Tensor Algebra

Continuum Mechanics Chapter 1 Introduction to Vectors and Tensors C. Agelet de Saracibar ETS Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Cataluña (UPC), Barcelona, Spain International Center for Numerical Methods in Engineering (CIMNE), Barcelona, Spain

Introduction to Vectors and Tensors > Contents

Contents Contents 1. 2. 3. 4. 5. 6.

Introduction Algebra of vectors Algebra of tensors Higher order tensors Differential operators Integral theorems

September 9, 2016

Carlos Agelet de Saracibar

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Introduction to Vectors and Tensors > Introduction

Introduction Tensors Continuum mechanics deals with physical properties of materials, either solids or fluids, which are independent of any particular coordinate system in which they are observed. Those physical properties are mathematically represented by tensors. Tensors are mathematical objects which have the required property of being independent of any particular coordinate system.

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Introduction 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, strain rate  Third-order tensors, i.e. piezoelectric tensor  Fourth-order tensors, i.e. elastic constitutive tensor, elastoplastic constitutive tensor First- and higher-order tensors may be expressed in terms of their components in a given coordinate system. September 9, 2016


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Introduction Notation on Printed Documents     

Zero-order tensors: α , a, A First-order tensors: α , a, A Second-order tensors: α , a, A Third-order tensors:  Fourth-order tensors: 

Notation on Hand-written Documents  Zero-order tensors: α , a, A  First-order tensors: α , a , A  Second-order tensors: α , a , A September 9, 2016


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Introduction to Vectors and Tensors > Algebra of Vectors

Algebra of Vectors Scalar A physical quantity, completely described by a single real number, such as the temperature, density or pressure, is called a scalar and is designated by α , a, A . A scalar may be viewed as a zero-order tensor.

Vector A vector is a directed line element in space. A model for physical quantities having both direction and length, such as velocity, acceleration or force, is called a vector and is designated by. α , a, A. A vector may be viewed as a first-order tensor. September 9, 2016


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Algebra of Vectors Sum of Vectors The sum of vectors yields a new vector, based on the parallelogram law of addition. The sum of vectors has the following properties,

u+ v = v+u

( u + v ) + w =u + ( v + w ) u+0= u u + ( −u ) =0 where 0 denotes the unique zero vector with unspecified direction and zero length. September 9, 2016


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Algebra of Vectors Scalar Multiplication Let u be a vector and α be a scalar. The scalar multiplication α u gives a new vector with the same direction as u if α > 0 or with the opposite direction to u if α < 0. The scalar multiplication has the following properties,

(αβ ) u = α ( β u ) (α + β ) u =α u + β u α (u + v ) = α u + α v

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Algebra of Vectors Dot Product The dot (or scalar or inner) product of two vectors u and v, denoted by u ⋅ v, is a scalar given by,

u⋅v

u v cos θ ( u, v ) , 0 ≤ θ ( u, v ) ≤ π

where u is the norm (or length or magnitude) of the vector u, which is a non-negative real number defined as,

u = (u ⋅ u )

12

≥0

and θ ( u, v ) is the angle between the non-zero vectors u and v when their origins coincide. September 9, 2016


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Algebra of Vectors Dot Product The dot (or scalar or inner) product of vectors has the following properties,

u ⋅ v = v ⋅u 0 u⋅0 = u ⋅ (α v + β w )= α ( u ⋅ v ) + β ( u ⋅ w ) u ⋅u > 0 ⇔ u ≠ 0 u ⋅ u= 0 ⇔ u= 0 ⋅ v 0, u ≠ 0, v ≠ 0 ⇔ u ⊥ v u=

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Algebra of Vectors Unit Vectors A vector e is called a unit vector if its norm (or lenght or magnitude) is equal to 1,

e =1

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Algebra of Vectors Orthogonal Vectors A non-zero vector u is said to be orthogonal (or perpendicular) to a non-zero vector v if,

⋅ v 0, u ≠ 0, v ≠ 0 ⇔ θ ( u, = u= v) π 2 ⇔ u ⊥ v

Orthonormal Vectors A unit vector u is said to be orthonormal (or perpendicular) to a unit vector v if,

u ⋅ v = 0,

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u = 1, v = 1 ⇔ θ ( u, v ) = π 2 ⇔ u ⊥ v


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Algebra of Vectors Projection The projection of a vector u along a direction given by a unit vector e is a (positive or negative) scalar quantity defined as,

= u ⋅e

u cos θ ( u, e ) , 0 ≤ θ ( u, e ) ≤ π

u θ ( u, e ) e

u cos θ ( u, e ) September 9, 2016


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Algebra of Vectors Cartesian Basis Let us consider a three-dimensional Euclidean space with a fixed set of three right-handed orthonormal basis vectors e1 , e 2 , e3 , denoted as Cartesian basis, satisfying the following properties, x3 ,

e1 ⋅ e 2 = e1 ⋅ e3 = e 2 ⋅ e3 = 0

e3

O e1

e2 x2 ,

= e1

e= 2

1 e= 3

x1 ,

A fixed set of three unit vectors which are mutually orthogonal form a so called orthonormal basis vectors, collectively denoted as {ei }, and define an orthonormal system. September 9, 2016


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Algebra of Vectors Cartesian Components Any vector u in the three-dimensional Euclidean space is represented uniquely by a linear combination of the basis vectors e1 , e 2 , e3 , i.e.

u = u1e1 + u2e 2 + u3e3 where the three real numbers u1 , u2 , u3 are the uniquely determined Cartesian components of vector u along the given directions e1 , e 2 , e3, respectively. Using matrix notation, the Cartesian components of the vector u can be collected into a vector of components [u ] ∈  3 given by,

[u ] = [u1 September 9, 2016

u2

u3 ]


T

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Algebra of Vectors Index Notation Using index notation the vector u can be written as,

u = ∑ i =1 ui ei 3

Leaving out the summation symbol, using the so called Einstein notation, the vector u can be written in short form as,

u = ui ei where we have adopted the summation convention introduced by Einstein.

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Algebra of Vectors Index Notation  The index that is summed over is said to be a dummy (or summation) index.  The same repeated index can appear only twice.  An index that is not summed over in a given term is called a free (or live) index.

vi = ai b j c j = ai ( b1c1 + b2 c2 + b3c3 ) v = ai bi c j d j = ( a1b1 + a2b2 + a3b3 )( c1d1 + c2 d 2 + c3d3 ) v= ai b j c j ck d k= ai ( b1c1 + b2 c2 + b3c3 )( c1d1 + c2 d 2 + c3d3 ) i

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Algebra of Vectors Kronecker Delta The Kronecker delta δ ij is defined as,

1 if i = j δ ij =  0 if i ≠ j The Kronecker delta satisfies the following properties,

= δ ii 3,= δ ijδ jk δ ik= , δ ij u j ui Note that the Kronecker delta plays the role of a replacement operator.

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Algebra of Vectors Dot Product The dot (or scalar or inner) product of two orthonormal basis vectors ei and e j , denoted as ei ⋅ e j , is a scalar quantity (taking values either 0 or 1) and can be conveniently written in terms of the Kronecker delta as,

ei ⋅ e j = δ ij

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Algebra of Vectors Components of a Vector

Taking the basis {ei }, the projection of a vector u onto the basis vector ei yields the i-th component of u ,

u ⋅ ei=

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(u e ) ⋅ e = j

j

i

u j e j ⋅ ei= u jδ ji= ui


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Algebra of Vectors Dot Product

Taking the basis {ei }, the dot (or scalar or inner) product of two vectors u and v, denoted as u ⋅ v , is a scalar quantity and using index notation can be written as,

u ⋅ v= ui ei ⋅ v j e j = ui v j ei ⋅ e j = ui v jδ ij = ui vi = u1v1 + u2 v2 + u3v3

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Algebra of Vectors Euclidean Norm

Taking the basis {ei }, the Euclidean norm (or length or magnitude) of a vector u, denoted as u , is a non-negative scalar quantity and, using index notation, can be written as,

u = (u ⋅ u )

12

=

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(u

2 1

= ( uiei ⋅ u je j )

12

= ( uiui )

12

)

2 12 3

+u +u 2 2

= ( uiu jδ ij )

12


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Assignment 1.1 Assignment 1.1 [Classwork] Write in index form the following expressions,

(a ⋅ b ) c v a (b ⋅ c) = v= ( a ⋅ b )( c ⋅ d ) v = a ( b ⋅ c )( c ⋅ d ) v=

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Algebra of Vectors Cross Product The cross (or vector) product of two vectors u and v , denoted as u × v, 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,

u× v

u v sin θ ( u, v ) e, 0 ≤ θ ( u, v ) ≤ π

where e is a unit vector perpendicular to the plane defined by the two vectors, u ⋅ e= 0, v ⋅ e= 0, in the direction given by the right-hand rule, .

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Algebra of Vectors Norm of the Cross Product of two Vectors The norm (or magnitude or length) of the cross (or vector) product of two vectors u and v, denoted as u × v , measures the area spanned by the two vectors and is given by,

= u× v

u v sin θ ( u, v ) , 0 ≤ θ ( u, v ) ≤ π

u× v v

u× v u

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Algebra of Vectors Permutation Symbol

The permutation (or alternating or Levi-Civita) symbol ε ijk is defined as,

+1 for even permutations of (i, j , k ), i.e. 123, 231, 312  ε ijk =  −1 for odd permutations of (i, j , k ), i.e. 213, 132, 321  0 if there is repeated index 

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Algebra of Vectors Permutation Symbol

The permutation (or alternating or Levi-Civita) symbol ε ijk may be written in terms of the Kronecker delta and has the following properties,

ε ijk

 δ ip  δ i1 δ i 2 δ i 3   δ , ε ε = det det δ δ j2 j3  ijk pqr δ jp  j1 δ kp δ k1 δ k 2 δ k 3  

δ iq δ ir   δ jq δ jr  δ kq δ kr 

ε ijpε klp = δ ik δ jl − δ ilδ jk , ε ikl ε jkl = 2δ ij , ε ijk ε ijk = 2δ ii = 6 i − j )( j − k )( k − i ) ( ε ijk = 2

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Algebra of Vectors Cross Product The cross (or vector) product of two right-handed orthonormal basis vectors satisfies the following properties, x3 ,

e1 ×= e 2 e3 , e 2 ×= e3 e1 , e3 ×= e1 e 2 , e 2 × e1 = −e3 , e3 × e 2 = −e1 , e1 × e3 = −e 2 ,

e3

O

e2

e1 x1 ,

x2 ,

e1 × e1 = e 2 × e 2 = e3 × e3 = 0.

Taking the right-handed orthonormal basis {ei }, the cross (or vector) product of two basis vectors ei and e j , denoted as ei × e j , can defined as,

ei × e j = ε ijk e k September 9, 2016


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Algebra of Vectors Cross Product

Taking the basis {ei } , the cross (or vector) product of two vectors u and v, denoted as u × v , is another vector, perpendicular to the plane defined by the two vectors, and can be written in index form as,

u × v= ui ei × v j e j = ui v j ei × e j = ui v j ε ijk e k = ε ijk ui v j e k = ( u2v3 − u3v2 ) e1 + ( u3v1 − u1v3 ) e2 + ( u1v2 − u2v1 ) e3 e1 e 2 = det u1 u2  v1 v2 September 9, 2016

e3  u3  v3  Carlos Agelet de Saracibar

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Algebra of Vectors Cross Product The cross (or vector) product of vectors has the following properties,

u × v =− ( v × u ) u ×= v 0, u ≠ 0, v ≠ 0 ⇔ u || v

(α u ) × v = u × (α v ) = α ( u × v ) u×(v + w) = u× v + u× w

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Assignment 1.2 Assignment 1.2 Using the expression,

u× v

2

u

2

v sin 2 θ ( u, v ) , 0 ≤ θ ( u, v ) ≤ π 2

obtain the following relationships between the permutation symbol and the Kronecker delta,

ε ijpε klp = δ ik δ jl − δ ilδ jk , ε ipqε jpq = 2δ ij , ε ijk ε ijk = 2δ ii = 6

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Algebra of Vectors Box Product The box (or triple scalar) product of three right-handed orthonormal basis vectors ei , e j , e k , denoted as ei ⋅ ( e j × e k ) , is a scalar quantity (taking values either 1, -1 or 0) and can be conveniently written in terms of the permutation symbol as,

ε ijk =ei ⋅ ( e j × e k ) =( ei × e j ) ⋅ e k

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Algebra of Vectors Box Product

Taking the orthonormal basis {ei }, the box (or triple scalar) product of three vectors u, v, w , denoted as u ⋅ ( v × w ), is a scalar quantity and using index notation can be written as,

u ⋅ ( v × w ) = ui ei ⋅ ( v j e j × wk e k ) = ui v j wk ei ⋅ ( e j × e k ) = ε ijk ui v j wk  u1 = det  v1  w1

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u2 v2 w2

u3  v3  w3 


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Algebra of Vectors Box Product The box (or triple scalar) product of three vectors u, v, w represents the volume of the parallelepiped spanned by the three vectors,

V =u ⋅ ( v × w )

v×w u

V =u ⋅ ( v × w ) w

v September 9, 2016


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Algebra of Vectors Box Product The box (or triple scalar) product has the following properties,

u ⋅ ( v × w ) =w ⋅ ( u × v ) =v ⋅ ( w × u ) u ⋅ ( v × w ) =−u ⋅ ( w × v ) u ⋅ ( u × v ) =v ⋅ ( u × v ) =0

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Algebra of Vectors Triple Vector Product

Taking the orthonormal basis {ei } , the triple vector product of three vectors u, v, w , denoted as u × ( v × w ) , is a vector and using index notation can be written as,

u×(v × w) = ε ijk ui ( ε pqj v p wq ) e k = ε kij ε pqj ui v p wq e k

(δ

=

δ − δ kqδ ip ) ui v p wq e k

kp iq

= ui vk wi e k − ui vi wk e k

= (u ⋅ w ) v − (u ⋅ v ) w

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Algebra of Vectors Triple Vector Product

Taking the orthonormal basis {ei } , the triple vector product of three vectors u, v, w , denoted as ( u × v ) × w , is a vector and using index notation can be written as,

ε ijk ( ε pqi u p vq ) w j e k (u × v ) × w = = ε jkiε pqi u p vq w j e k

(δ

=

δ − δ jqδ kp ) u p vq w j e k

jp kq

= u j w j vk e k − v j w j uk e k = (u ⋅ w ) v − ( v ⋅ w ) u September 9, 2016


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Algebra of Vectors Triple Vector Product The triple vector product has the following properties,

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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Introduction to Vectors and Tensors > Algebra of Tensors

Algebra of Tensors Second-order Tensors A second-order tensor A may be thought of as a linear operator that acts on a vector u generating a vector v, defining a linear transformation that assigns a vector v to a vector u,

v = Au The second-order unit tensor, denoted as 1 , satisfies the following identity,

u = 1u

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Algebra of Tensors Second-order Tensors As linear operators defining linear transformations, secondorder tensors have the following properties,

A (α u + v )= α Au + Av

(α A ) u = α ( Au ) ( A ± B ) u =Au ± Bu

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Algebra of Tensors Tensor Product The tensor product (or dyad) of two vectors u and v , denoted as u ⊗ v , is a second-order tensor which linearly transforms a vector w into a vector with the direction of u following the rule,

( u ⊗ v ) w =u ( v ⋅ w ) =( v ⋅ w ) u A dyadic is a linear combination of dyads.

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Algebra of Tensors Tensor Product The tensor product (or dyad) has the following linear properties,

( u ⊗ v )(α w + x )= α ( u ⊗ v ) w + ( u ⊗ v ) x (α u + β v ) ⊗ w = α u ⊗ w + β v ⊗ w u(v ⋅ w) (u ⊗ v ) w = (v ⋅ w)u = ( u ⊗ v )( w ⊗ x ) = ( v ⋅ w ) u ⊗ x = u ⊗ x ( v ⋅ w ) A (u ⊗ v= ) ( Au ) ⊗ v Generally, the tensor product (or dyad) is not commutative, i.e.

u ⊗ v ≠ v ⊗u

( u ⊗ v )( w ⊗ x ) ≠ ( w ⊗ x )( u ⊗ v ) September 9, 2016


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Algebra of Tensors Tensor Product The tensor product (or dyad) of two orthonormal basis vectors ei and e j , denoted as ei ⊗ e j , is a second-order tensor and can be thought as a linear operator such that,

( e ⊗ e ) e =( e i

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j

k

j

⋅ e k ) ei = δ jk ei


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Algebra of Tensors Second-order Tensors Any second-order tensor A may be represented by a linear combination of dyads formed by the Cartesian basis {ei }

= A Aij ei ⊗ e j where the nine real numbers, represented by Aij , are the uniquely determined Cartesian components of the tensor A with respect to the dyads formed by the Cartesian basis {ei }, represented by ei ⊗ e j , which constitute a basis second-order tensor for A . The second-order unit tensor may be written as,

1 = δ ij ei ⊗ e j = ei ⊗ ei September 9, 2016


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Algebra of Tensors Cartesian Components The ij-th Cartesian component of the second-order tensor A, denoted as Aij , can be written as,

ei ⋅ Ae j = ei ⋅ ( Akl e k ⊗ el ) e j = ei ⋅ ( Aklδ lj e k ) = ei ⋅ ( Akj e k ) = Akj ei ⋅ e k = Akjδ ik = Aij The ij-th Cartesian component of the second-order unit tensor 1 is the Kronecker delta,

ei ⋅ 1e j = ei ⋅ e j = δ ij

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Algebra of Tensors Matrix of Components Using matrix notation, the Cartesian components of any secondorder tensor A may be collected into a 3 x 3 matrix of components, denoted by [ A ] , given by,

= A Aij ei ⊗ e j

 A11 A A = [ ]  21  A31

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A12 A22 A32


A13  A23  A33 

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Algebra of Tensors Matrix of Components Using matrix notation, the Cartesian components of the secondorder unit tensor 1 may be collected into a 3 x 3 matrix of components, denoted as [1], given by,

1 = δ ij ei ⊗ e j = ei ⊗ ei

1 0 0  [1] = 0 1 0 0 0 1 

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Algebra of Tensors Second-order Tensors The tensor product (or dyad) of the vectors u and v, denoted as u ⊗ v , may be represented by a linear combination of dyads formed by the Cartesian basis {ei }

u ⊗= v ui v j ei ⊗ e j where the nine real numbers, represented by ui v j , are the uniquely determined Cartesian components of the tensor u ⊗ v with respect to the dyads formed by the Cartesian basis {ei }, represented by ei ⊗ e j .

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Algebra of Tensors Matrix of Components Using matrix notation, the Cartesian components of any secondorder tensor u ⊗ v may be collected into a 3 x 3 matrix of components, denoted as [u ⊗ v ], given by,

u ⊗= v ui v j ei ⊗ e j

 u1v1 u1v2 u v u v [u ⊗ v ] =  2 1 2 2 u3v1 u3v2

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u1v3  u2 v3  u3v3 

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Algebra of Tensors Second-order Tensors Using index notation, the linear transformation v = Au can be written as,

v = Au vi ei = ( Aij ei ⊗ e j ) ( uk e k ) = Aij uk ( ei ⊗ e j ) e k = Aij uk δ jk ei vi ei = Aij u j ei yielding

= v Au = , vi Aij u= [ v] j,

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[ A ][u ]

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Algebra of Tensors Positive Semi-definite Second-order Tensor A second-order tensor A is said to be a positive semi-definite tensor if the following relation holds for any non-zero vector u ≠ 0

u ⋅ Au ≥ 0 ∀u ≠ 0

Positive Definite Second-order Tensor A second-order tensor A is said to be a positive-definite tensor if the following relation holds for any non-zero vector u ≠ 0

u ⋅ Au > 0 ∀u ≠ 0

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Algebra of Tensors Transpose of a Second-order Tensor The unique transpose of a second-order tensor A , denoted as AT , is governed by the identity,

v ⋅ AT u = u ⋅ Av ∀u, v T v= A v= u j Aji vi i ij u j i A ji u j

∀u j , vi

The Cartesian components of the transpose of a second-order tensor A satisfy, T T AijT := A = e ⋅ A ( ) i e j =e j ⋅ Aei =Aji ij

Using index notation the transpose of A may be written as,

= AT Aji ei ⊗ e j September 9, 2016


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Algebra of Tensors Transpose of a Second-order Tensor The matrix of components of the transpose of a second-order tensor A is equal to the transpose of the matrix of components of A and takes the form,

A Aij ei ⊗ e j = AT Aji ei ⊗ e j =  A11 T T  A  [ = = A ]  A12  A13 September 9, 2016

A21 A22 A23


A31  A32  A33  56


Algebra of Tensors Symmetric Second-order Tensor A second-order tensor A is said to be symmetric if the following relation holds,

A = AT Using index notation, the Cartesian components of a symmetric second-order tensor A satisfy,

A= Aij ei ⊗ e j = Aji ei ⊗ e j = AT ,

Aij = Aji

Using matrix notation, the matrix of components of a symmetric second-order tensor A satisfies,

= A  [ A ] = T

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[ A]


T

57


Algebra of Tensors Dot Product The dot product of two second-order tensors A and B, denoted as AB , is a second-order tensor such that,

= ( AB ) u A ( Bu ) ∀u

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Algebra of Tensors Dot Product The components of the dot product AB along an orthonormal basis {ei } read,

ei ⋅ ( AB ) e j = ei ⋅ A ( Be j ) Cij = ( AB )ij = ei ⋅ A ( Bkj e k ) = ei ⋅ Ae k Bkj = = Aik Bkj = C AB = , Cij Aik Bkj

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Algebra of Tensors Dot Product The dot product of second-order tensors has the following properties,

AB ) C (=

A= ( BC ) ABC

A 2 = AA AB ≠ BA

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Algebra of Tensors Trace

The trace of a dyad u ⊗ v, denoted as tr ( u ⊗ v ) , is a scalar quantity given by,

tr ( u ⊗ v ) = u ⋅ v = ui vi The trace of a second-order tensor A , denoted as tr A , is a scalar quantity given by,

A tr ( Aij ei ⊗ e= Aij tr ( ei ⊗ e j ) tr = j) = Aij ( ei ⋅ e j )= Aijδ ij = Aii

= A11 + A22 + A33 = tr [ A ] September 9, 2016


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Algebra of Tensors Trace The trace of a second-order tensor has the following properties,

tr A = tr AT tr ( AB ) = tr ( BA ) tr ( A + B ) = tr A + tr B tr (α A ) = α tr ( A )

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Algebra of Tensors Double Dot Product The double dot product of two second-order tensors A and B , denoted as A : B , is a scalar quantity defined as, T T = A : B tr= A B tr B ( ) ( A) T T = tr= AB tr BA ( ) ( )

= B:A or using index notation,

A= : B A= B= B:A ij Bij ij Aij

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64


Algebra of Tensors Double Dot Product The double dot product has the following properties,

A= : 1 1= : A tr A A:B = B:A = A : ( BC )

T = B ( A) : C

T AC ( ):B

A : ( u ⊗ v ) =u ⋅ Av =( u ⊗ v ) : A

( u ⊗ v ) : ( w ⊗ x ) = ( u ⋅ w )( v ⋅ x ) ( ei ⊗ e j ) : ( ek ⊗ el ) = ( ei ⋅ ek ) ( e j ⋅ el ) = δ ikδ jl September 9, 2016


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Algebra of Tensors Euclidean Norm The euclidean norm of a second-order tensor A is a nonnegative scalar quantity, denoted as A , given by,

= A

September 9, 2016

= : A) (A ( Aij Aij ) 12


12

≥0

66


Algebra of Tensors Determinant The determinant of a second-order tensor A is a scalar quantity, denoted as det A , given by,

det A = det [ A ]  A11 = det  A21  A31

A12 A22 A32

A13  A23  A33 

1 = ε= ε ijk ε pqr Api Aqj Ark ijk A1i A2 j A3 k 6

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Algebra of Tensors Determinant The determinant of a second-order tensor has the following properties,

det AT = det A det (α A ) = α 3 det A det ( AB ) = det A det B

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68


Algebra of Tensors Singular Tensors A second-order tensor A is said to be singular if and only if its determinant is equal to zero, i.e.

A is singular ⇔ det A = 0

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Algebra of Tensors Inverse of a Second-order Tensor For a non-singular second-order tensor A , i.e. det A ≠ 0, there exist a unique non-singular inverse second-order tensor, denoted as A −1 , satisfying, −1 AA ( )=u

−1 A ( A )=u 1u= u ∀u

yielding, −1 −1 AA = A= A 1,

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−1 Aik= Akj−1 Aik= Akj δ ij


70


Algebra of Tensors Inverse of a Second-order Tensor The inverse second-order tensor satisfies the following properties,

(A ) = A A ) A ) (= (= −1 −1

T

−1

−1 T

A −T

(α A ) = α −1A −1 −1 −1 −1 AB B A = ( ) −1

A −2 = A −1A −1 det A −1 = ( det A ) September 9, 2016

−1


71


Algebra of Tensors Orthogonal Second Order Tensor An orthogonal second-order tensor Q is a linear operator that preserves the norms and the angles between two vectors,

Qu = ( u ⋅ Q Qu ) T

12

= (u ⋅ u )

12

=u

∀u

Qu ⋅ Qv = u ⋅ QT Qv = u ⋅ v   ∀u, v T Qv ⋅ Qu =v ⋅ Q Qu =v ⋅ u  and the following relations hold,

QT Q = QQT = 1, Q −1 = QT , det Q = ±1 September 9, 2016


72


Algebra of Tensors Rotation Second Order Tensor A rotation tensor R is a proper orthogonal second-order tensor, i.e., is a linear operator that preserves the lengths (norms) and the angles between two vectors,

Ru = ( u ⋅ R Ru ) T

12

= (u ⋅ u )

12

=u

∀u

Ru ⋅ Rv =u ⋅ R T Rv =u ⋅ v   ∀u, v T Rv ⋅ Ru =v ⋅ R Ru =v ⋅ u  and satisfies the following expressions, T T R= R RR = 1, = R −1 R T , det = R 1

September 9, 2016


73


Algebra of Tensors Symmetric/Skew-symmetric Additive Split A second-order tensor A can be uniquely additively split into a symmetric second-order tensor S and a skew-symmetric second-order tensor W, such that,

A= S + W, S=

A+A )= ( 2

1

W= (A − A 1

2

September 9, 2016

T

T

Sij =

T

S ,

−W )=

Aij = Sij + Wij

T

A ( 2

1

ij

+ Aji ) = S ji

, Wij = −W ji ( Aij − Aji ) =


1

2

74


Algebra of Tensors Symmetric/Skew-symmetric Second-order Tensors The matrices of components of a symmetric second-order tensor S and a skew-symmetric second-order tensor W, are given by,

 S11  [S ] =  S12  S13

September 9, 2016

S12 S 22 S 23

S13   S 23  , S33 

 0  [ W ] =  −W12  −W13


W12 W13   0 W23  −W23 0 

75


Algebra of Tensors Skew-symmetric Second-order Tensors The double dot product of a symmetric second-order tensor S and a skew-symmetric second-order tensor W gives,

S : W = SijWij = S12W12 + S13W13 + S 23W23 − S12W12 − S13W13 − S 23W23 =0

September 9, 2016


76


Algebra of Tensors Skew-symmetric Second-order Tensor A skew-symmetric second-order tensor W can be defined by means of an axial (or dual) vector ω , such that,

Wu =ω × u ∀u W = 2ω and using index notation the following relations hold,

W= Wij ei ⊗ e j = −ε ijk ωk ei ⊗ e j , Wij = −ε ijk ωk 1

1

2

2

ω= ωk e k = − ε ijkWij e k , ωk = − ε ijkWij September 9, 2016


77


Algebra of Tensors Skew-symmetric Second-order Tensor Using matrix notation and index notation the following relations hold,

W12 W13   0 −ω3 ω2   0  −W = ω  W = − 0 W 0 ω [ ]  12 23  1  3  −W13 −W23 0   −ω2 ω1 0 

[ω ] = [ω1 , ω2 , ω3 ]

T

W = September 9, 2016

2 2 (W122 + W132 + W= 23 )

= [ −W23 , W13 , −W12 ]

T

2 2 ω12 + ω32 + ω = 3


2ω 78


Algebra of Tensors Spherical Second-order Tensor A spherical second-order tensor A is defined as,

= A α= 1, Aij αδ ij and has the following properties,

A α= Aii αδ = tr= tr 1 3α , = 3α ii α 0 0  0 α 0 A = [ ]    0 0 α 

September 9, 2016


79


Algebra of Tensors Deviatoric Second-order Tensor A second-order tensor A is said to be deviatoric if the following condition holds,

= tr A 0,= Aii 0

September 9, 2016


80


Algebra of Tensors Spherical/Deviatoric Additive Split A second-order tensor A can be uniquely additively split into a spherical second-order tensor A esf and a deviatoric secondorder tensor dev A, such that,

A= A esf + dev A, A

esf

Aij = Aijesf + ( dev A )ij

= A ( tr A ) 1,

1

esf ij

3

1 3

Akk δ ij

A − ( tr A ) 1, ( dev A )ij = dev A = Aij − Akk δ ij

September 9, 2016

1

1

3

3


81


Algebra of Tensors Eigenvalues/Eigenvectors Given a symmetric second-order tensor A , its eigenvalues λi ∈  and associated eigenvectors ni, which define an orthonormal basis along the principal directions, satisfy the following equation (Einstein notation does not applies here),

An= λi ni , i

( A − λi 1) n=i

0

In order to get a non-trivial solution, the tensor A − λi 1 has to be singular, i.e., its determinant, defining the characteristic polynomial, has to be equal to zero, yielding,

p ( λi )= det ( A − λi 1)= 0 September 9, 2016


82


Algebra of Tensors Eigenvalues/Eigenvectors Given a symmetric second-order tensor A , its eigenvalues λi ∈  and associated eigenvectors ni, which define an orthonormal basis along the principal directions, characterize the physical nature of the tensor. The eigenvalues do not depend on the particular orthonormal basis chosen to define the components of the tensor and, therefore, the solution of the characteristic polynomial does not depend on the system of reference in which the components of the tensor are given.

September 9, 2016


83


Algebra of Tensors Characteristic Polynomial and Principal Invariants The characteristic polynomial of a second-order tensor A takes the form,

−λ 3 + I1λ 2 − I 2 λ + I 3 = p (λ ) = det ( A − λ1) = 0 where the coefficients of the polynomial are the principal scalar invariants of the tensor (invariants in front of an arbitrary rotation of the orthonormal basis) given by,

I1 ( A ) = tr A = λ1 + λ2 + λ3 I2 ( A ) =

1 2

2 2 tr A − tr A ( ) = λ1λ2 + λ2λ3 + λ3λ1

I 3= = A λ1λ2 λ3 ( A ) det September 9, 2016


84


Algebra of Tensors Eigenvalues/Eigenvectors Given a deviatoric part of a symmetric second-order tensor dev A , its eigenvalues λi′ ∈  and asociated eigenvectors n′i , which define an orthonormal basis along the principal directions, satisfy the following equation (Einstein notation does not applies here),

An′i λi′n′i , dev=

′ ′ ( dev A − λ= i 1) n i

0

In order to get a non-trivial solution, the tensor dev A − λi′1 has to be singular, i.e., its determinant, defining the characteristic polynomial, has to be equal to zero, yielding,

p= λi′1) 0 ( λi′) det ( dev A − = September 9, 2016


85


Algebra of Tensors Characteristic Polynomial and Principal Invariants The characteristic polynomial of the deviatoric part of a secondorder tensor dev A takes the form,

−λ ′3 + I1′λ ′2 − I 2′λ ′ + I 3′ = p (λ′ ) = det ( dev A − λ ′1) = 0 where the coefficients of the polynomial are the pincipal scalar invariants of the deviatoric part of the tensor (invariants in front of an arbitrary rotation of the orthonormal basis) given by,

I1′ (= dev A ) tr= dev A 0 − tr ( dev A ) = − ( λ1′λ1′ + λ2′λ2′ + λ3′λ3′ ) I 2′ ( dev A ) = 1

2

2

= I 3= dev A λ1′λ2′λ3′ ( dev A ) det September 9, 2016

1

2


86


Algebra of Tensors Eigenvalues/Eigenvectors The eigenvalues/eigenvectors problem for a symmetric secondorder tensor A and its the deviatoric part dev A satisfy the following relations (Einstein notation does not applies here),

An= λi ni , i An′i λi′n′i , dev=

( A − λi 1) n=i 0 ′ ′ ( dev A − λ= i 1) n i

0

Then,

 1    0 ( dev A − λi′1) n′i = ( A − λi 1) ni =  A −  λi′ + 3 ( tr A )  1  n′i =    

yielding,

λi = λi′ + ( tr A ) , ni = n′i 1

3

September 9, 2016


87


Algebra of Tensors Spectral Decomposition The spectral decomposition of the secod-order unit tensor takes the form,

1 = δ ij ei ⊗ e j = ei ⊗ ei For a second-order unit tensor, all three eigenvalues are equal to one, λ= λ= λ= 1, and any direction is a principal direction. 1 2 3 Then we may take any orthonormal basis as principal directions and the spectral decomposition can be written,

1 = ∑ i=1,3 ni ⊗ ni = ni ⊗ ni

September 9, 2016


88


Algebra of Tensors Spectral Decomposition The spectral decomposition of a symmetric second-order tensor A takes the form (Einstein notation does not applies here),

A =

∑

i =1,3

λi ni ⊗ ni

λ2 , the spectral If there are two equal eigenvalues λ= λ= 1 decomposition takes the form,

A = λ (1 − n3 ⊗ n3 ) + λ3n3 ⊗ n3 If the three eigenvalues are equal λ= λ= λ= λ3 , the spectral 1 2 decomposition takes the form,

A = λ1 = λ ∑ i =1,3 ni ⊗ ni = λni ⊗ ni September 9, 2016


89

Introduction to Vectors and Tensors > Higher-order Tensors

Third-order Tensors Third-order Tensors A third-order tensor, denoted as  , may be written as a linear combination of tensor products of three orthonormal basis vectors, denoted as ei ⊗ e j ⊗ e k , such that,

=

∑ ∑ ∑ 3

3

3

=i 1 =j 1 = k 1

ijk ei ⊗ e j ⊗ e= ijk ei ⊗ e j ⊗ e k k

A third-order tensor has 33 = 27 components.

September 9, 2016


90


Fourth-order Tensors Fourth-order Tensors A fourth-order tensor, denoted as  , may be written as a linear combination of tensor products of four orthonormal basis vectors, denoted as ei ⊗ e j ⊗ e k ⊗ el , such that,

 =

∑ ∑ ∑ ∑ 3

3

3

3

 ijkl ei ⊗ e j ⊗ e k ⊗ e=  ijkl ei ⊗ e j ⊗ e k ⊗ el l

=i 1 =j 1 = k 1 =l 1

A fourth-order tensor has 34 = 81 components.

September 9, 2016


91


Fourth-order Tensors Double Dot Product with Second-order Tensors The double dot product of a fourth-order tensor  with a second-order tensor A is another second-order tensor B given by,

B=  : A=  ijkl Akl ei ⊗ e j = Bij ei ⊗ e j , Bij =  ijkl Akl

September 9, 2016


92


Fourth-order Tensors Transpose of a Fourth-order Tensor The transpose of a fourth-order tensor  is uniquely defined as a fourth-order tensor T such that for arbitrary second-order tensors A and B the following relationship holds, T T , A :  : B B : T := A, Aij B B = A   ijkl kl kl klij ij ijkl  klij

The transpose of the transpose of a fourth-order tensor  is the same fourth-order tensor  ,

( )

T T

September 9, 2016

=


93


Fourth-order Tensors Tensor Product of two Second-order Tensors The tensor product of two second-order tensors A and B is a fourth order tensor  given by,

 = A ⊗ B = Aij Bkl ei ⊗ e j ⊗ e k ⊗ el =  ijkl ei ⊗ e j ⊗ e k ⊗ el ,

 ijkl = ( A ⊗ B )ijkl =Aij Bkl

The transpose of the fourth-order tensor = A ⊗ B is given by,

 = ( A ⊗ B ) = B ⊗ A = Bij Akl ei ⊗ e j ⊗ e k ⊗ el T

T

T =  ijkl ei ⊗ e j ⊗ e k ⊗ el ,



T ijkl

September 9, 2016

= ( A ⊗ B )ijkl = ( B ⊗ A )ijkl =Bij Akl T


94


Fourth-order Tensors Fourth-order Identity Tensors Let us consider the following fourth-order identity tensors,

=  δ ik δ jl ei ⊗ e j ⊗ e k ⊗ el =  δ ilδ jk ei ⊗ e j ⊗ e k ⊗ el ˆ= 1 (  +  )= 1 (δ δ + δ δ ) e ⊗ e ⊗ e ⊗ e ik jl il jk i j k l 2 2

September 9, 2016


95


Fourth-order Tensors Fourth-order Identity Tensors The fourth-order identity tensor  satisfies the following expressions,

 : A= δ ik δ jl Akl ei ⊗ e j = Aij ei ⊗ e j = A, A :=  Aijδ ik δ jl e k ⊗ e= Akl e k ⊗ e= A l l The fourth-order identity tensor  is symmetric, satisfying,

A:=  : B A= : B B= : A B :  : A T ⇒  =   T A::B = B: :A 

September 9, 2016


96


Fourth-order Tensors Fourth-order Identity Tensors The fourth-order identity tensor  satisfies the following expressions,

 : A= δ ilδ jk Akl ei ⊗ e j= Aji ei ⊗ e j= AT T A : = Aijδ ilδ jk Akl e k ⊗ e= A e ⊗ e = A l lk k l

The fourth-order identity tensor  is symmetric, satisfying,

: BT B= : AT B :  : A  A :=  : B A=  ⇒ T A:  :B = B:  :A 

September 9, 2016


= T

97


Fourth-order Tensors Fourth-order Identity Tensors The fourth-order identity tensor ˆ satisfies the following expressions,

ˆ : A =1 (  +  ) : A =1 ( A + AT ) 2 2 1 1 T ˆ A : = A : (  +  )= A + A ( ) 2 2 The fourth-order identity tensor ˆ is symmetric, satisfying, 1 1  T T ˆ ˆ : : : + = A :  := B A : (B + B= B A A B  A ) ( )  T ˆ ˆ 2 2 ⇒ =     A : ˆ : B = B : ˆ T : A  September 9, 2016


98


Fourth-order Tensors Fourth-order Deviatoric Projection Operator Tensor The fourth-order deviatoric projection operator tensor dev is defined as,

1

dev =−  1 ⊗ 1, 3

( )

dev ijkl

1

=δ ik δ jl − δ ijδ kl 3

The deviatoric part of a second-order tensor A can be obtained as, 1  1  − 1 ⊗ 1  : A = A − ( tr A ) 1, dev A = dev : A =  3  3 

( dev A )ij

1 1   Aij − ( Akk ) δ ij = ( dev )ijkl Akl =  δ ik δ jl − δ ijδ kl  Akl = 3 3  

September 9, 2016


99


Algebra of Tensors Algebra of Tensors

a = u ⋅ v = ui vi , a = ui vi w = u × v = ε ijk u j vk ei , wi = ε ijk u j vk a =u ⋅ ( v × w ) =ε ijk ui v j wk , a =ε ijk ui v j wk A = u ⊗ v = ui v j ei ⊗ e j ,

Aij = ui v j

v Au = = Aij u j ei , = vi Aij u j C= AB = Aik Bkj ei ⊗ e j , Cij = Aik Bkj = a A= a Aij Bij : B Aij Bij ,= B= :A =  ijkl Akl ei ⊗ e j , Bij =  ijkl Akl

 = A ⊗ B = Aij Bkl ei ⊗ e j ⊗ e k ⊗ el ,  ijkl = Aij Bkl September 9, 2016


100

Introduction to Vectors and Tensors > Differential Operators

Differential Operators Nabla The nabla vector differential operator, denoted as ∇ , is defined as,

∂ ∇ = ei ∂xi

Laplacian

The laplacian scalar differential operator, denoted as ∆ , is defined as,

∂2 ∆= 2 ∂xi September 9, 2016


101


Differential Operators Hessian The hessian symmetric second-order tensor differential operator, denoted as ∇ ⊗ ∇, is defined as,

∂2 = ∇ ⊗∇ ei ⊗ e j ∂xi ∂x j

September 9, 2016


102


Differential Operators Divergence, Curl and Gradient The divergence differential operator div (·) is defined as,

∂ () div (  ) = ∇ ⋅ (  ) = ⋅ ei ∂xi The curl differential operator curl (·) is defined as,

∂ () curl (  ) = ∇ × (  ) = ei × ∂xi The gradient differential operator grad (·) is defined as,

∂ () ∂ () grad (  ) = ∇ ⊗ (  ) = ⊗ ei = ∇ (  ) = ei ∂xi ∂xi September 9, 2016


103


Differential Operators Laplacian The laplacian scalar differential operator can be expressed as the div (grad (·)) operator,

∂2 ∆ (  ) = ∇ ⋅∇ (  ) = 2 (  ) = div grad (  ) ∂xi

Hessian The hessian symmetric second order-tensor differential operator can be expressed as the grad (grad (·)) operator,

∂2 () ∇= ⊗ ∇ () = ei ⊗ e j grad grad (  ) ∂xi ∂x j September 9, 2016


104


Differential Operators Gradient of a Scalar Field The gradient differential operator grad (·) is defined as,

∂ () ∂ () grad (  ) = ∇ ⊗ (  ) = ⊗ ei = ∇ (  ) = ei ∂xi ∂xi The gradient of a scalar field is a vector field defined as,

∂φ grad φ =∇φ = ei =φ,i ei ∂xi

September 9, 2016


105


Differential Operators Laplacian of a Scalar Field

The laplacian differential operator ∆ (  ) is defined as,

∂2 () ∆ (  ) = div grad (  ) = ∇ ⋅∇ (  ) = ∂xi2 The laplacian of a scalar field is a scalar field defined as,

∂ 2φ ∆φ = div grad φ = ∇ ⋅∇φ = 2 = φ,ii ∂xi

September 9, 2016


106


Differential Operators Hessian of a Scalar Field

The hessian differential operator ∇ ⊗ ∇ (  ) is defined as,

∂2 () ∇= ⊗ ∇ () ei ⊗ e j ∂xi ∂x j The hessian of a scalar field is a symmetric second-order tensor field defined as,

∂ 2φ ∇ ⊗∇ = φ ei ⊗= e j φ,ij ei ⊗ e j ∂xi ∂x j

September 9, 2016


107


Differential Operators Gradient of a Vector Field The gradient differential operator grad (·) is defined as,

∂ () ∂ () grad (  ) = ∇ ⊗ (  ) = ⊗ ei = ∇ (  ) = ei ∂xi ∂xi The gradient of a vector field is a second-order tensor field defined as,

∂ui ∂u ∇ ⊗ u = ⊗ e j = ei ⊗ e j = grad u = ui , j ei ⊗ e j ∂x j ∂x j

September 9, 2016


108


Differential Operators Curl of a Vector Field The curl differential operator curl (·) is defined as,

∂ () curl (  ) = ∇ × (  ) = ei × ∂xi The curl of a vector field is a vector field defined as,

∂uk ∂uk ∂u curl u= ∇ × u= e j × = e j × e k= ε ijk e= ε ijk uk , j ei i ∂x j ∂x j ∂x j If the curl of a vector field is zero, the vector field is said to be curl-free and there is a scalar field such that,

curl u = 0 ⇒ ∃φ | u = grad φ September 9, 2016


109


Differential Operators Curl of the Gradient of a Scalar Field The curl differential operator curl (·) and gradient differential operator grad (·) are defined, respectively, as,

∂ () ∂ () ⊗ ei curl (  ) = ∇ × (  ) = ei × , grad (  ) = ∇ ⊗ (  ) = ∂xi ∂xi The gradient of a scalar field is a vector field defined as,

∂φ grad φ =∇φ = ei =φ,i ei ∂xi The curl of the gradient of a vector field is a null vector,

curl grad φ = ∇ × ∇φ = ε ijkφ,kj ei = 0 September 9, 2016


110


Differential Operators Divergence of a Vector Field The divergence differential operator div (·) is defined as,

∂ () div (  ) = ∇ ⋅ (  ) = ⋅ ei ∂xi The divergence of a vector field is a scalar field defined as,

∂ui ∂ui ∂ui ∂u div u = ∇ ⋅ u = ⋅ e j = ei ⋅ e j = δ ij = = ui ,i ∂x j ∂x j ∂x j ∂xi If the divergence of a vector field is zero, the vector field is said to be solenoidal or div-free and there is a vector field such that,

div u = 0 ⇒ ∃ v | u = curl v September 9, 2016


111


Differential Operators Divergence of the Curl of a Vector Field The divergence differential operator div (·) and curl differential operator curl (·) are defined, respectively, as,

∂ () ∂ () div (  ) = ∇ ⋅ (  ) = ⋅ ei , curl (  ) = ∇ × (  ) = ei × ∂xi ∂xi The curl of a vector field is a vector field defined as,

∂uk ∂uk ∂u e k= ε ijk e= curl u= ∇ × u= e j × = e j × ε ijk uk , j ei i ∂x j ∂x j ∂x j The divergence of a curl of a vector field is a null scalar,

div curl u = ∇ ⋅ ( ∇ × u ) = ε ijk uk , ji = 0 September 9, 2016


112


Differential Operators Laplacian of a Vector Field

The laplacian differential operator ∆ (  ) is defined as,

∂2 () ∆ (  ) = ∇ ⋅∇ (  ) = ∂xi2 The laplacian of a vector field is a vector field defined as,

∂ 2ui ∆u = ∇ ⋅∇u = 2 ei = ui , jj ei ∂x j

September 9, 2016


113


Differential Operators Divergence of a Second Order Tensor Field The divergence differential operator div (·) is defined as,

∂ () div (  ) = ∇ ⋅ (  ) = ⋅ ei ∂xi

The divergence of a second-order tensor field is a vector field defined as,

div A = ∇ ⋅ A

∂Aik ∂Aik ∂A ei ⊗ e k ⋅ e j = = ⋅e j = δ kj ei ∂x j ∂x j ∂x j ∂Aij = = ei Aij , j ei ∂x j September 9, 2016


114


Differential Operators Curl of a Second-order Tensor Field The curl differential operator curl (·) is defined as,

∂ () curl (  ) = ∇ × (  ) = ei × ∂xi

The curl of a second-order tensor field is a second-order tensor field defined as,

curl A = ∇ × A ∂Akj ∂A =el × =el × ek ⊗ e j ∂xl ∂xl = ε lki September 9, 2016

∂Akj ∂xl

ei ⊗ e j ε lki Akj ,l ei ⊗ e j =


115


Differential Operators Gradient of a Second-order Tensor Field The gradient differential operator grad (·) is defined as,

∂ () ∂ () grad (  ) = ∇ ⊗ (  ) = ⊗ ei = ∇ (  ) = ei ∂xi ∂xi

The gradient of a second-order tensor field is a third-order tensor field defined as,

grad A = ∇ ⊗ A ∂A = ⊗ ek ∂xk =

September 9, 2016

∂Aij ∂xk

ei ⊗ e j ⊗ e= Aij ,k ei ⊗ e j ⊗ e k k Carlos Agelet de Saracibar

116


Differential Operators Differential Operators

grad φ =∇φ =φ,i ei , ∆φ =∇ ⋅∇φ =φ,ii , ∇ ⊗ ∇φ =φ,ij ei ⊗ e j div u = ∇ ⋅ u = ui ,i grad u = ∇ ⊗ u = ui , j ei ⊗ e j , curl u = ∇ × u = ε ijk uk , j ei ∆u = ∇ ⋅∇u = ui , jj ei div A = ∇ ⋅ A = Aij , j ei grad A = ∇ ⊗ A = Aij ,k ei ⊗ e j ⊗ e k curl A = ∇ × A = ε lki Akj ,l ei ⊗ e j September 9, 2016


117


Assignments Assignment 1.3

Establish the following identities involving a smooth scalar field φ and a smooth vector field v,

September 9, 2016

(1)

div (φ= v ) φ div v + v ⋅ grad φ

(2)

grad (φ v ) = v ⊗ grad φ + φ grad v


118


Assignments Assignment 1.4 [Classwork] Establish the following identities involving the smooth scalar fields φ and ψ , smooth vector fields u and v , and a smooth second order tensor field A,

(1) (2)

T div ( A= v)

(3)

div ( u × v ) = v ⋅ curl u − u ⋅ curl v

(4)

div ( u= ⊗ v)

(5) September 9, 2016

div= (φ A ) φ div A + A grad φ

( div A ) ⋅ v + A : grad v

( grad u ) v + u div v grad = (φψ ) ( grad φ )ψ + φ ( gradψ ) Carlos Agelet de Saracibar

120


Assignments Assignment 1.5 [Homework] Establish the following identities involving the smooth scalar field φ , and smooth vector fields u and v , T T grad u v + grad v)u ( ) (

(1)

grad= (u ⋅ v )

(2)

curl (φ= v ) grad φ × v + φ curl v

(3) curl ( u × = v ) u div v − v div u + ( grad u ) v − ( grad v ) u (4) ∆v grad ( div v ) − curl ( curl v ) = (5) ∆ ( u ⋅ v ) =

September 9, 2016

( ∆u ) ⋅ v + 2 grad u : grad v + u ⋅ ∆v Carlos Agelet de Saracibar

123


Assignments Assignment 1.6 [Classwork] Given the vector v =v ( x ) =x1 x2 x3e1 + x1 x2e 2 + x1e3 determine div v, curl v, grad v, ∆v.

September 9, 2016


127

Introduction to Vectors and Tensors > Integral Theorems

Integral Theorems Divergence or Gauss Theorem Given a vector field u in a volume V with closed boundary surface ∂V and outward unit normal to the boundary n, the divergence (or Gauss) theorem reads,

∫

V

div u dV=

∫ ∇ ⋅ u dV= ∫ V

∂V

u ⋅ n dS

Given a second-order tensor field A in a volume V with closed boundary surface ∂V and outward unit normal to the boundary n the divergence (or Gauss) theorem reads,

∫

V

September 9, 2016

div A dV=

∫ ∇ ⋅ A dV= ∫ V


∂V

An dS

130

Introduction to Vectors and Tensors > Integral Theorems

Integral Theorems Curl or Stokes Theorem Given a vector field u in a surface S with closed boundary ∂S and outward unit normal to the surface n , the curl (or Stokes) theorem reads,

∫ ( curl u ) ⋅ n dS =∫ ( ∇ × u ) ⋅ n dS =∫ S

S

∂S

u ⋅ dr

where the curve of the line integral must have positive orientation, such that dr points counter-clockwise when the unit normal points to the viewer, following the right-hand rule.

September 9, 2016


131

Tensor Algebra

Tensor Algebra

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