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Curves, surfaces, and volumes

We will be dealing with regions of space, V , having volume that may be bound by surfaces, S, having area. Regions of the surface may be bound by a closed curve, C , having length.

Surfaces are defined by one relationship between the spatial coordinates.

x 3 = f ( x 1 , x 2 ) , o r F ( x 1 , x 2 , x 3 ) = 0 , o r F ( x ) = 0

Alternatively, a pair of surface coordinates, u 1 , u 2 can define a surface.

x i = x i ( u 1 , u 2 ) , i = 1 , 2 , 3 o r x = x ( u 1 , u 2 )

Each point on the surface that has continuous first derivatives has associated with it the normal vector, n , a unit vector that is perpendicular or normal to the surface and is outwardly directed if it is a closed surface. Fluid-fluid interfaces need to also be characterized by the mean curvature, H, at each point on the surface to describe the normal component of the momentum balance across the interface. The flux of a vector, f , across a differential element of the surface is denoted as follows, i.e. the normal component of the flux vector multiplied by the differential area.

f · n d a

Curves are defined by two relationships between the spatial coordinates or by the intersection of two surfaces.

f 1 ( x 1 , x 2 , x 3 ) = 0 a n d f 2 ( x 1 , x 2 , x 3 ) = 0 , o r f 1 ( x ) = 0 a n d f 2 ( x ) = 0

Alternatively, a curve in space can be parameterized by a single parameter, such as the distance along the curve, s or time, t .

x = x ( s )

The tangent vector is a unit vector that is tangent to each point on the curve.

τ = d x ( s ) / d s

The component of a vector, f , tangent to a differential element of a curve is denoted as follows.

f · τ d s

If the parameter along the curve is time, the differential of position with respect to time is the velocity vector and the differential of velocity is acceleration.

v = d x d t a = d v d t MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaaCODaiabg2da9maalaaabaGaamizaiaahIhaaeaacaWGKbGaamiDaaaaaeaacaWHHbGaeyypa0ZaaSaaaeaacaWGKbGaaCODaaqaaiaadsgacaWG0baaaaaaaa@419A@

Coordinate systems

Scalars, vectors, and tensors are physical entities that are independent of the choice of coordinate systems. However, the components of vectors and tensors depend on the choice of coordinate systems. The algebra and calculus of vectors and tensors will be illustrated here with Cartesian coordinate systems but these operations are valid with any coordinate system. The student is suggested to read Aris to learn about curvilinear coordinate systems. Bird, Stewart, and Lightfoot express the components of the relevant vector and tensor equations in Cartesian, cylindrical polar, and spherical polar coordinate systems.

Cartesian coordinates have coordinate axes that have the same direction in the entire space and the coordinate values have the units of length. Curvilinear coordinates, in general, may have coordinate axis that are in different directions at different locations in space and have coordinate values that may not have the units of length, e.g., θ in the cylindrical polar system. If ( y 1 , y 2 , y 3 ) are Cartesian coordinates and ( x 1 , x 2 , x 3 ) are curvilinear coordinates, a differential length is related to the differential of the coordinates by the following relations.

d s 2 = k = 1 3 d y k d y k d y k = y k x i d x i y k x 1 d x 1 + y k x 2 d x 2 + y k x 3 d x 3 d s 2 = k = 1 3 ( y k x i d x i ) ( y k x j d x j ) = g i j d x i d x j g i j = k = 1 3 ( y k x i ) ( y k x j ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@C8B9@

where g ij are components of the metric tensor which transforms differential of the coordinates to differential of length. Summation is understood for repeated indices. Calculus in a curvilinear coordinate system will require the metric tensor.

Questions & Answers

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.
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, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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