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f ( x ) = 1 1 + 25 x 2 .

As we increase the degree of the approximation, and supposedly increase the accuracy of the approximation, the following figure is produced:

The red curve is the function we are trying to approximate on a set of evenly spaced points.The blue curve is a 5 th order approximation, and the green curve is a 9 th order approximation. As we can see, the approximation actually becomes worse; this is a pitfall we would like to avoid.

Chebyshev points, fourier matrices, and plots

To sidestep the possibility of suffering Runge's Phenomenon, we can redefine our grid using Chebyshev points instead of equally spaced points. The Chebyshev points cluster near the boundary of the grid (in the two dimensional case, the interval), and are more widely spaced in the center of the grid. This forces the interpolation to be much more accurate on the edges of the grid, bounding the error that can occur. In fact, using this method causes the interpolation to converge extremely rapidly.

To handle periodicity in the θ variable, we borrow some theory about Fourier Discretization Matrices from Trefethen's Spectral Methods in MATLAB [link] . These matrices allow us to solve our differential equation on the interior of the cylinder while leaving the boundary conditions fixed.

With this framework, our program allows us to input a height h , a radius function r ( t ) , and two functions f ( θ ) = ϕ ( θ , 0 ) and g ( θ ) = ϕ ( θ , h ) that describe the boundary conditions, and finds a very close approximation of a function ϕ ( θ , t ) which satisfies Equation [link] . Some images produced by this program can be seen in Section (10).

Limitations and future work

Decisions made at the beginning of our research period have, of course, led our work in a particular direction. As is often the case, these decisions placed certain limitations on our results that were not foreseen at the time. The limitations are not insurmountable, nor do they invalidate any of the above, but are worth thinking about.

We have primarily discussed a function ϕ : [ 0 , 2 π ] × [ 0 , h ] R , periodic in θ , as a representative for a vector field on a surface S . Certainly, every such function defines a vector field via the relation given in equation [link] . However, this vector field is not uniquely defined: ϕ n = ϕ 0 + 2 n π describes the same vector field as ϕ 0 for all integers n .

Furthermore, not every vector field can be described by such a ϕ . By requiring ϕ to be periodic, the vector field V ( ϕ ) must “untwist" as much as it “twists." This is convenient, for multiple reasons. Periodicity simplifies many of the integration by parts steps taken in our proofs. It also prevents a situation in which the top and bottom boundary conditions have different numbers of twists. Given such conditions, it is topologically impossible to define a continuous vector field between them. However, placing a periodicity requirement on ϕ also rules out perfectly valid vector fields: for example, a field with exactly one twist on every horizontal slice.

It would not be overly difficult, we imagine, to rethink our results to accommodate continuous vector vields that cannot be represented by periodic ϕ . One result which is slightly troublesome is uniqueness on a short cylinder (Theorem 1). Given boundary conditions that are constantly “straight up" on the bottom and “straight down" on the top, and a vector field V between them, one can construct a vector field V ˜ with identical energy to V by having V ˜ rotate clockwise wherever V rotates counterclockwise, and vice versa.

Perhaps another early limiting choice which was looking specifically at unit-length vector fields on unit-height cylinders with unit radius. Fixing three quantities greatly restricts the problem, and while this was certainly conductive to finding early results, we were misled to search for analogues in more general cases. Again, uniqueness comes to mind; somehow, the ratio between vector length and cylinder radius creates a special value between 8 and 10 . Once we turned our attention to surfaces of general radius and height, we did not find such a point, and it is quite possible that there is no equivalent.

There are a number of directions in which we can continue our research. Resolving the above questions is one such direction. We would also like to answer the questions about existence of solutions to Equation [link] that have nagged us since the beginning: is there always a vector field which attains the minimal energy value on any given surface? We have conjectured many “limiting results," as in "Limiting Results on the Cylinder" , that we suspect are true but have not proven yet.

Another problem that we have only just begun to examine is, in a way, a reversal of the current problem. Given a function ϕ ( θ , t ) : [ 0 , 2 π ] × [ 0 , h ] R , can we find a radius function r ( t ) : [ 0 , h ] ( 0 , ) describing a surface of rotation S such that the energy of the vector field described by ϕ on S is minimal over all surfaces of height h ? Since energy is inversely proportional to radius, r ( t ) tends toward for all t ; we must somehow constrain r . Bounding its derivative and fixing the surface area of S are two approaches we have considered. This problem seems to have plenty of potential for future research.

Numerical plots

In this section we reproduce some numerical results as described in Section (7).

First, the plot of ϕ and surface for a cylinder of height 1 with boundary conditions f ( θ ) = c o s ( θ ) and g ( θ ) = s i n ( θ )

Second, the plot of ϕ and surface with radius r ( t ) = 2 + s i n ( t ) and boundary conditions f ( θ ) = c o s ( θ ) and g ( θ ) = s i n ( θ )


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 round 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 Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundation grant DMS-0739420. We would like to thank the faculty mentors, Bob Hardt, Leo Rosales, and Mike Wolf, and our graduate student mentors, Chris Davis and Renee Laverdiere, for all their help this summer. Without their assistance, this paper would be noticeably shorter. We are also indebted to a number of members of the math department who volunteered advice, notably Frank Jones and Rolf Ryham. Nearly all of the numerical approximation work is thanks to Mark Embree, who, after hearing us present our work, accomplished in an hour what we had been unable to do all summer. We also thank the undergraduate members of this group, Yan Digilov, Bill Eggert, Michael Jauch, Rob Lewis, and Hector Perez.

Questions & Answers

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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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