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If a and b are two vectors, the set of nine products, a i b j =A ij , is a second order tensor, for

A ¯ p q = a ¯ p b ¯ q = l i p a i l j q b j = l i p l j q ( a i b j ) = l i p l j q A i j MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@6E07@

An important example of this is the momentum flux tensor. If ρ is the density and v is the velocity, ρ v i is the i th component in the direction Oi . The rate at which this momentum crosses a unit area normal to Oj is the tensor, ρ v i v j .

Scalar multiplication and addition

If α is a scalar and A a second order tensor, the scalar product of α and A is a tensor α A each of whose components is α times the corresponding component of A .

The sum of two second order tensors is a second order tensor each of whose components is the sum of the corresponding components of the two tensors. Thus the ij th component of A + B is A ij + B ij . Notice that the tensors must be of the same order to be added; a vector can not be added to a second order tensor. A linear combination of tensors results from using both scalar multiplication and addition. α A + ß B is the tensor whose ij th component is α A ij + ß B ij . Subtraction may therefore be defined by putting α = 1 , ß = -1 .

Any second order tensor can be represented as the sum of a symmetric part and an antisymmetric part. For

A i j = 1 2 ( A i j + A j i ) + 1 2 ( A i j - A j i ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaaiaadgeadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4kaSIaamyqamaaBaaaleaacaWGQbGaamyAaaqabaaakiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaGaamyqamaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsislcaWGbbWaaSbaaSqaaiaadQgacaWGPbaabeaaaOGaayjkaiaawMcaaaaa@4DFB@

and changing i and j in the first factor leaves it unchanged but changes the sign of the second. Thus,

A = 1 2 ( A + A ' ) + 1 2 ( A - A ' ) MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahgeacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaabmaabaGaaCyqaiabgUcaRiaahgeacaWHNaaacaGLOaGaayzkaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaaiaahgeacqGHsislcaWHbbGaaC4jaaGaayjkaiaawMcaaaaa@4510@

represents A as the sum of a symmetric tensor and antisymmetric tensor.

Contraction and multiplication

As in vector operations, summation over repeated indices is understood with tensor operations. The operation of identifying two indices of a tensor and so summing on them is known as contraction . A ii is the only contraction of A ij ,

A i i = A 11 + A 22 + A 33 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadgeadaWgaaWcbaGaamyAaiaadMgaaeqaaOGaeyypa0JaamyqamaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWkcaWGbbWaaSbaaSqaaiaaikdacaaIYaaabeaakiabgUcaRiaadgeadaWgaaWcbaGaaG4maiaaiodaaeqaaaaa@42DB@

and this is no longer a tensor of the second order but a scalar, or a tensor of order zero. The scalar A ii is known as the trace of the second order tensor A . The notation tr A is sometimes used. The contraction operation in computing the trace of a tensor A is analogous to the operation in the calculation the magnitude of a vector a , i.e., | a | 2 = a a = a 1 a 1 + a 2 a 2 + a 3 a 3

If A and B are two second order tensors, we can form 81 numbers from the products of the 9 components of each. The full set of these products is a fourth order tensor. Contracted products result in second order or zero order tensors. We will not have an occasion to use products of tensors in our course.

The product A ij a j of a tensor A and a vector a is a vector whose i th component is A ij a j . Another possible product of these two is A ij a I . These may be written A a and a A , respectively. For example, the diffusive flux of a quantity is computed as the contracted product of the transport coefficient tensor and the potential gradient vector, e.g., q = - k T

The vector of an antisymmetric tensor

We showed earlier that a second order tensor can be represented as the sum of a symmetric part and an antisymmetric part. Also, an antisymmetric tensor is characterized by three numbers. We will later show that the antisymmetric part of the velocity gradient tensor represents the local rotation of the fluid or body. Here, we will develop the relation between the angular velocity vector, ω , introduced earlier and the corresponding antisymmetric tensor.

Recall that the relative velocity between a pair of points in a rigid body was described as follows.

Δ v = ω × Δ x MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aejaahAhacqGH9aqpcqaHjpWDcaaMc8Uaey41aqRaeuiLdqKaaCiEaaaa@4128@

We wish to define a tensor Ω that also can determine the relative velocity.

Δ v = ω × Δ x = Δ x Ω

The following relation between the components satisfies this relation.

Ω i j = ε i j k ω k ω k = 1 2 ε i j k Ω i j MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeuyQdC1aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iabew7aLnaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOGaaGPaVlabeM8a3naaBaaaleaacaWGRbaabeaaaOqaaiabeM8a3naaBaaaleaacaWGRbaabeaakiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaeqyTdu2aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGccaaMc8UaeuyQdC1aaSbaaSqaaiaadMgacaWGQbaabeaaaaaa@5309@

Written in matrix notation these are as follows.

ω = [ ω 1 ω 2 ω 3 ] , Ω = [ 0 ω 3 - ω 2 - ω 3 0 ω 1 ω 2 - ω 1 0 ] MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@60C8@

The notation vec Ω is sometimes used for ω . In summary, an antisymmetric tensor is completely characterized by the vector, vec Ω .

Canonical form of a symmetric tensor

We showed earlier that any second order tensor can be represented as a sum of a symmetric part and an antisymmetric part. The symmetric part is determined by 6 numbers. We now seek the properties of the symmetric part. A theorem in linear algebra states that a symmetric matrix with real elements can be transformed by its eigenvectors to a diagonal matrix with real elements corresponding the eigenvalues. (see Appendix A of Aris.) If the eigenvalues are distinct, then the eigenvector directions are orthogonal. The eigenvectors determine a coordinate system such that the contracted product of the tensor with unit vectors along the coordinate axis is a parallel vector with a magnitude equal to the corresponding eigenvalue. The surface described by the contracted product of all unit vectors in this transformed coordinate system is an ellipsoid with axes corresponding to the coordinate directions.

The eigenvalues and the scalar invariants of a second order tensor can be determined from the characteristic equation.

det ( A i j - λ δ i j ) = ψ - λ Φ + λ 2 Θ - λ 3 where Θ = A 11 + A 22 + A 33 = t r A Φ = A 22 A 33 - A 23 A 32 + A 33 A 11 - A 31 A 13 + A 11 A 22 - A 12 A 21 ψ = det A MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=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@9F4B@

    Assignment 2.1

  1. Relative velocity of points in a rigid body. If x and y are two points inside a rigid body that is translating and rotating, determine the relation between the relative velocity of these two points as a function of their relative positions. If x and y are points on a line parallel to the axis of rotation, what is their relative velocity? If x and y are points on opposite sides of the axis of rotation but with equal distance, r, what is their relative velocity? Draw diagrams.
  2. Prove that: a •( b × c ) = ( a × b )• c
  3. Show a •( b × c ) vanishes identically if two of the three vectors are proportional of one another.
  4. Show that if a is coplanar with b and c , then a •( b × c ) is zero.
  5. Prove that: a × ( b × c ) = ( a c ) b - ( a b ) c MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaahggacqGHxdaTdaqadaqaaiaahkgacqGHxdaTcaWHJbaacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaacaWHHbGaeyOiGCRaaC4yaaGaayjkaiaawMcaaiaaykW6caWHIbGaeyOeI0YaaeWaaeaacaWHHbGaeyOiGCRaaCOyaaGaayjkaiaawMcaaiaaykW6caWHJbaaaa@4EFE@
  6. Prove that the contracted product of a tensor A and a vector a , A a , transforms under a rotation of coordinates as a vector.
  7. Show that you get the same result for relative velocity whether you use ω or Ω for the rotation of a rigid body.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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
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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