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Example 2. Again let S be the unit cylinder, and define the tangent vector field V ( θ , t ) = ( - sin ( θ ) , cos ( θ ) , 0 ) (equivalently, ϕ ( θ , t ) = 0 ). V is the unit vector field horizontally tangent (in a counterclockwise direction) to the surface. We see from equation [link] that E ( V ) = 2 π . See the following figure.

Example 3. Let S be a frustum with base radius 2, top radius 1, and unit height: S = ( ( 2 - t ) cos ( θ ) , ( 2 - t ) sin ( θ ) , t ) . Let V again be the unit vector field horizontally tangent to S : V ( θ , t ) = ( - sin ( θ ) , cos ( θ ) , 0 ) . From equation [link] , we can calculate

E ( V ) = 0 1 0 2 π 2 2 - t cos 2 ( θ ) + sin 2 ( θ ) d θ d t = 2 2 π 0 1 1 2 - t d t = 2 2 π log 2 .

See the following figure.

Minimizing energy

Now that we have a solid concept of the energy of a vector field, we can answer a number of questions. Specifically, we are interested in finding an expression for the unit vector field with minimal energy on a given surface (if it exists), with specified boundary conditions. To do so, we use a technique from the calculus of variations.

Calculus of variations: an interlude

A common problem in calculus of variations asks a question very similar to ours: given a collection of paths y ( x ) : x [ a , b ] and a function L ( x , y ( x ) , y ' ( x ) ) , which path y minimizes the cost functional J [ y ] = a b L ( x , y ( x ) , y ' ( x ) ) d x ?

Suppose that y is such a minimizer. For any “perturbation" η ( x ) with η ( a ) = η ( b ) = 0 , we can consider the cost J ( ϵ ) = J [ y + ϵ η ] = a b L ( x , y ( x ) + ϵ η ( x ) , y ' ( x ) + ϵ η ' ( x ) ) d x . We calculate d J d ϵ and evaluate at ϵ = 0 . If y is a minimizing path, then ϵ = 0 should be a critical point of J ( ϵ ) . Supposing that 0 = d J d ϵ | ϵ = 0 , we obtain the famous Euler-Lagrange equation:

L y - d d x L y ' = 0 .

Any path y which minimizes the cost functional J must satisfy this differential equation. Note that the condition is necessary, not sufficient- not every function which satisfies [link] will produce minimal cost.

Our application: the unit cylinder

We wish to apply a similar technique in our situation. For now, we restrict our attention to the cylinder with unit radius, and only consider vector fields with unit length. Thus, it becomes convenient to use the angle notation mentioned in section 1.2: any vector field V ( θ , t ) : 0 θ 2 π , 0 t h in consideration can be represented by the angle ϕ ( θ , t ) that V ( θ , t ) makes with the horizontal tangent vector.

Suppose that ϕ is the angle representation of the unit vector field with minimal energy on the cylinder. Let η : [ 0 , 2 π ] × [ 0 , h ] R be a perturbation with η ( θ , 0 ) = η ( θ , h ) = 0 . Mimicking the calculus of variations technique, we want to plug into our cost functional, equation [link] . Dropping the ( θ , t ) arguments:

J ( ϵ ) = J [ y + ϵ η ] = 0 h 0 2 π cos ( ϕ + ϵ η ) 2 + ϕ θ + ϵ η θ 2 + ϕ t + ϵ η t 2 d θ d t 0 = d J d ϵ ( 0 ) = 0 h 0 2 π - 2 η cos ( ϕ ) sin ( ϕ ) + 2 ϕ θ η θ + ϕ t η t d θ d t

Since η is periodic in θ and η ( θ , 0 ) = η ( θ , h ) = 0 , integration by parts on the right-hand terms yields

0 = 0 h 0 2 π η Δ ϕ + sin ( 2 ϕ ) 2 d θ d t .

This expression holds for all perturbations η . Thus, we can deduce

Δ ϕ + sin ( 2 ϕ ) 2 = 0 .

Any function ϕ which describes the vector field of minimal energy on the unit cylinder will satisfy this equation. Assuming that ϕ is rotationally symmetric, or independent of θ (sometimes a reasonable assumption, we will see later), equation [link] becomes an ODE. Unfortunately, it is not a friendly ODE to solve. One can find approximate solutions by considering the Taylor series expansion of sin ( 2 x ) 2 , but such solutions are nonsatisfactory. A far better method of approximation is explored in "Computer Approximations" . And even if we cannot solve the equation analytically, we can still determine properties of solutions.

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?
what is a peer
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.
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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