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Continuum Mechanics
C. Agelet de SaracibarETS Ingenieros de Caminos, Canales y Puertos, Universidad Politcnica de Catalua (UPC), Barcelona, Spain
International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain
Chapter 1Introduction to Vectors and Tensors
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Contents
1. Introduction
2. Algebra of vectors
3. Algebra of tensors
4. Higher order tensors5. Differential operators
6. Integral theorems
Introduction to Vectors and Tensors > Contents
Contents
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Tensors
Continuum mechanics deals withphysical propertiesof
materials, such as solids, liquids, gases or intermediate states,
which are independent of any particular coordinate system in
which they are observed.
Thosephysical properties are mathematically represented by
tensors.
Tensorsare mathematical objectswhich have the required
property of being independentof any particular coordinate
system.
Introduction to Vectors and Tensors > Introduction
Introduction
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Tensors
According to their tensorial order, tensors may be classified as:
Zero-order tensors: scalars, i.e. density, temperature, pressure
First-order tensors: vectors, i.e. velocity, acceleration, force
Second-order tensors, i.e. stress, strain
Third-order tensors, i.e. permutation
Fourth-order tensors, i.e. elastic constitutive tensor
First- and higher-order tensors may be expressed in terms of
their componentsin a given coordinate system.
Introduction to Vectors and Tensors > Introduction
Introduction
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Scalar
A physical quantity, completely described by a single real
number, such as the temperature, densityorpressure, is called a
scalarand is designated by .
A scalarmay be viewed as a zero-order tensor.
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
October 20, 2012 Carlos Agelet de Saracibar 5
Vector
A vectoris a directed line element in space. A model for physical
quantities having both direction and length, such as velocity,
accelerationorforce, is called a vectorand is designated by
A vectormay be viewed as afirst-order tensor.
, , .u v w
, ,
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Sum of Vectors
The sumof vectors yields a new vector, based on the parallelo-
gram law of addition.
The sumof vectors has the following properties,
where denotes the unique zero vectorwith unspecified
direction and zero length.
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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u v v uu v w u v w
u 0 u
u u 00
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Scalar Multiplication
Let be a vector and be a scalar. The scalar multiplication
gives a new vector with the same direction as if or with
the opposite direction to if .
The scalar multiplicationhas the following properties,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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u u
u u u
u v u v
u u
u 00u
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Dot Product
The dot (or scalaror inner) productof two vectors and ,
denoted by , is a scalar given by,
where is the norm(or lengthor magnitude)of the vector ,
which is a non-negative real number defined as,
and is the anglebetween the non-zero vectors and
when their origins coincide.
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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cos , , 0 , u v u v u v u v
u
u v
u v
u
1 2
0 u u u
, u v u v
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Dot Product
The dot (or scalaror inner) productof vectors has the following
properties,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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0
0
00, ,
u v v u
u 0
u v w u v u w
u u u 0
u u u 0
u v u 0 v 0 u v
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Orthogonal Vectors
A non-zero vector is said to be orthogonal(or perpendicular)
to a non-zero vector if,
Unit Vectors
A vector is called a unit vectorif its norm (or lenght or
magnitude) is equal to 1,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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u
0, , , 2 u v u 0 v 0 u v u v
v
e
1e
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Projection
The projectionof a vector along the direction of a unit vector
is defined as,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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u e
cos , , 0 , u e u u e u e
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Cartesian Basis
Let us consider a three-dimensional Euclidean spacewith a fixed
set of three right-handed orthonormal basis vectors ,
denoted as Cartesian basis, satisfying the following properties,
A fixed set of three unit vectors which are mutually orthogonalform a so called orthonormal basis vectors, collectively denoted
as ,and define an orthonormal system.
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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1 2 3, ,e e e
1 2 1 3 2 3
1 2 3
0
1
e e e e e e
e e e
ie
1e
2e
3e
1,x
2,x
3,x
O
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Cartesian Components
Any vector in the three-dimensional Euclidean spaceis
represented uniquelyby a linear combination of the basis
vectors , i.e.
where the three real numbers are the uniquely
determined Cartesian components of vector along the given
directions , respectively.
Using matrix notation, the Cartesian components of the vectorcan be collected into a vector of components given by,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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u
1 2 3, ,e e e
1 1 2 2 3 3u u u
u e e e
1 2 3, ,u u uu
1 2 3, ,e e e
u
1 2 3T
u u uu
3u
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Index Notation
Using index notationthe vector can be written as,
or in short form, leaving out the summation symbol, simply as
where we have adopted the summation convention introduced
by Einstein.
The index that is summed over is said to be a dummy(or
summation) index. An index that is not summed over in a given
term is called a free(or live) index.
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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u
i iuu e
3
1 i ii u
u e
d d l b f
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Kronecker Delta
The Kronecker delta is defined as,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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ij
1 if
0 ifij
i j
i j
The Kronecker delta satisfies the following properties,
Note that the Kronecker delta also serves as a replacement
operator.
3, ,ii ij jk ik ij j iu u
I d i V d T Al b f V
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Dot Product
The dot (or scalaror inner) product of two orthonormal basis
vectors and , denoted as , is a scalar quantity and can
be conveniently written in terms of the Kronecker delta as,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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ie
i j ij e e
je i je e
I t d ti t V t d T Al b f V t
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Components of a Vector
Taking the basis , the projectionof a vector onto the basis
vector yields the i-th componentof ,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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u
ie
i j j i j ji iu u u u e e e
u
ie
I t d ti t V t d T Al b f V t
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Dot Product
Taking the basis , the dot(or scalaror inner) productof two
vectors and , denoted as , is a scalar quantity and using
index notationcan be written as,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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u
ie
1 1 2 2 3 3
i i j j i j i j i j ij i iu v u v u v u vu v u v u v
u v e e e e
v u v
I t d ti t V t d T > Al b f V t
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Euclidean Norm
Taking the basis , the Euclidean norm(or lengthor
magnitude) of a vector , denoted as , is a scalar quantity
and using index notationcan be written as,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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ie
1 2 1 21 2 1 2
1 22 2 2
1 2 3
i i j j i j ij i iu u u u u u
u u u
u u u e e
u u
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Assignment 1.1
Write in index form the following expressions,
Introduction to Vectors and Tensors > Algebra of Vectors
Assignment 1.1
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v
v a b c
v a b c
a b c d
v a b c c d
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Assignment 1.1
Write in index form the following expressions,
Introduction to Vectors and Tensors > Algebra of Vectors
Assignment 1.1
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,
,
,
,
j j i i i i i j j i
i j j i i i i i j j
i i j j i i j j
i j j k k i i i i i j j k k
a b c v v a b c
a b c v v a b c
v a b c d v a b c d
a b c c d v v a b c c d
v a b c e e
v a b c e e
a b c d
v a b c c d e e
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Cross Product
The cross(or vector) productof two vectors and , denoted
as , is another vector which is perpendicular to the plane
defined by the two vectors and its norm (or length) is given by
the area of the parallelogram spanned by the two vectors,
where is a unit vector perpendicular to the plane defined by
the two vectors, , in the direction given by the
right-hand rule, .
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
October 20, 2012 Carlos Agelet de Saracibar 22
u v
u v
sin , , 0 , u v u v u v e u v
e
0, 0 u e v e
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Norm of the Cross Product of two Vectors
The norm (or magnitudeor length) of the cross (or vector)
product of two vectors and , denoted as , measures
the area spanned by the two vectors and is given by,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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sin , , 0 , u v u v u v u v
u v u v
u v
u v
u
v
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Permutation Symbol
The permutation (or alternatingor Levi-Civita) symbol is
defined as,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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ijk
1 if 123, 231 312
1 if 213,132 3210 if there is a repeated index
ijk
ijk or
ijk or
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Permutation Symbol
The permutation (or alternatingor Levi-Civita) symbol may
be written in terms of the Kronecker deltaand the following
expressions hold,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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ijk
2 2 6, ,ijp pqk ip jq iq jp ijk pjk ip ijk ijk ii
1 2 3
1 2 3
1 2 3
det , det
i i i ip iq ir
ijk j j j ijk pqr jp jq jr
k k k kp kq kr
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Cross Product
Taking the right-handed orthonormal basis , the cross (or
vector)product of two basis vectors and , denoted as
is defined as,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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jeie
i j ijk k e e e
1 2 3 2 3 1 3 1 2
2 1 3 3 2 1 1 3 2
1 1 2 2 3 3 .
, , ,
, , ,
0
e e e e e e e e e
e e e e e e e e e
e e e e e e
The cross (or vector)product of two right-handed orthonormal
basis vectorssatisfies the following properties,
ie,i je e
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Cross Product
Taking the basis , the cross(or vector) productof two
vectors and , denoted as , is another vector and can be
written in index form as,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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2 3 3 2 1 3 1 1 3 2 1 2 2 1 3
1 2 3
1 2 3
1 2 3
det
i i j j i j i j i j ijk k ijk i j k u v u v u v u vu v u v u v u v u v u v
u u uv v v
u v e e e e e e
e e e
e e e
ieu v u v
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Cross Product
The cross (or vector) product of vectors has the following
properties,
Introduction to Vectors and Tensors > Algebra of Vectors
Algebra of Vectors
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, , ||
u v v u
u v 0 u 0 v 0 u v
u v u v u v
u v w u v u w
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Assignment 1.2
Using the expression,
obtain the following relationships between the permutationsymbol and the Kronecker delta,
t oduct o to ecto s a d e so s geb a o ecto s
Assignment 1.2
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2 2 6, ,ijp klp ik jl il jk ipq jpq ij ijk ijk ii
2 2 2 2sin , , 0 , u v u v u v u v
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Box Product
The box (or triple scalar) product of the three right-handed
orthonormal basis vectors , denoted as , is a
scalar quantity and can be conveniently written in terms of the
permutation symbol as,
g
Algebra of Vectors
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i j k i j k ijk
ikj i k j i k j
e e e e e e
e e e e e e
, ,i j ke e e i j k e e e
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Box Product
Taking the orthonormal basis , the box(or triple scalar)
productof three vectors , denoted as , is a
scalar quantity and using index notation can be written as,
g
Algebra of Vectors
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1 2 3
1 2 3
1 2 3
det
i i j j k k i j k i j k
ijk i j k
u v w u v wu v w
u u u
v v vw w w
u v w e e e e e e
ie, ,u v w u v w
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Box Product
The box (or triple scalar) product of three vectors
represents the volumeof the parallelepiped spanned by the
three vectors forming a right-handedtriad,
g
Algebra of Vectors
October 20, 2012 Carlos Agelet de Saracibar 34
V
u v w
, ,u v w
V u v w
v
w
u
v w
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Box Product
The box (or triple scalar) product has the following properties,
Algebra of Vectors
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0
u v w w u v v w u
u v w u w v
u u v v u v
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Triple Vector Product
Taking the orthonormal basis , the triple vectorproductof
three vectors , denoted as , is a vector and
using index notation can be written as,
Algebra of Vectors
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ijk i pqj p q k kij pqj i p q k
kp iq kq ip i p q k
i k i k i i k k
u v w u v w
u v w
u v w u v w
u v w e e
e
e e
u w v u v w
ie, ,u v w u v w
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Triple Vector Product
Taking the orthonomal basis , the triple vectorproductof
three vectors , denoted as , is a vector and
using index notation can be written as,
Algebra of Vectors
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ijk pqi p q j k jki pqi p q j k
jp kq jq kp p q j k
j j k k j j k k
u v w u v w
u v w
u w v v w u
u v w e e
e
e e
u w v v w u
ie, ,u v w u v w
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Triple Vector Product
The triple vectorproducthas the following properties,
Algebra of Vectors
October 20, 2012 Carlos Agelet de Saracibar 38
u v w u w v u v w
u v w u w v v w u
u v w u v w
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Second-order Tensors
A second-order tensor may be thought of as a linear operator
that acts on a vector generating a vector , defining a linear
transformationthat assigns a vector to a vector ,
The second-order unit tensor, denoted as , satisfies the
following identity,
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 39
A
u v
v Au
v u
1
u 1u
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Second-order Tensors
As linear operatorsdefining linear transformations, second-
order tensors have the following properties,
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 40
A u v Au Av
A u Au
A B u Au Bu
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Tensor Product
The tensor product(or dyad)of two vectors and , denoted
as , is a second-order tensor which linearly transformsa
vector into a vector with the direction of following the rule,
A dyadicis a linear combination of dyads.
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 41
u vu v
u
u v w u v w v w u
w
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Tensor Product
The tensor product (or dyad) has the following linear properties,
Generally, the tensor product (or dyad) is notcommutative, i.e.
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 42
u v w x u v w u v x
u v w u w v w
u v w v w u u v w
u v w x v w u x u x v w
A u v Au v
u v v u
u v w x w x u v
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Tensor Product
The tensor product(or dyad) of two orthonormal basis vectors
and , denoted as , is a second-order tensor and can be
thought as a linear operatorsuch that,
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 43
i je ei
e
je
i j k jk i e e e e
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Second-order Tensors
Any second-order tensor may be represented by a linear
combination of dyads formed by the Cartesian basis
where the nine real numbers, represented by , are theuniquely determined Cartesian components of the tensor
with respect to the dyads formed by the Cartesian basis ,
represented by , which constitute a basis second-order
tensor for .The second-order unit tensormay be written as,
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 44
A
ij i jA A e e
ijAA
ie
i je e ie
A
ij i j i i 1 e e e e
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Cartesian Components
The ij-th Cartesian component of the second-order tensor ,
denoted as , can be written as,
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 45
A
i j i kl k l j i kl lj k
i kj k kj i k kj ik ij
A A
A A A A
e Ae e e e e e e
e e e e
ijA
The ij-th Cartesian component of the second-order unit tensor
is the Kronecker delta,
1
i j i j ij e 1e e e
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Matrix of Components
Using matrix notation, the Cartesian components of any second-
order tensor may be collected into a 3 x 3 matrix of compo-
nents, denoted by , given by,
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 46
A
A
11 12 13
21 22 23
31 32 33
A A AA A A
A A A
A
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Matrix of Components
Using matrix notation, the Cartesian components of the second-
order unit tensor may be collected into a 3 x 3 matrix of
components, denoted as , given by,
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 47
1 0 00 1 0
0 0 1
1
1
1
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Second-order Tensors
The tensor product (or dyad) of the vectors and , denoted as
, may be represented by a linear combination of dyads
formed by the Cartesian basis
where the nine real numbers, represented by , are the
uniquely determined Cartesian components of the tensor
with respect to the dyads formed by the Cartesian basis ,
represented by .
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 48
i j i ju v u v e e
i ju v
ie
i je e ie
u v
u v
u v
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Matrix of Components
Using matrix notation, the Cartesian components of any second-
order tensor may be collected into a 3 x 3 matrix of
components, denoted as , given by,
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 49
1 1 1 2 1 3
2 1 2 2 2 3
3 1 3 2 3 3
u v u v u vu v u v u v
u v u v u v
u v
u v
u v
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Second-order Tensors
Using index notation, the linear transformation can be
written as,
yielding
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 50
v Au
ij i j k k ij k i j k ij k jk iij j i i i
A u A u A u
A u v
v Au e e e e e e e
e e
, i ij jv A u v Au
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Positive Semi-definite Second-order Tensor
Asecond-order tensor is said to be a positive semi-definite
tensor if the following relation holds for any non-zero vector
Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 51
A
0 u Au u 0
Positive Definite Second-order Tensor
Asecond-order tensor is said to be a positive-definite tensor
if the following relation holds for any non-zero vector
0 u Au u 0
u 0
A
u 0
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Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 52
Transpose of a Second-order Tensor
The unique transpose of a second-order tensor , denoted as
is governed by the identity,
A
,
,
T
T
i ij j i ji j j ji i j iv A u v A u u A v u v
v A u u Av u v
,TA
The Cartesian componentsof the transpose of a a second-order
tensor satisfy,
Using index notationthe transpose of may be written as, :
T T T
ij i j j i jiijA A
A e A e e Ae
A
A
T
ji i jA A e e
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Algebra of Tensors
October 20, 2012 Carlos Agelet de Saracibar 53
Transpose of a Second-order Tensor
The matrix of components of the transpose of a second-order
tensor is equal to the transpose of the matrix of components
of and takes the form,
A
11 21 31
12 22 32
13 23 33
TTA A AA A A
A A A
A A
A
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Symmetric Second-order Tensor
A second-order tensor is said to be symmetricif the following
relation holds,
Using index notation, the Cartesian components of a symmetricsecond-order tensor satisfy,
Using matrix notation, the matrix of components of a symmetricsecond-order tensor satisfies,
A
TA A
A
,Tij i j ji i j ij jiA A A A A e e e e A
A
TT A A A
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Dot Product
Thedot product of two second-order tensors and , denoted
as , is a second-order tensor such that,
A B
AB u A Bu u
AB
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Dot Product
Thecomponents of the dot product along an orthonormal
basis read,
i j i jij
i kj k i k kj
ik kj
B B
A B
AB e AB e e A Be
e A e e Ae
AB
ie
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Dot Product
Thedot product of second-order tensors has the following
properties,
2
AB C A BC ABC
A AA
AB BA
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Trace
The traceof a dyad , denoted as , is a scalar
quantity given by,u v
tr i iu v u v u v
tr u v
The traceof a second-order tensor , denoted as , is a scalarquantity given by,
11 22 33
tr tr tr
tr
ij i j ij i j
ij i j ij ij
ii
A A
A A
A
A A A
A e e e e
e e
A
A trA
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Trace
The traceof a second-order tensor has the following properties,
tr tr
tr tr
tr tr tr
tr tr
T
A A
AB BA
A B A B
A A
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Double Dot Product
Thedouble dot product of two second-order tensors and ,
denoted as , is defined as,
or using index notation,
: tr tr
tr tr
:
T T
T T
A B A B B A
AB BA
B A
A B
:A B
: :ij ij ij ijA B B A A B B A
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Double Dot Product
Thedouble dot product has the following properties,
: : tr
: :
: : :
: :
:
:
T T
i j k l i k j l ik jl
A 1 1 A A
A B B A
A BC B A C AC B
A u v u Av u v A
u v w x u w v x
e e e e e e e e
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Euclidean Norm
Theeuclidean norm of a second-order tensor is a non-
negative scalar quantity, denoted as , given by,
A
1 21 2
: 0ij ijA A A A A
A
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Determinant
The determinant of a second-order tensor is a scalar quantity,
denoted as , given by,
A
11 12 13
21 22 23
31 32 33
1 2 3
det det
det
16
ijk i j k ijk pqr pi qj rk
A A AA A A
A A A
A A A A A A
A A
det A
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Determinant
The determinant of a second-order tensor has the following
properties,
3
det det
det det
det det det
T
A A
A A
AB A B
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Singular Tensors
A second-order tensor is said to be singular if and only if its
determinant is equal to zero, i.e.
is singular det 0 A A
A
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Inverse of a Second-order Tensor
For anon-singular second-order tensor , i.e. , there
exista unique non-singular inverse second-order tensor,
denoted as ,satisfying,
yielding,
A
1 1
AA u A A u 1u u u
det 0A
1A
1 1 1 1, ik kj ik kj ijA A A A AA A A 1
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Inverse of a Second-order Tensor
The inverse second-order tensor satisfies the following
properties,
11
1 1
1 1 1
1 1 1
2 1 1
11det det
TT T
A A
A A A
A A
AB B A
A A A
A A
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Orthogonal Second Order Tensor
Anorthogonal second-order tensor is a linear operator that
preserves the norms and the angles given by two vectors,
and the following relations hold,
Q
,
T
T
Qu Qv u Q Qv u vu v
Qv Qu v Q Qu v u
1 2 1 2T Qu u Q Qu u u u u
1, , det 1T T T Q Q QQ 1 Q Q Q
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Rotation Second Order Tensor
A rotation tensor is a proper orthogonal second-order tensor,
i.e., is a linear operator that preserves the lengths (norms) and
the angles given by two vectors,
and satisfies the following expressions,
R
1, , det 1T T T R R RR 1 R R R
,
T
T
Ru Rv u R Rv u vu v
Rv Ru v R Ru v u
1 2 1 2T
Ru u R Ru u u u u
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Symmetric/Skew-symmetric Additive Split
Asecond-order tensor can be uniquely additively split into a
symmetricsecond-order tensor and a skew-symmetric
second-order tensor , such that,
A
, ij ij ijA S W A S W
1 1
2 2
1 1
2 2
,
,
T T
ij ij ji ji
T T
ij ij ji ji
S A A S
W A A W
S A A S
W A A W
S
W
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Symmetric/Skew-symmetric Second-order Tensors
The matrices of components of a symmetricsecond-order
tensor and a skew-symmetric second-order tensor , are
given by,
W
11 12 13 12 13
12 22 23 12 23
13 23 33 13 23
0, 0
0
S S S W W S S S W W
S S S W W
S W
S
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Skew-symmetric Second-order Tensor
Askew-symmetric second-order tensor can be defined by
means of an axial (or dual) vector , such that,
W
and using index notation the following relations hold,
1 1
2 2
,
,
ij i j ijk k i j ij ijk k
k k ijk ij k k ijk ij
W W
W W
W e e e e
e e
2
Wu u u
W
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Skew-symmetric Second-order Tensor
Using matrix notationand index notationthe following relations
hold,
12 13 3 2
12 23 3 1
13 23 2 1
1 2 3 23 13 12
0 0
0 00 0
, , , ,T T
W W
W WW W
W W W
W
2 2 2 2 2 212 13 23 1 3 32 2 2W W W W
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Spherical Second-order Tensor
Asecond-order tensor is said to be spherical if the following
relations hold,
A
0 0
0 0
0 0
A
,
tr tr 3 , 3
ij ij
ii ii
A
A
A 1
A 1
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Deviatoric Second-order Tensor
Asecond-order tensor is said to be deviatoricif the following
condition holds,
A
tr 0, 0iiA A
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Spherical/Deviatoric Additive Split
Asecond-order tensor can be uniquely additively split into a
spherical second-order tensor and a deviatoric second-
order tensor , such that,
A
1 1
3 3
1 1
3 3
dev , dev
tr ,
dev tr , dev
esf esf
ij ij ij
esf esf
ij kk ij
ij kk ijij
A A A
A A
A A A
A A A
A A 1
A A A 1
esfA
dev A
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Eigenvalues/Eigenvectors
Given a symmetric second-order tensor , its eigenvalues
and associated eigenvectors , which define an
orthonormal basis along the principal directions, satisfy the
following equation (Einstein notation does not applies here),
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A
in
,i i i i i An n A 1 n 0
In order to get a non-trivial solution, the tensor has to
be singular, i.e., its determinant, defining the characteristic
polynomial, has to be equal to zero, yielding,
det 0i ip A 1
iA 1
i
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Eigenvalues/Eigenvectors
Given a symmetric second-order tensor , its eigenvalues
and associated eigenvectors , which define an
orthonormal basis along the principal directions, characterize
the physical nature of the tensor.
The eigenvaluesdo not depend on the particular orthonormal
basis chosen to define the components of the tensor and,
therefore, the solution of the characteristic polynomial do not
depends on the system of reference in which the components ofthe tensor are given.
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A
i in
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Characteristic Polynomial and Principal Invariants
The characteristic polynomialof a second-order tensor takes
the form,
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where the coefficients of the polynomial are the principal scalarinvariantsof the tensor (invariants in front of an arbitrary
rotation of the orthonormal basis) given by,
3 21 2 3det 0i ip I I I A 1
A
1 1 2 3
2 2
2 1 2 2 3 3 1
3 1 2 3
1
2
tr
tr tr
det
I
I
I
A A
A A A
A A
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Eigenvalues/Eigenvectors
Given a deviatoric part of a symmetric second-order tensor
, its eigenvalues and asociated eigenvectors ,
which define an orthonormal basis along the principal directions,
satisfy the following equation (Einstein notation does not applies
here),
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dev Ai in
dev , devi i i i i An n A 1 n 0
In order to get a non-trivial solution, the tensor has
to be singular, i.e., its determinant, defining the characteristic
polynomial, has to be equal to zero, yielding,
det dev 0i ip A 1
dev iA 1
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Characteristic Polynomial and Principal Invariants
The characteristic polynomialof the deviatoric part of a second-order tensor takes the form,
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where the coefficients of the polynomial are the pincipal scalarinvariantsof the deviatoric part of the tensor (invariants in front
of an arbitrary rotation of the orthonormal basis) given by,
3 21 2 3det dev 0i ip I I I A 1
1
2
2 1 1 2 2 3 3
3 1 2 3
1 1
2 2
dev tr dev 0
dev tr dev
dev det dev
I
I
I
A A
A A
A A
dev A
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Eigenvalues/Eigenvectors
The eigenvalues/eigenvectorsproblem for a symmetric second-order tensor and its the deviatoric part satisfy the
following relations (Einstein notation does not applies here),
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dev A
dev , devi i i i i An n A 1 n 0
1
3
dev tr i i i i
A 1 n A A 1 n 0
,i i i i i An n A 1 n 0
A
Then,
1
3tr ,i i i i A n n
yielding,
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For a second-order unit tensor, all three eigenvalues are equal toone, , and any direction is a principal direction.
Then we may take any orthonormal basis as principal directions
and the spectral decomposition can be written,
Spectral Decomposition
The spectral decomposition of the secod-order unit tensor takesthe form,
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ij i j i i 1 e e e e
1 2 3 1
1,3 i i i ii 1 n n n n
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Spectral Decomposition
The spectral decomposition of a symmetric second-order tensortakes the form (Einstein notation does not applies here),
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A
1,3 i i ii
A n n
If there are two equal eigenvalues , the spectraldecomposition takes the form,
1 2
3 3 3 3 3 A 1 n n n n
1 2 3
1,3 i i i ii
A 1 n n n n
If the three eigenvalues are equal , the spectraldecomposition takes the form,
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Third-order Tensors
A third-order tensor, denoted as , may be written as a linearcombination of tensor products of three orthonormal basis
vectors, denoted as , such that,
A third-order tensor has components.
Third order Tensors
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3 3 3
1 1 1 ijk i j k ijk i j k i j k
e e e e e e
i j k e e e
33 27
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Fourth-order Tensors
A fourth-order tensor, denoted as , may be written as a linearcombination of tensor products of four orthonormal basis
vectors, denoted as , such that,
A fourth-order tensor has components.
Fourth order Tensors
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3 3 3 3
1 1 1 1 ijkl i j k l ijkl i j k l i j k l e e e e e e e e
i j k l e e e e
43 81
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Double Dot Product with Second-order Tensors
The double dot productof a fourth-order tensor with asecond-order tensor is another second-order tensor given
by,
Fourth order Tensors
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: ,ijkl kl i j ij i j ij ijkl kl
A B B A B A e e e e
A B
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Transpose of a Fourth-order Tensor
The transpose of a fourth-order tensor is uniquely defined asa fourth-order tensor such that for arbitrary second-order
tensors and the following relationship holds,
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BA
T
: : : : , ,T T Tij ijkl kl kl klij ij ijkl klij
A B B A A B B A
The transpose of the transpose of a fourth-order tensor is
the same fourth-order tensor ,
T
T
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Tensor Product of two Second-order Tensors
The tensor product of two second-order tensors and is afourth order tensor given by,
Fourth order Tensors
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,ij kl i j k l ijkl i j k l
ijkl ij kl ijkl
A B
A B
A B e e e e e e e e
A B
BA
The transpose of the fourth-order tensor is given by, A B
,
TT
ij kl i j k l
T
ijkl i j k l
TT
ijkl ij kl ijkl ijkl
B A
B A
A B B A e e e e
e e e e
A B B A
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Fourth-order Identity Tensors
Let us consider the following fourth-order identity tensors,
Fourth order Tensors
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1 1
2 2
ik jl i j k l
il jk i j k l
ik jl il jk i j k l
e e e e
e e e e
e e e e
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Fourth-order Identity Tensors
The fourth-order identity tensor satisfies the followingexpressions,
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: ,
:
ik jl kl i j ij i j
ij ik jl k l kl k l
A A
A A
A e e e e A
A e e e e A
: : : : : :
: : : :
T
T
A B A B B A B A
A B B A
The fourth-order identity tensor is symmetric, satisfying,
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Fourth-order Identity Tensors
The fourth-order identity tensor satisfies the followingexpressions,
Fourth order Tensors
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:
:
T
il jk kl i j ji i j
Tij il jk kl k l lk k l
A A
A A A
A e e e e A
A e e e e A
: : : : : :
: : : :
T TT
T
A B A B B A B A
A B B A
The fourth-order identity tensor is symmetric, satisfying,
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Fourth-order Identity Tensors
The fourth-order identity tensor satisfies the followingexpressions,
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1 1 : :
2 21 1
: :2 2
T
T
A A A A
A A A A
1 1 : : : : : : 2 2
: : : :
T T
T
T
A B A B B B A A B A
A B B A
The fourth-order identity tensor is symmetric, satisfying,
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Fourth-order Deviatoric Projection Operator Tensor
The fourth-order deviatoric projection operator tensor isdefined as,
October 20, 2012 Carlos Agelet de Saracibar 95
dev
1 1
3 3,dev dev ik jl ij kl ijkl
1 1
The deviatoric part of a second-order tensor can be obtained
as,
A
1 1
3 3
1 1
3 3
dev : : tr ,
dev
dev
dev kl ik jl ij kl kl ij kk ijij ijklA A A A
A A 1 1 A A A 1
A
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Nabla
The nabla vector differential operator, denoted as , is definedas,
Using Einstein notation,
p
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1,3 ii
ix
e
iix
e
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Laplacian
The laplacian scalar differential operator, denoted as , isdefined as,
p
October 20, 2012 Carlos Agelet de Saracibar 97
2
2
ix
Hessian
The hessian symmetric second-order tensor differential
operator, denoted as ,is defined as,
2
i j
i jx x
e e
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Divergence, Curl and Gradient
The divergence differential operator div () is defined as,
p
October 20, 2012 Carlos Agelet de Saracibar 98
div iix
e
The curl differential operator curl () is defined as,
curl iix
e
The gradient differential operator grad () is defined as,
grad i ii ix x
e e
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Laplacian
The laplacian scalar differential operator can be expressed as thediv (grad ()) operator,
p
October 20, 2012 Carlos Agelet de Saracibar 99
2
2 divgrad
ix
Hessian
The hessian symmetric second order-tensor differential operator
can be expressed as the grad (grad ()) operator,
2
gradgradi ji jx x
e e
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Gradient of a Scalar Field
The gradient differential operator grad () is defined as,
p
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The gradient of a scalar fieldis a vector field defined as,
grad i ii ix x
e e
,grad i i iix
e e
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Laplacian of a Scalar Field
The laplacian differential operator is defined as,
p
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The laplacian of a scalar fieldis a scalar field defined as,
2
,2 ii
ix
2
2
ix
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Hessian of a Scalar Field
The hessian differential operator is defined as,
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The hessian of a scalar fieldis a symmetric second-order tensor
field defined as,
2
,i j ij i ji jx x
e e e e
2
i j
i jx x
e e
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Gradient of a Vector Field
The gradient differential operator grad () is defined as,
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The gradient of a vector fieldis a second-order tensor fielddefined as,
grad i ii ix x
e e
,grad i
j i j i j i j
j j
uu
x x
uu u e e e e e
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Curl of a Vector Field
The curl differential operator curl () is defined as,
October 20, 2012 Carlos Agelet de Saracibar 104
The curl of a vector fieldis a vector field defined as,
,curl k k
j j k ijk i ijk k j i
j j j
u uu
x x x
uu u e e e e e
curl iix
e
If the curl of a vector fieldis zero, the vector field is said to becurl-freeand there is a scalar field such that,
curl 0 | grad u u
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Curl of the Gradient of a Scalar Field
The curl differential operator curl () and gradient differentialoperator grad () are defined, respectively, as,
October 20, 2012 Carlos Agelet de Saracibar 105
The gradient of a scalar fieldis a vector field defined as,
,grad i i i
i
x
e e
curl , gradi ii ix x
e e
The curl of the gradient of a vector fieldis a null vector,
,curlgrad ijk kj i e 0
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Divergence of a Vector Field
The divergence differential operator div () is defined as,
October 20, 2012 Carlos Agelet de Saracibar 106
The divergence of a vector fieldis a scalar field defined as,
,div i i i
j i j ij i i
j j j i
u u uu
x x x x
uu u e e e
div iix
e
If the divergence of a vector fieldis zero, the vector field is said
to be solenoidalor div-freeand there is a vector field such that,
div 0 | curl u v u v
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Divergence of the Curl of a Vector Field
The divergence differential operator div () and curl differentialoperator curl () are, respectively, defined as,
October 20, 2012 Carlos Agelet de Saracibar 107
The curl of a vector fieldis a vector field defined as,
,curl k k
j j k ijk i ijk k j i
j j j
u uu
x x x
uu u e e e e e
div , curli ii ix x
e e
The divergence of a curl of a vector fieldis a null scalar,
,divcurl 0ijk k jiu u u
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Laplacian of a Vector Field
The laplacian differential operator is defined as,
October 20, 2012 Carlos Agelet de Saracibar 108
The laplacian of a vector fieldis a vector field defined as,
2
,2
ii i jj i
j
uu
x
u u e e
2
2
ix
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Divergence of a Second Order Tensor Field
The divergence differential operator div () is defined as,
October 20, 2012 Carlos Agelet de Saracibar 109
The divergence of a second-order tensor fieldis a vector field
defined as,
,
div
ik ik j i k j kj i
j j j
ij
i ij j i
j
A A
x x x
AA
x
A A
Ae e e e e
e e
div iix
e
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Curl of a Second-order Tensor Field
The curl differential operator curl () is defined as,
October 20, 2012 Carlos Agelet de Saracibar 110
The curl of a second-order tensor fieldis a second-order tensorfield defined as,
,
curl
kj
l l k jl l
kj
lki i j lki kj l i j
l
A
x x
AA
x
A A
Ae e e e
e e e e
curl iix
e
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Gradient of a Second-order Tensor Field
The gradient differential operator grad () is defined as,
October 20, 2012 Carlos Agelet de Saracibar 111
The gradient of a second-order tensor fieldis a third-ordertensor field defined as,
grad i ii ix x
e e
,
grad
kk
ij
i j k ij k i j k
k
x
AA
x
A A
Ae
e e e e e e
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Assignment 1.3
Establish the following identities involving a smooth scalar fieldand a smooth vector field ,
October 20, 2012 Carlos Agelet de Saracibar 112
(1)
(2)
div div grad
grad grad grad
v v v
v v v
v
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Assignment 1.3
Establish the following identities involving a smooth scalar fieldand a smooth vector field ,
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(1)
(2)
div div grad
grad grad grad
v v v
v v v
v
, ,,
(1) div div grad
div div gradi i i i iiv v v
v v v
v v v
, ,,
(2) grad grad grad
grad grad gradi i j i jj ij ijijv v v
v v v
v v v
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Assignment 1.4 [Classwork]
Establish the following identities involving the smooth scalarfields and , smooth vector fields and , and a smooth
second order tensor field ,
October 20, 2012 Carlos Agelet de Saracibar 114
u
(1)
(2)
(3)
(4)
(5)
div div grad
div div : grad
div curl curl
div grad divgrad grad grad
T
A A A
A v A v A v
u v v u u v
u v u v u v
vA
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Assignment 1.4 [Classwork]
Establish the following identities involving the smooth scalarfields and , smooth vector fields and , and a smooth
second order tensor field ,
October 20, 2012 Carlos Agelet de Saracibar 115
u vA
, ,,
(1) div div grad
div div gradij ij j ij j i ii jA A A
A A A
A A A
, ,,
(2) div div : grad
div div : grad
T
T
ji j ji i j ji j iiA v A v A v
A v A v A v
A v A v A v
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, ,,, ,
(3) div curl curl
div
curl curl
ijk j k ijk j i k ijk j k ii
k kij j i j jik k i
u v u v u v
v u u v
u v v u u v
u v
v u u v
, ,,
(4) div grad div
div grad divi j i j j i j j ii iju v u v u v u
u v u v u v
u v u v v
, ,,
(5) grad grad gradgrad grad gradi ii i ii
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Assignment 1.5 [Homework]
Establish the following identities involving the smooth scalarfield , and smooth vector fields and ,
October 20, 2012 Carlos Agelet de Saracibar 117
u v
(1)
(2)
(3)
(4)
(5)
grad grad grad
curl grad curl
curl div div grad grad
grad div curl curl
2grad : grad
T T
u v u v v u
v v v
u v u v v u u v v u
v v v
u v u v u v u v
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Assignment 1.5 [Homework]
Establish the following identities involving the smooth scalarfield , and smooth vector fields and ,
October 20, 2012 Carlos Agelet de Saracibar 118
u v
, ,,
(1) grad grad grad
grad
grad grad
grad grad
grad grad
T T
j j j i j j j ii i
j jji ji
T T
j jij ij
T T
i
u v u v u v
v u
v u
u v u v v u
u v
u v
u v
u v v u
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, ,,
(2) curl grad curl
curl
grad curl
ijk k ijk j k ijk k jji
i
v v v
v v v
v
v v
, ,
, ,
, ,
, , , ,
(3) curl div div grad grad
curl
grad div div grad
ijk ijk klm l mk j ji
ijk klm l j m ijk klm l m j
il jm im jl l j m il jm im jl l m j
i j j j j i i j j j i j
i ii i
u v
u v u v
u v u v
u v u v u v u v
v u
u v u v v u u v v u
u v u v
u v u v v u
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, ,
, ,, ,
, , , , ,
(4) grad div curl curl
, grad div
curl curl curl
i i jj j jii i
ijk ijk klm m l ijk klm m ljk ji j
il jm im jl m lj j ij i jj j ji i jj
v v v
v v
v v v v v
v v v
v v
v v
, ,, , ,
, , , ,
(5) 2grad : grad
2
2grad : grad
i i i i i j i i i jjj j j
i jj i i j i j i i jj
u v u v u v u v
u v u v u v
u v u v u v u v
u v
u v u v u v
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Assignment 1.6 [Classwork]
Given the vector determine
October 20, 2012 Carlos Agelet de Saracibar 121
div ,curl ,grad , .v v v v 1 2 3 1 1 2 2 1 3x x x x x x v v x e e e
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Assignment 1.6 [Classwork]
Given the vector determine
October 20, 2012 Carlos Agelet de Saracibar 122
div ,curl ,grad , .v v v v 1 2 3 1 1 2 2 1 3x x x x x x v v x e e e
, 2 3 1div i iv x x x v
1 2 3
1 2 3
1 2 3
3,2 2,3 1 1,3 3,1 2 2,1 1,2 3
1 2 2 2 1 3 3
curl det
1
v v v
v v v v v v
x x x x x
e e e
v
e e e
e e
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Assignment 1.6 [Classwork]
Given the vector determine
October 20, 2012 Carlos Agelet de Saracibar 123
div ,curl ,grad , .v v v v 1 2 3 1 1 2 2 1 3x x x x x x v v x e e e
,
2 3 1 3 1 2
2 1
grad
grad 0
1 0 0
i j i jv
x x x x x xx x
v e e
v
,i i i jj iv v v e e 0
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Divergence Theorem
Given a vector field in a volume V with closed boundarysurface and outward unit normal to the boundary , the
divergence(or Gauss) theoremreads,
Given a second-order tensor field in a volume V with closed
boundary surface and outward unit normal to the boundary
the divergence(or Gauss) theoremreads,
uV n
divV V V
dS dV dV
u n u u
V nA
divV V V
dS dV dV
A n A A
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Curl Theorem
Given a vector field in a surface S with closed boundaryand outward unit normal to the surface , the curl(or Stokes)
theoremreads,
where the curve of the line integral must have positive
orientation, such that drpoints counter-clockwise when the unit
normal points to the viewer, following the right-hand rule.
u Sn
rotS S S
d dS dS
u r u n u n