1. SSG 805 Mechanics of Continua
INSTRUCTOR: Fakinlede OA oafak@unilag.edu.ng oafak@hotmail.com
ASSISTED BY: Akano T manthez@yahoo.com
Oyelade AO oyelade.akintoye@gmail.com
Department of Systems Engineering, University of Lagos
2. ๏ช Available to beginning graduate students in
Engineering
๏ช Provides a background to several other courses such
as Elasticity, Plasticity, Fluid Mechanics, Heat Transfer,
Fracture Mechanics, Rheology, Dynamics, Acoustics,
etc. These courses are taught in several of our
departments. This present course may be viewed as
an advanced introduction to the modern approach to
these courses
๏ช It is taught so that related graduate courses can build
on this modern approach.
Purpose of the Course
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3. ๏ช The slides here are quite extensive. They are meant to
assist the serious learner. They are NO SUBSTITUTES
for the course text which must be read and followed
concurrently.
๏ช Preparation by reading ahead is ABSOLUTELY
necessary to follow this course
๏ช Assignments are given at the end of each class and
they are due (No excuses) exactly five days later.
๏ช Late submission carry zero grade.
What you will need
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4. oafak@unilag.edu.ng 12/31/2012
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Scope of Instructional Material
Course Schedule:
The read-ahead materials are from Gurtin except the
part marked red. There please read Holzapfel. Home
work assignments will be drawn from the range of
pages in the respective books.
Slide Title Slides Weeks Text Read Pages
Vectors and Linear Spaces 70 2 Gurtin 1-8
Tensor Algebra 110 3 Gurtin 9-37
Tensor Calculus 119 3 Gurtin 39-57
Kinematics: Deformation & Motion 106 3 Gurtin 59-123
Theory of Stress & Heat Flux 43 1 Holz 109-129
Balance Laws 58 2 Gurtin 125-205
5. The only remedy for late submission is that you fight for
the rest of your grade in the final exam if your excuse is
considered to be genuine. Ordinarily, the following will
hold:
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Examination
Evaluation Obtainable
Home work 50
Midterm 25
Final Exam 25
Total 100
6. ๏ช This course was prepared with several textbooks and
papers. They will be listed below. However, the main
course text is: Gurtin ME, Fried E & Anand L, The
Mechanics and Thermodynamics of Continua,
Cambridge University Press, www.cambridge.org
2010
๏ช The course will cover pp1-240 of the book. You can
view the course as a way to assist your reading and
understanding of this book
๏ช The specific pages to be read each week are given
ahead of time. It is a waste of time to come to class
without the preparation of reading ahead.
๏ช This preparation requires going through the slides
and the area in the course text that will be covered.
Course Texts
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7. ๏ช The software for the Course is Mathematica 9 by
Wolfram Research. Each student is entitled to a
licensed copy. Find out from the LG Laboratory
๏ช It your duty to learn to use it. Students will find some
examples too laborious to execute by manual
computation. It is a good idea to start learning
Mathematica ahead of your need of it.
๏ช For later courses, commercial FEA Simulations
package such as ANSYS, COMSOL or NASTRAN will be
needed. Student editions of some of these are
available. We have COMSOL in the LG Laboratory
Software
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8. ๏ช Gurtin, ME, Fried, E & Anand, L, The Mechanics and
Thermodynamics of Continua, Cambridge University Press,
www.cambridge.org 2010
๏ช Bertram, A, Elasticity and Plasticity of Large
Deformations, Springer-Verlag Berlin Heidelberg, 2008
๏ช Tadmore, E, Miller, R & Elliott, R, Continuum Mechanics
and Thermodynamics From Fundamental Concepts to
Governing Equations, Cambridge University Press,
www.cambridge.org , 2012
๏ช Nagahban, M, The Mechanical and Thermodynamical
Theory of Plasticity, CRC Press, Taylor and Francis Group,
June 2012
๏ช Heinbockel, JH, Introduction to Tensor Calculus and
Continuum Mechanics, Trafford, 2003
Texts
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9. ๏ช Bower, AF, Applied Mechanics of Solids, CRC Press, 2010
๏ช Taber, LA, Nonlinear Theory of Elasticity, World Scientific,
2008
๏ช Ogden, RW, Nonlinear Elastic Deformations, Dover
Publications, Inc. NY, 1997
๏ช Humphrey, JD, Cadiovascular Solid Mechanics: Cells,
Tissues and Organs, Springer-Verlag, NY, 2002
๏ช Holzapfel, GA, Nonlinear Solid Mechanics, Wiley NY, 2007
๏ช McConnell, AJ, Applications of Tensor Analysis, Dover
Publications, NY 1951
๏ช Gibbs, JW โA Method of Geometrical Representation of
the Thermodynamic Properties of Substances by Means
of Surfaces,โ Transactions of the Connecticut Academy of
Arts and Sciences 2, Dec. 1873, pp. 382-404.
Texts
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10. ๏ช Romano, A, Lancellotta, R, & Marasco A, Continuum
Mechanics using Mathematica, Fundamentals,
Applications and Scientific Computing, Modeling and
Simulation in Science and Technology, Birkhauser, Boston
2006
๏ช Reddy, JN, Principles of Continuum Mechanics, Cambridge
University Press, www.cambridge.org 2012
๏ช Brannon, RM, Functional and Structured Tensor Analysis
for Engineers, UNM BOOK DRAFT, 2006, pp 177-184.
๏ช Atluri, SN, Alternative Stress and Conjugate Strain
Measures, and Mixed Variational Formulations Involving
Rigid Rotations, for Computational Analysis of Finitely
Deformed Solids with Application to Plates and Shells,
Computers and Structures, Vol. 18, No 1, 1984, pp 93-116
Texts
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11. ๏ช Wang, CC, A New Representation Theorem for Isotropic
Functions: An Answer to Professor G. F. Smith's Criticism
of my Papers on Representations for Isotropic Functions
Part 1. Scalar-Valued Isotropic Functions, Archives of
Rational Mechanics, 1969 pp
๏ช Dill, EH, Continuum Mechanics, Elasticity, Plasticity,
Viscoelasticity, CRC Press, 2007
๏ช Bonet J & Wood, RD, Nonlinear Mechanics for Finite
Element Analysis, Cambridge University Press,
www.cambridge.org 2008
๏ช Wenger, J & Haddow, JB, Introduction to Continuum
Mechanics & Thermodynamics, Cambridge University
Press, www.cambridge.org 2010
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Texts
12. ๏ช Li, S & Wang G, Introduction to Micromechanics and
Nanomechanics, World Scientific, 2008
๏ช Wolfram, S The Mathematica Book, 5th Edition
Wolgram Media 2003
๏ช Trott, M, The Mathematica Guidebook, 4 volumes:
Symbolics, Numerics, Graphics &Programming,
Springer 2000
๏ช Sokolnikoff, IS, Tensor Analysis, Theory and
Applications to Geometry and Mechanics of
Continua, John Wiley, 1964
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Texts
14. ๏ช Continuum Mechanics can be thought of as the grand
unifying theory of engineering science.
๏ช Many of the courses taught in an engineering
curriculum are closely related and can be obtained as
special cases of the general framework of continuum
mechanics.
๏ช This fact is easily lost on most undergraduate and
even some graduate students.
Unified Theory
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16. Continuum Mechanics views matter as continuously
distributed in space. The physical quantities we are
interested in can be
โข Scalars or reals, such as time, energy, power,
โข Vectors, for example, position vectors, velocities, or
forces,
โข Tensors: deformation gradient, strain and stress
measures.
Physical Quantities
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17. ๏ช Since we can also interpret scalars as zeroth-order tensors,
and vectors as 1st-order tensors, all continuum mechanical
quantities can generally be considered as tensors of
different orders.
๏ช It is therefore clear that a thorough understanding of
Tensors is essential to continuum mechanics. This is NOT
always an easy requirement;
๏ช The notational representation of tensors is often
inconsistent as different authors take the liberty to express
themselves in several ways.
Physical Quantities
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18. ๏ช There are two major divisions common in the literature:
Invariant or direct notation and the component notation.
๏ช Each has its own advantages and shortcomings. It is
possible for a reader that is versatile in one to be
handicapped in reading literature written from the other
viewpoint. In fact, it has been alleged that
๏ช โContinuum Mechanics may appear as a fortress surrounded
by the walls of tensor notationโ It is our hope that the
course helps every serious learner overcome these
difficulties
Physical Quantities
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19. ๏ช The set of real numbers is denoted by R
๏ช Let R ๐
be the set of n-tuples so that when ๐ = 2,
R 2
we have the set of pairs of real numbers. For
example, such can be used for the ๐ฅ and
๐ฆ coordinates of points on a Cartesian coordinate
system.
Real Numbers & Tuples
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20. A real vector space V is a set of elements (called
vectors) such that,
1. Addition operation is defined and it is commutative
and associative underV : that is, ๐ + ๐ฏ โV, ๐ + ๐ฏ =
๐ฏ + ๐, ๐ + ๐ฏ + ๐ = ๐ + ๐ฏ + ๐, โ ๐, ๐ฏ, ๐ โ V.
Furthermore,V is closed under addition: That is, given
that ๐, ๐ฏ โ V, then ๐ = ๐ + ๐ฏ = ๐ฏ + ๐, โ ๐ โ V.
2. V contains a zero element ๐ such that ๐ + ๐ = ๐ โ
๐ โV. For every ๐ โ V, โ โ ๐: ๐ + โ๐ = ๐.
3. Multiplication by a scalar. For ๐ผ, ๐ฝ โ R and ๐, ๐ โ V,
๐ผ๐ โ V , 1๐ = ๐, ๐ผ ๐ฝ๐ = ๐ผ๐ฝ ๐, ๐ผ + ๐ฝ ๐ = ๐ผ๐ +
๐ฝ๐, ๐ผ ๐ + ๐ฏ = ๐ผ๐ + ๐ผ๐ฏ.
Vector Space
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21. ๏ช We were previously told that a vector is something
that has magnitude and direction. We often represent
such objects as directed lines. Do such objects
conform to our present definition?
๏ช To answer, we only need to see if the three conditions
we previously stipulated are met:
Magnitude & Direction
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22. From our definition of the Euclidean space, it is easy to see
that,
๐ฏ = ๐ผ1๐1 + ๐ผ2๐2 + โฏ + ๐ผ๐๐๐
such that, ๐ผ1, ๐ผ2, โฏ , ๐ผ๐ โ R and ๐1, ๐2, โฏ , ๐๐ โ E, is also a
vector. The subset
๐1, ๐2, โฏ , ๐๐ โ E
is said to be linearly independent or free if for any choice of
the subset ๐ผ1, ๐ผ2, โฏ , ๐ผ๐ โ R other than 0,0, โฏ , 0 , ๐ฏ โ 0.
If it is possible to find linearly independent vector systems of
order ๐ where ๐ is a finite integer, but there is no free system
of order ๐ + 1, then the dimension ofE is ๐. In other words,
the dimension of a space is the highest number of linearly
independent members it can have.
Dimensionality
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23. ๏ช Any linearly independent subset ofE is said to form a
basis forE in the sense that any other vector inE can
be expressed as a linear combination of members of
that subset. In particular our familiar Cartesian
vectors ๐, ๐ and ๐ is a famous such subset in three
dimensional Euclidean space.
๏ช From the above definition, it is clear that a basis is not
necessarily unique.
Basis
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24. 1. Addition operation for a directed line segment is defined by
the parallelogram law for addition.
2. V contains a zero element ๐ in such a case is simply a point
with zero length..
3. Multiplication by a scalar ๐ผ. Has the meaning that
0 < ๐ผ โค 1 โ Line is shrunk along the same direction by ๐ผ
๐ผ > 1 โ Elongation by ๐ผ
Negative value is same as the above with a change of direction.
Directed Line
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25. ๏ช Now we have confirmed that our original notion of a
vector is accommodated. It is not all that possess
magnitude and direction that can be members of a
vector space.
๏ช A book has a size and a direction but because we
cannot define addition, multiplication by a scalar as
we have done for the directed line segment, it is not a
vector.
Other Vectors
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26. Complex Numbers. The set C of complex numbers is a real
vector space or equivalently, a vector space over R.
2-D Coordinate Space. Another real vector space is the set of
all pairs of ๐ฅ๐ โ R forms a 2-dimensional vector space overR
is the two dimensional coordinate space you have been
graphing on! It satisfies each of the requirements:
๐ = {๐ฅ1, ๐ฅ2}, ๐ฅ1, ๐ฅ2 โ R, ๐ = {๐ฆ1, ๐ฆ2}, ๐ฆ1, ๐ฆ2 โ R.
๏ช Addition is easily defined as ๐ + ๐ = {๐ฅ1 + ๐ฆ1, ๐ฅ2 + ๐ฆ2} clearly
๐ + ๐ โ R 2 since ๐ฅ1 + ๐ฆ1, ๐ฅ2 + ๐ฆ2 โ R. Addition operation creates other
members for the vector space โ Hence closure exists for the operation.
๏ช Multiplication by a scalar: ๐ผ๐ = {๐ผ๐ฅ1, ๐ผ๐ฅ2}, โ๐ผ โ R.
๏ช Zero element: ๐ = 0,0 . Additive Inverse: โ๐ = โ๐ฅ1, โ๐ฅ2 , ๐ฅ1, ๐ฅ2 โ R
Type equation here.A standard basis for this : ๐1 = 1,0 , ๐2 = 0,1
Type equation here.Any other member can be expressed in terms of
this basis.
Other Vectors
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27. ๐ โD Coordinate Space. For any positive number ๐, we may
create ๐ โtuples such that, ๐ = {๐ฅ1, ๐ฅ2, โฆ , ๐ฅ๐} where
๐ฅ1, ๐ฅ2, โฆ , ๐ฅ๐ โ R are members of R ๐
โ a real vector space
over theR. ๐ = {๐ฆ1, ๐ฆ2, โฆ , ๐ฆ๐}, ๐ฆ1, ๐ฆ2, โฆ , ๐ฆ๐ โ R. ๐ + ๐ =
{๐ฅ1 + ๐ฆ1, ๐ฅ2 + ๐ฆ2, โฆ , ๐ฅ๐ + ๐ฆ๐},. ๐ + ๐ โ R ๐
since ๐ฅ1 +
๐ฆ1, ๐ฅ2 + ๐ฆ2, โฆ , ๐ฅ๐ + ๐ฆ๐ โ R
Addition operation creates other members for the vector
space โ Hence closure exists for the operation.
๏ช Multiplication by a scalar: ๐ผ๐ = {๐ผ๐ฅ1, ๐ผ๐ฅ2, โฆ , ๐ผ๐ฅ๐}, โ๐ผ โ
๐ .
๏ช Zero element ๐ = 0,0, โฆ , 0 . Additive Inverse โ๐ =
โ๐ฅ1, โ๐ฅ2, โฆ , โ๐ฅ๐ , ๐ฅ1, ๐ฅ2, โฆ , ๐ฅ๐ โ R
There is also a standard basis which is easily proved to be
linearly independent:
๐1 = 1,0, โฆ , 0 , ๐2 = 0,1, โฆ , 0 , โฆ , ๐๐ = {0,0, โฆ , 0}
N-tuples
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28. Let R๐ร๐
denote the set of matrices with entries that are
real numbers (same thing as saying members of the real
spaceR, Then, R๐ร๐
is a real vector space. Vector
addition is just matrix addition and scalar multiplication is
defined in the obvious way (by multiplying each entry by
the same scalar). The zero vector here is just the zero
matrix. The dimension of this space is ๐๐. For example, in
R3ร3
we can choose basis in the form,
๏ช
1 0 0
0 0 0
0 0 0
,
0 1 0
0 0 0
0 0 0
, โฆ,
0 0 0
0 0 0
0 0 1
Matrices
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29. Forming polynomials with a single variable ๐ฅ to order ๐
when ๐ is a real number creates a vector space. It is left as
an exercise to demonstrate that this satisfies all the three
rules of what a vector space is.
The Polynomials
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30. A mapping from the Euclidean Point spaceE to the
product space R 3 is called a global chart.
๐: E โ R ๐
So that โ ๐ฅ โ ๐, โ {๐1 ๐ฅ , ๐2 ๐ฅ , ๐3(๐ฅ)} which are the
coordinates identified with each ๐ฅ.
๏ช A global chart is sometimes also called a coordinate
system.
Global Chart
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31. ๏ช An Inner-Product (also called a Euclidean Vector)
SpaceE is a real vector space that defines the scalar
product: for each pair ๐, ๐ฏ โ E, โ ๐ โ R such that,
๐ = ๐ โ ๐ฏ = ๐ฏ โ ๐ . Further, ๐ โ ๐ โฅ 0, the zero value
occurring only when ๐ = 0. It is called โEuclideanโ
because the laws of Euclidean geometry hold in such
a space.
๏ช The inner product also called a dot product, is the
mapping
" โ ": V ร V โ R
from the product space to the real space.
Euclidean Vector Space
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32. ๏ช A mapping from a vector space is also called a
functional; a term that is more appropriate when we
are looking at a function space.
๏ช A linear functional ๐ฏ โ: V โ R on the vector spaceV
is called a convector or a dual vector. For a finite
dimensional vector space, the set of all convectors
forms the dual spaceV โ ofV . If V is an Inner
Product Space, then there is no distinction between
the vector space and its dual.
Co-vectors
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33. Magnitude The norm, length or magnitude of ๐,
denoted ๐ is defined as the positive square root of
๐ โ ๐ = ๐ 2. When ๐ = 1, ๐ is said to be a unit
vector. When ๐ โ ๐ฏ = 0, ๐ and ๐ฏ are said to be
orthogonal.
Direction Furthermore, for any two vectors ๐ and ๐ฏ, the
angle between them is defined as,
cosโ1
๐ โ ๐ฏ
๐ ๐ฏ
The scalar distance ๐ between two vectors ๐ and ๐ฏ
๐ = ๐ โ ๐ฏ
Magnitude & Direction Again
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34. ๏ช A 3-D Euclidean space is a Normed space because the
inner product induces a norm on every member.
๏ช It is also a metric space because we can find distances
and angles and therefore measure areas and volumes
๏ช Furthermore, in this space, we can define the cross
product, a mapping from the product space
" ร ": V ร V โ V
Which takes two vectors and produces a vector.
3-D Euclidean Space
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35. Without any further ado, our definition of cross product
is exactly the same as what you already know from
elementary texts. We simply repeat a few of these for
emphasis:
1. The magnitude
๐ ร ๐ = ๐ ๐ sin ๐ (0 โค ๐ โค ๐)
of the cross product ๐ ร ๐ is the area ๐ด(๐, ๐)
spanned by the vectors ๐ and ๐. This is the area of
the parallelogram defined by these vectors. This
area is non-zero only when the two vectors are
linearly independent.
2. ๐ is the angle between the two vectors.
3. The direction of ๐ ร ๐ is orthogonal to both ๐ and ๐
Cross Product
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36. Cross Product
The area ๐ด(๐, ๐) spanned by the vectors ๐ and ๐. The unit
vector ๐ in the direction of the cross product can be
obtained from the quotient,
๐ร๐
๐ร๐
.
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37. The cross product is bilinear and anti-commutative:
Given ๐ผ โ R , โ๐, ๐, ๐ โ V ,
๐ผ๐ + ๐ ร ๐ = ๐ผ ๐ ร ๐ + ๐ ร ๐
๐ ร ๐ผ๐ + ๐ = ๐ผ ๐ ร ๐ + ๐ ร ๐
So that there is linearity in both arguments.
Furthermore, โ๐, ๐ โ V
๐ ร ๐ = โ๐ ร ๐
Cross Product
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38. The trilinear mapping,
[ , ,] โถ V ร V ร V โ R
From the product set V ร V ร V to real space is defined by:
[ u,v,w] โก ๐ โ ๐ ร ๐ = ๐ ร ๐ โ ๐
Has the following properties:
1. [ a,b,c]=[b,c,a]=[c,a,b]=โ[ b,a,c]=โ[c,b,a]=โ[a,c,b]
HW: Prove this
1. Vanishes when a, b and c are linearly dependent.
2. It is the volume of the parallelepiped defined by a, b and c
Tripple Products
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40. ๏ช We introduce an index notation to facilitate the
expression of relationships in indexed objects.
Whereas the components of a vector may be three
different functions, indexing helps us to have a
compact representation instead of using new
symbols for each function, we simply index and
achieve compactness in notation. As we deal with
higher ranked objects, such notational conveniences
become even more important. We shall often deal
with coordinate transformations.
Summation Convention
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41. ๏ช When an index occurs twice on the same side of any
equation, or term within an equation, it is understood
to represent a summation on these repeated indices
the summation being over the integer values
specified by the range. A repeated index is called a
summation index, while an unrepeated index is called
a free index. The summation convention requires that
one must never allow a summation index to appear
more than twice in any given expression.
Summation Convention
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42. ๏ช Consider transformation equations such as,
๐ฆ1 = ๐11๐ฅ1 + ๐12๐ฅ2 + ๐13๐ฅ3
๐ฆ2 = ๐21๐ฅ1 + ๐22๐ฅ2 + ๐23๐ฅ3
๐ฆ3 = ๐31๐ฅ1 + ๐32๐ฅ2 + ๐33๐ฅ3
๏ช We may write these equations using the summation symbols as:
๐ฆ1 = ๐1๐๐ฅ๐
๐
๐=1
๐ฆ2 = ๐2๐๐ฅ๐
๐
๐=1
๐ฆ3 = ๐2๐๐ฅ๐
๐
๐=1
Summation Convention
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43. ๏ช In each of these, we can invoke the Einstein
summation convention, and write that,
๐ฆ1 = ๐1๐๐ฅ๐
๐ฆ2 = ๐2๐๐ฅ๐
๐ฆ3 = ๐2๐๐ฅ๐
๏ช Finally, we observe that ๐ฆ1, ๐ฆ2, and ๐ฆ3 can be
represented as we have been doing by ๐ฆ๐, ๐ = 1,2,3
so that the three equations can be written more
compactly as,
๐ฆ๐ = ๐๐๐๐ฅ๐, ๐ = 1,2,3
Summation Convention
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44. Please note here that while ๐ in each equation is a dummy
index, ๐ is not dummy as it occurs once on the left and in each
expression on the right. We therefore cannot arbitrarily alter
it on one side without matching that action on the other side.
To do so will alter the equation. Again, if we are clear on the
range of ๐, we may leave it out completely and write,
๐ฆ๐ = ๐๐๐๐ฅ๐
to represent compactly, the transformation equations above.
It should be obvious there are as many equations as there are
free indices.
Summation Convention
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45. If ๐๐๐ represents the components of a 3 ร 3 matrix ๐, we can
show that,
๐๐๐๐๐๐ = ๐๐๐
Where ๐ is the product matrix ๐๐.
To show this, apply summation convention and see that,
for ๐ = 1, ๐ = 1, ๐11๐11 + ๐12๐21 + ๐13๐31 = ๐11
for ๐ = 1, ๐ = 2, ๐11๐12 + ๐12๐22 + ๐13๐32 = ๐12
for ๐ = 1, ๐ = 3, ๐11๐13 + ๐12๐23 + ๐13๐33 = ๐13
for ๐ = 2, ๐ = 1, ๐21๐11 + ๐22๐21 + ๐23๐31 = ๐21
for ๐ = 2, ๐ = 2, ๐21๐12 + ๐22๐22 + ๐23๐32 = ๐22
for ๐ = 2, ๐ = 3, ๐21๐13 + ๐22๐23 + ๐23๐33 = ๐23
for ๐ = 3, ๐ = 1, ๐31๐11 + ๐32๐21 + ๐33๐31 = ๐31
for ๐ = 3, ๐ = 2, ๐31๐12 + ๐32๐22 + ๐33๐32 = ๐32
for ๐ = 3, ๐ = 3, ๐31๐13 + ๐32๐23 + ๐33๐33 = ๐33
Summation Convention
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Department of Systems Engineering, University of Lagos 45
46. The above can easily be verified in matrix notation as,
๐๐ =
๐11 ๐12 ๐13
๐21 ๐22 ๐23
๐31 ๐32 ๐33
๐11 ๐12 ๐13
๐21 ๐22 ๐23
๐31 ๐32 ๐33
=
๐11 ๐12 ๐13
๐21 ๐22 ๐23
๐31 ๐32 ๐33
= ๐
In this same way, we could have also proved that,
๐๐๐๐๐๐ = ๐๐๐
๏ช Where ๐ is the product matrix ๐๐T. Note the
arrangements could sometimes be counter intuitive.
Summation Convention
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Department of Systems Engineering, University of Lagos 46
47. Suppose our basis vectors ๐ ๐, ๐ = 1,2,3 are not only not
unit in magnitude, but in addition are NOT orthogonal.
The only assumption we are making is that ๐ ๐ โ V , ๐ =
1,2,3 are linearly independent vectors.
With respect to this basis, we can express vectors
๐ฏ, ๐ฐ โ V in terms of the basis as,
๐ฏ = ๐ฃ๐ ๐ ๐, ๐ฐ = ๐ค๐ ๐ ๐
Where each ๐ฃ๐ is called the contravariant component of
๐ฏ
Vector Components
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Department of Systems Engineering, University of Lagos 47
48. Clearly, addition and linearity of the vector space โ
๐ฏ + ๐ฐ = (๐ฃ๐+๐ค๐)๐ ๐
Multiplication by scalar rule implies that if ๐ผ โ R , โ๐ฏ โ
V ,
๐ผ ๐ฏ = (๐ผ๐ฃ๐)๐ ๐
Vector Components
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Department of Systems Engineering, University of Lagos 48
49. For any basis vectors ๐ ๐ โ V , ๐ = 1,2,3 there is a
reciprocal basis defined by the reciprocity relationship:
๐ ๐ โ ๐ ๐ = ๐ ๐ โ ๐ ๐ = ๐ฟ๐
๐
Where ๐ฟ๐
๐
is the Kronecker delta
๐ฟ๐
๐
=
0 if ๐ โ ๐
1 if ๐ = ๐
Let the fact that the above equations are actually nine
equations each sink. Consider the full meaning:
Reciprocal Basis
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50. Kronecker Delta: ๐ฟ๐๐, ๐ฟ๐๐
or ๐ฟ๐
๐
has the following properties:
๐ฟ11 = 1, ๐ฟ12 = 0, ๐ฟ13 = 0
๐ฟ21 = 0, ๐ฟ22 = 1, ๐ฟ23 = 0
๐ฟ31 = 0, ๐ฟ32 = 0, ๐ฟ33 = 1
As is obvious, these are obtained by allowing the indices to attain
all possible values in the range. The Kronecker delta is
defined by the fact that when the indices explicit values
are equal, it has the value of unity. Otherwise, it is zero.
The above nine equations can be written more compactly
as,
๐ฟ๐
๐
=
0 if ๐ โ ๐
1 if ๐ = ๐
Kronecker Delta
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51. ๏ช For any โ๐ฏ โ V ,
๐ฏ = ๐ฃ๐๐ ๐ = ๐ฃ๐๐ ๐
Are two related representations in the reciprocal bases.
Taking the inner product of the above equation with the
basis vector ๐ ๐, we have
๐ฏ โ ๐ ๐ = ๐ฃ๐๐ ๐ โ ๐ ๐ = ๐ฃ๐๐ ๐ โ ๐ ๐
Which gives us the covariant component,
๐ฏ โ ๐ ๐ = ๐ฃ๐
๐๐๐ = ๐ฃ๐๐ฟ๐
๐
= ๐ฃ๐
Covariant components
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52. In the same easy manner, we may evaluate the
contravariant components of the same vector by taking
the dot product of the same equation with the
contravariant base vector ๐ ๐:
๐ฏ โ ๐ ๐ = ๐ฃ๐๐ ๐ โ ๐ ๐ = ๐ฃ๐๐ ๐ โ ๐ ๐
So that,
๐ฏ โ ๐ ๐ = ๐ฃ๐๐ฟ๐
๐
= ๐ฃ๐๐๐๐ = ๐ฃ๐
Contravariant Components
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53. The nine scalar quantities, ๐๐๐ as well as the nine related
quantities ๐๐๐ play important roles in the coordinate
system spanned by these arbitrary reciprocal set of
basis vectors as we shall see.
They are called metric coefficients because they metrize
the space defined by these bases.
Metric coefficients
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55. ๏ช While the above equations might look arbitrary at first, a
closer look shows there is a simple logic to it all. In fact,
note that whenever the value of an index is repeated, the
symbol has a value of zero. Furthermore, we can see that
once the indices are an even arrangement (permutation)
of 1,2, and 3, the symbols have the value of 1, When we
have an odd arrangement, the value is -1. Again, we desire
to avoid writing twenty seven equations to express this
simple fact. Hence we use the index notation to define the
Levi-Civita symbol as follows:
๏ช ๐๐๐๐ =
1 if ๐, ๐ and ๐ are an even permutation of 1,2 and 3
โ1 if ๐, ๐ and ๐ are an odd permutation of 1,2 and 3
0 In all other cases
Levi Civita Symbol
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56. Given that ๐ = det ๐๐๐ of the covariant metric coefficients, It is
not difficult to prove that
๐ ๐ โ ๐ ๐ ร ๐ ๐ = ๐๐๐๐ โก ๐๐๐๐๐
๏ช This relationship immediately implies that,
๐ ๐ ร ๐ ๐ = ๐๐๐๐๐ ๐.
๏ช The dual of the expression, the equivalent contravariant
equivalent also follows from the fact that,
๐ 1 ร ๐ 2 โ ๐ 3 = 1/ ๐
Cross Product of Basis Vectors
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57. This leads in a similar way to the expression,
๐ ๐
ร ๐ ๐
โ ๐ ๐
=
๐๐๐๐
๐
= ๐๐๐๐
It follows immediately from this that,
๐ ๐ ร ๐ ๐ = ๐๐๐๐๐ ๐
Cross Product of Basis Vectors
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58. ๏ช Given that, ๐ ๐, ๐ 2 and ๐ 3 are three linearly
independent vectors and satisfy ๐ ๐ โ ๐ ๐ = ๐ฟ๐
๐
, show
that ๐ 1 =
1
๐
๐ 2 ร ๐ 3, ๐ 2 =
๐
๐ฝ
๐ 3 ร ๐ 1, and ๐ 3 =
1
๐
๐ 1 ร
๐ 2, where ๐ = ๐ 1 โ ๐ 2 ร ๐ 3
Exercises
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Department of Systems Engineering, University of Lagos 58
59. It is clear, for example, that ๐ 1 is perpendicular to ๐ 2 as
well as to ๐ 3 (an obvious fact because ๐ 1 โ ๐ 2 = 0 and
๐ 1 โ ๐ 3 = 0), we can say that the vector ๐ 1 must
necessarily lie on the cross product ๐ 2 ร ๐ 3 of ๐ 2 and
๐ 3. It is therefore correct to write,
๐ 1 =
๐
๐ฝ
๐ 2 ร ๐ 3
Where ๐โ1is a constant we will now determine. We can
do this right away by taking the dot product of both
sides of the equation (5) with ๐ 1 we immediately
obtain,
๐ 1 โ ๐ 1 = ๐โ1 ๐ 1 โ ๐ 2 ร ๐ 3 = 1
So that, ๐ = ๐ 1 โ ๐ 2 ร ๐ 3
the volume of the parallelepiped formed by the three
vectors ๐ 1, ๐ 2, and ๐ 3 when their origins are made to
coincide.
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60. Show that ๐๐๐๐๐๐ = ๐ฟ๐
๐
First note that ๐ ๐ = ๐๐๐๐ ๐ = ๐๐๐๐ ๐.
Recall the reciprocity relationship: ๐ ๐ โ ๐ ๐
= ๐ฟ๐
๐
. Using the above, we can write
๐ ๐ โ ๐ ๐
= ๐๐๐ผ๐ ๐ผ
โ ๐๐๐ฝ
๐ ๐ฝ
= ๐๐๐ผ๐๐๐ฝ
๐ ๐ผ
โ ๐ ๐ฝ
= ๐๐๐ผ๐๐๐ฝ๐ฟ๐ฝ
๐ผ
= ๐ฟ๐
๐
which shows that
๐๐๐ผ๐๐๐ผ
= ๐๐๐๐๐๐
= ๐ฟ๐
๐
As required. This shows that the tensor ๐๐๐ and ๐๐๐
are inverses of each other.
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Department of Systems Engineering, University of Lagos 60
61. Show that the cross product of vectors ๐ and ๐ in general coordinates is
๐๐
๐๐
๐๐๐๐๐ ๐
or ๐๐๐๐
๐๐๐๐๐ ๐ where ๐๐
, ๐๐
are the respective contravariant
components and ๐๐, ๐๐ the covariant.
Express vectors ๐ and ๐ as contravariant components: ๐ = ๐๐
๐ ๐, and
๐ = ๐๐
๐ ๐. Using the above result, we can write that,
๐ ร ๐ = ๐๐๐ ๐ ร ๐๐๐ ๐
= ๐๐๐๐๐ ๐ ร ๐ ๐ = ๐๐๐๐๐๐๐๐๐ ๐.
Express vectors ๐ and ๐ as covariant components: ๐ = ๐๐๐ ๐ and ๐ = ๐๐๐ ๐.
Again, proceeding as before, we can write,
๐ ร ๐ = ๐๐๐ ๐ ร ๐๐๐ ๐
= ๐๐๐๐๐๐๐๐๐ ๐
Express vectors ๐ as a contravariant components: ๐ = ๐๐
๐ ๐ and ๐ as
covariant components: ๐ = ๐๐๐ ๐
๐ ร ๐ = ๐๐
๐ ๐ ร ๐๐๐ ๐
= ๐๐๐๐ ๐ ๐ ร ๐ ๐
= ๐ โ ๐ ๐ ๐ ร ๐ ๐
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63. Show that ๐ฟ๐๐๐
๐๐๐
= 2๐ฟ๐
๐
Contracting one more index, we have:
๐๐๐๐๐๐๐๐
= ๐ฟ๐๐๐
๐๐๐
= ๐ฟ๐
๐
๐ฟ๐
๐
โ ๐ฟ๐
๐
๐ฟ๐
๐
= 3๐ฟ๐
๐
โ ๐ฟ๐
๐
= 2๐ฟ๐
๐
These results are useful in several situations.
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64. Show that ๐ฎ โ ๐ฎ ร ๐ฏ = 0
๐ฎ ร ๐ฏ = ฯต๐๐๐๐ข๐๐ฃ๐๐ ๐
๐ฎ โ ๐ฎ ร ๐ฏ = ๐ข๐ผ๐ ๐ผ โ ฯต๐๐๐๐ข๐๐ฃ๐๐ ๐
= ๐ข๐ผ ฯต๐๐๐๐ข๐๐ฃ๐ ๐ฟ๐
๐ผ
= ฯต๐๐๐๐ข๐๐ฃ๐๐ข๐
= 0
On account of the symmetry and antisymmetry
in ๐ and ๐.
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65. Show that ๐ฎ ร ๐ฏ = โ ๐ฏ ร ๐ฎ
๐ฎ ร ๐ฏ = ฯต๐๐๐๐ข๐๐ฃ๐๐ ๐
= โฯต๐๐๐
๐ข๐๐ฃ๐๐ ๐ = โฯต๐๐๐
๐ฃ๐๐ข๐๐ ๐
= โ ๐ฏ ร ๐ฎ
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