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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

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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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