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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 of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed around the study of a common problem. This module investigates the ``kinetic energy" of unit-length vector fields on surfaces of rotation, focusing mainly on minimizing energy given a surface and boundary conditions. Questions of existence and uniqueness are explored. This work was studied in the Rice University VIGRE 2009 Summer Undergraduate Internship Program.

Introduction

In his classic 1710 treatise, the Théodicée , philosopher and mathematician Gottfried Leibniz set out to explore the problem of evil. Faced with the question of how an omniscient and completely benevolent deity could create a world in which pain and misfortune were ever-present, Leibniz proposed the theory of the “best of all possible worlds." Our world, he argued, must have been chosen by God because it maximizes “goodness" overall. Were anything different, even just slightly– say, the death toll of World War II were lower– then in the grand scheme of things, the world would actually be worse.

Leibniz's argument foresaw, at least philosophically, the development of the calculus of variations in mathematics. One can associate a “cost" with a “path" between two “points," which depends on how the path changes in time and space. Here the words “path," “point," and “cost" are used very abstractly. In the Théodicée , Leibniz took as a “path" the sequence of events in the world, between the beginning and end of time; the “cost" of this path is its “goodness." (Defining “goodness," of course, is another problem in itself.) Calculus of variations aims to determine, given such a cost, which path will minimize it. Seen from this perspective, Leibniz imagined God as the ultimate mathematician.

This summer, our VIGRE group has spent eight weeks applying techniques from the calculus of variations to a problem similar to Leibniz's, albeit of slightly reduced scope. On a surface of rotation (imagine a shape made with clay on a pottery wheel), one can define a unit-length vector field (imagine an infinite collection of arrows, all of equal length, such that one arrow is tacked to every point on the shape). There is a sense of “energy" to this vector field; a field in which every arrow points in the same direction is boring compared to one in which the arrows spin wildly. We take this “energy" to be our cost; our paths are the possible vector fields that can be placed on the surface. Our goal is to determine, given a specific shape and a vector field on its boundaries, what vector field on the rest of the shape has minimal energy. In examining this and related questions, we have touched on topics from a number of fields, including functional analysis, differential geometry, and topology.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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