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x ( s + d s ) - x ( s ) d s = x ˙ ( s ) + O ( d s ) and x ( s + d s ) - 2 x ( s ) + x ( s - d s ) d s 2 = x ¨ ( s ) + O ( d s 2 )

Thus, in the limit when the points are coincident, the plane reaches a limiting position defined by the first two derivatives x ˙ ( s ) and x ¨ ( s ) . This limiting plane is called the osculating plane and the curve appears to lie in this plane in the intermediate neighborhood of the point. To prove this statement: (1) A plane is defined by the two vectors, x ˙ ( s ) and x ¨ ( s ) , if they are not co-linear. (2) The coordinates of the three points on the curve in the previous two equations are a linear combination of x ( s ) , x ˙ ( s ) and x ¨ ( s ) , thus they line in the plane.

Now x ˙ = τ so x ¨ = τ ˙ and since τ τ = 1 ,

d τ τ d s = 0 = τ ˙ τ + τ τ ˙ = 2 τ τ ˙ τ τ ˙ = 0

so that the vector τ ˙ is at right angles to the tangent. Let 1 / ρ denote the magnitude of τ ˙ .

τ ˙ τ ˙ = 1 ρ 2 and ν = ρ τ ˙

Then ν is a unit normal and defines the direction of the so-called principle normal to the curve.

To interpret ρ , we observe that the small angle d θ between the tangents at s and s + d s is given by

cos d θ = τ ( s ) τ ( s + d s ) 1 - 1 2 d θ 2 + . . . = τ τ + τ τ ˙ d s + 1 2 τ τ ¨ d s 2 + . . . = 1 - 1 2 τ ˙ τ ˙ d s 2 + . . .

since τ τ ˙ = 0 and so τ τ ¨ + τ ˙ τ ˙ = 0 . Thus,

ρ = d s d θ

is the reciprocal of the rate of change of the angle of the tangent with arc length, i.e., ρ is the radius of curvature. Its reciprocal 1 / ρ is the curvature, κ d θ / d s = 1 / ρ .

A second normal to the curve may be taken to form a right-hand system with τ and ν . This is called the unit binormal,

β = τ × ν

Line integrals

If F ( x ) is a function of position and C is a curve composed of connected arcs of simple curves, x = x ( t ) , a t b or x = x ( s ) , a s b , we can define the integral of F along C as

C F ( x ) d t = a b F x ( t ) d t or C F ( x ) d s = a b F x ( t ) x ˙ ( t ) x ˙ ( t ) 1 / 2 d t

Henceforth, we will assume that the curve has been parameterized with respect to distance along the curve, s .

The integral is from a to b . If the integral is in the opposite direction with opposite limits, then the integral will have the same magnitude but opposite sign. If x ( a ) = x ( b ) , the curve C is closed and the integral is sometimes written

C F [ x ( s ) ] d s

If the integral around any simple closed curve vanishes, then the value of the integral from any pair of points a and b is independent of path. To see this we take any two paths between a and b , say C 1 and C 2 , and denote by C the closed path formed by following C 1 from a to b and C 2 back from b to a .

C F d s = a b F d s C 1 + b a F d s C 2 = a b F d s C 1 - a b F d s C 2 = 0

If a ( x ) is any vector function of position, a τ is the projection of a tangent to the curve. The integral of a τ around a simple closed curve C is called the circulation of a around C .

C a τ d s = C a i x 1 ( s ) , x 2 ( s ) , x 3 ( s ) τ i d s

We will show later that a vector whose circulation around any simple closed curve vanishes is the gradient of a scalar.

Surface integrals

Many types of surfaces and considered in transport phenomena. Most often the surfaces are the boundaries of volumetric region of space where boundary conditions are specified. The surfaces could also be internal boundaries where the material properties change between two media. Finally the surface itself may the subject of interest, e.g. the statics and dynamics of soap films.

A proper mathematical treatment of surfaces requires some definitions. A closed surface is one which lies within a bounded region of space and has an inside and outside. If the normal to the surface varies continuously over a part of the surface, that part is called smooth . The surface may be made up of a number of subregions, which are smooth and are called piece-wise smooth . A closed curve on a surface, which can be continuously shrunk to a point, is called reducible . If all closed curves on a surface are reducible, the surface is called simply connected . The sphere is simply connected but a torus is not.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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
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Akash Reply
it is a goid question and i want to know the answer as well
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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.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar 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, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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