) = ϕ 0 , ϕ ( 1 ) = ϕ 1 , | ϕ | < 2 π , E ( ϕ ) < E + 1 } .

Lemma 1

Suppose that V k are a sequence of unit tangent vector fields with E ( V k ) E . Then we can replace this sequence by another sequence of vector fields V k so that when we write the V k using an angle fucntion ϕ k , ϕ k is constant in θ and so that the energies E ( V k ) still approach E .


Suppose that V k depends on θ . Then there exists a θ -independent V j such that E ( V j ) E ( V k ) . E ( V k ) E thus implies E ( V j ) E .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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