<< Chapter < Page Chapter >> Page >
d V = d x 1 d x 2 d x 3

Suppose, however, that it is convenient to describe the position by some other coordinates, say ξ 1 , ξ 2 , ξ 3 . We may ask what volume is to be associated with the three small changes d ξ 1 , d ξ 2 , d ξ 3 .

The change of coordinates must be given by specifying the Cartesian point x that is to correspond to a given set ξ 1 , ξ 2 , ξ 3 , by

x i = x i ( ξ 1 , ξ 2 , ξ 3 )

Then by partial differentiation the small differences corresponding to a change d ξ i are

d x i = x i ξ j d ξ j

Let d x ( j ) be the vectors with the components ( δ x i / δ ξ j ) d ξ j for j = 1 , 2 , a n d 3 . Then the volume element is

d V = d x ( 1 ) d x ( 2 ) × d x ( 3 ) = ε i j k x i ξ 1 d ξ 1 x j ξ 2 d ξ 2 x k ξ 3 d ξ 3 = J d ξ 1 d ξ 2 d ξ 3

where

J = x 1 , x 2 , x 3 ξ 1 , ξ 2 , ξ 3 = ε i j k x i ξ 1 x j ξ 2 x k ξ 3

is called the Jacobian of the transformation of variables

Vector fields

When the components of a vector or tensor depend on the coordinates we speak of a vector or tensor field. The flow of a fluid is a perfect realization of a vector field for at each point in the region of flow we have a velocity vector v ( x ) . If the flow is unsteady then the velocity depends on the time as well as position, v = v ( x , t ) .

Associated with any vector field a ( x ) are its trajectories , which is the name given to the family of curves everywhere tangent to the local vector a . They are solutions of the simultaneous equations

d x d s = a ( x ) ; that is d x i d s = a i ( x 1 , x 2 , x 3 )

Where s is a parameter along the trajectory. (It will be arc length if a is always a unit vector.) Streamlines of a steady flow are a realization of these trajectories. For a time dependent vector field the trajectories will also be time dependent. If C is any closed curve in the vector field and we take the trajectories through all points of C , they describe a surface known as the vector tube of the field. For flow fields, it is called a stream tube .

The vector operator -gradient of a scalar

The symbol (enunciated as "del") is used for the symbolic vector operator whose i t h component is δ / δ x i . Thus if operates on a scalar function of position ϕ ( x ) it produces a vector ϕ with components δ ϕ / δ x i .

ϕ x ¯ j = ϕ x i x i x ¯ j = l i j ϕ x i
grad ϕ = ϕ = ϕ , i = e ( 1 ) ϕ x 1 + e ( 2 ) ϕ x 2 + e ( 3 ) ϕ x 3

since a ¯ j = l i j a i so ϕ is a vector. The suffix notation, i for the partial derivative with respect to x i is a very convenient one and will be taken over for the generalization of this operation that must be made for non-Cartesian frame of reference. The notation "grad" for is often used and referred to as the gradient operator. Thus grad ϕ is the vector which is the gradient of the scalar. The gradient operator can also operate on higher order tensors and the operation raises the order by one. Thus the gradient of a vector a is grad a , a , or in component notation a i , j . is sometimes written δ / δ x and can be expanded as the vector operator

The suffix notation, i for the partial derivative with respect to x i is a very convenient one and will be taken over for the generalization of this operation that must be made for non-Cartesian frame of reference. The notation "grad" for is often used and referred to as the gradient operator. Thus grad ϕ is the vector which is the gradient of the scalar. The gradient operator can also operate on higher order tensors and the operation raises the order by one. Thus the gradient of a vector a is grad a , a , or in component notation a i , j . is sometimes written δ / δ x and can be expanded as the vector operator

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Transport phenomena' conversation and receive update notifications?

Ask