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    Topics covered in this chapter

  • Definition of a vector
  • Examples of vectors
  • Scalar multiplication
  • Addition of vectors – coplanar vectors
  • Unit vectors
  • A basis of non-coplanar vectors
  • Scalar product – orthogonality
  • Directional cosines for coordinate transformation
  • Vector product
  • Velocity due to rigid body rotations
  • Triple scalar product
  • Triple vector product
  • Second order tensors
  • Examples of second order tensors
  • Scalar multiplication and addition
  • Contraction and multiplication
  • The vector of an antisymmetric tensor
  • Canonical form of a symmetric tensor
Reading Assignment: Chapter 2 of Aris, Appendix A of BSL

The algebra of vectors and tensors will be described here with Cartesian coordinates so the student can see the operations in terms of its components without the complexity of curvilinear coordinate systems.

Definition of a vector

Suppose x i , i.e., ( x 1 , x 2 , x 3 ), are the Cartesian coordinates of a point P in a frame of reference, 0123 . Let 0 1 ¯ 2 ¯ 3 ¯ be another Cartesian frame of reference with the same origin but defined by a rigid rotation. The coordinates of the point P in the new frame of reference is x ¯ j where the coordinates are related to those in the old frame as follows.

x ¯ j = l i j x i = l 1 j x 1 + l 2 j x 2 + l 3 j x 3 x i = l i j x ¯ j = l i 1 x ¯ 1 + l i 2 x ¯ 2 + l i 3 x ¯ 3

where l ij are the cosine of the angle between the old and new coordinate systems. Summation over repeated indices is understood when a term or a product appears with a common index.

Cartesian Vector
A Cartesian vector, a, in three dimensions is a quantity with three components a 1 , a 2 , a 3 in the frame of reference 0123, which, under rotation of the coordinate frame to 0 1 ¯ 2 ¯ 3 ¯ , become components a ¯ 1 , a ¯ 2 , a ¯ 3 , where
a ¯ j = l i j a i MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadggagaqeamaaBaaaleaacaWGQbaabeaakiabg2da9iaadYgadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamyyamaaBaaaleaacaWGPbaabeaaaaa@3E14@

Examples of vectors

In Cartesian coordinates, the length of the position vector of a point from the origin is equal to the square root of the sum of the square of the coordinates. The magnitude of a vector, a , is defined as follows.

| a | = ( a i a i ) 1 / 2 MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaaemaabaGaaCyyaaGaay5bSlaawIa7aiabg2da9maabmaabaGaamyyamaaBaaaleaacaWGPbaabeaakiaadggadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVaGaaGOmaaaaaaa@42ED@

A vector with a magnitude of unity is called a unit vector . The vector, a /| a |, is a unit vector with the direction of a . Its components are equal to the cosine of the angle between a and the coordinate axis. Some special unit vectors are the unit vectors in the direction of the coordinate axis and the normal vector of a surface.

Scalar multiplication

If α is a scalar and a is a vector, the product α a is a vector with components, α a i , magnitude α| a |, and the same direction as a .

Addition of vectors – coplanar vectors

If a and b are vectors with components a i and b i , then the sum of a and b is a vector with components, a i + b i .

The order and association of the addition of vectors are immaterial.

a + b = b + a ( a + b ) + c = a + ( b + c )

The subtraction of one vector from another is the same as multiplying one by the scalar (-1) and adding the resulting vectors.

If a and b are two vectors from the same origin, they are colinear or parallel if one is a linear combination of the other, i.e., they both have the same direction. If a and b are two vectors from the same origin, then all linear combination of a and b are in the same plane as a and b , i.,e., they are coplanar . We will prove this statement when we get to the triple scalar product.

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
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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
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Damian Reply
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Daniel
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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.
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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
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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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