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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
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Stoney Reply
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Adin Reply
?
Kyle
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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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
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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.
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s.
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s. Reply
of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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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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