agelet.rmee.upc.eduagelet.rmee.upc.edu/master/chapter 1. tensor algebra v1.7.pdf · tensors ....

Continuum Mechanics

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

Chapter 1 Introduction to Vectors and Tensors

Contents 1. Introduction 2. Algebra of vectors 3. Algebra of tensors 4. Higher order tensors 5. Differential operators 6. Integral theorems

Introduction to Vectors and Tensors > Contents

Contents

September 9, 2016 Carlos Agelet de Saracibar 2

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.

Introduction to Vectors and Tensors > Introduction

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.


Introduction


Notation on Printed Documents Zero-order tensors: First-order tensors: Second-order tensors: Third-order tensors: Fourth-order tensors:

Notation on Hand-written Documents Zero-order tensors: First-order tensors: Second-order tensors:


Introduction


, ,a Aα, ,a Aα

, ,a Aα

, ,a Aα, ,a Aα

, ,a Aα

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 . A scalar may be viewed as a zero-order tensor.

Introduction to Vectors and Tensors > Algebra of Vectors

Algebra of Vectors


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. A vector may be viewed as a first-order tensor.

, ,a Aα

, , .a Aα

Sum of Vectors The sum of vectors yields a new vector, based on the parallelo-gram law of addition. The sum of vectors has the following properties, where denotes the unique zero vector with unspecified direction and zero length.


Algebra of Vectors


( ) ( )

( )

+ = +

+ + = + +

+ =

+ − =

u v v uu v w u v w

u 0 uu u 0

0

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 multiplication has the following properties,


Algebra of Vectors


( ) ( )( )( )

αβ α β

α β α β

α α α

=

+ = +

+ = +

u u

u u u

u v u v

u α αuu 0α >

0α <u

Dot Product The dot (or scalar or inner) product of two vectors and , denoted by , is a scalar given by, where is the norm (or length or magnitude) of the vector , which is a non-negative real number defined as, and is the angle between the non-zero vectors and when their origins coincide.


Algebra of Vectors


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

u

u v⋅u v

u

( )1 2 0= ⋅ ≥u u u

( ),θ u v u v

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


Algebra of Vectors


( ) ( ) ( )0

000, ,

α β α β

⋅ = ⋅⋅ =

⋅ + = ⋅ + ⋅

⋅ > ⇔ ≠⋅ = ⇔ =⋅ = ≠ ≠ ⇔ ⊥

u v v uu 0u v w u v u wu u u 0u u u 0u v u 0 v 0 u v

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


Algebra of Vectors


e

1=e

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

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


Algebra of Vectors


u

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

v

uv

( )0, 1, 1 , 2θ π⋅ = = = ⇔ = ⇔ ⊥u v u v u v u v

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


Algebra of Vectors


ue

( ) ( )cos , , 0 ,θ θ π⋅ = ≤ ≤u e u u e u e

u

e

( )cos ,θu u e

( ),θ u e

Cartesian Basis Let us consider a three-dimensional Euclidean space with 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 orthogonal form a so called orthonormal basis vectors, collectively denoted as , and define an orthonormal system.


Algebra of Vectors


1 2 3, ,e e e

1 2 1 3 2 3

1 2 3

01

⋅ = ⋅ = ⋅ =

= = =

e e e e e ee e e

{ }ie

1e

2e

3e

1,x

2 ,x

3,x

O

Cartesian Components Any vector in the three-dimensional Euclidean space is represented uniquely by 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 vector can be collected into a vector of components given by,


Algebra of Vectors


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 eu

[ ] [ ]1 2 3Tu u u=u

[ ] 3∈u

Index Notation Using index notation the vector can be written as, Leaving out the summation symbol, using the so called Einstein notation, the vector can be written in short form as, where we have adopted the summation convention introduced by Einstein.


Algebra of Vectors


u

i iu=u e

3

1 i iiu

==∑u e

u

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.


Algebra of Vectors


( )( )( )

( )( )

1 1 2 2 3 3

1 1 2 2 3 3 1 1 2 2 3 3

1 1 2 2 3 3 1 1 2 2 3 3

i i j j i

i i j j

i i j j k k i

v a b c a b c b c b c

v a b c d a b a b a b c d c d c d

v a b c c d a b c b c b c c d c d c d

= = + +

= = + + + +

= = + + + +

Kronecker Delta The Kronecker delta is defined as,


Algebra of Vectors


ijδ

1 if0 ifij

i ji j

δ=

= ≠

The Kronecker delta satisfies the following properties, Note that the Kronecker delta plays the role of a replacement operator.

3, ,ii ij jk ik ij j iu uδ δ δ δ δ= = =

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


Algebra of Vectors


ie

i j ijδ⋅ =e e

je i j⋅e e

Components of a Vector Taking the basis , the projection of a vector onto the basis vector yields the i-th component of ,


Algebra of Vectors


uie

( )i j j i j j i j ji iu u u uδ⋅ = ⋅ = ⋅ = =u e e e e e

u{ }ie

Dot Product Taking the basis , the dot (or scalar or inner) product of two vectors and , denoted as , is a scalar quantity and using index notation can be written as,


Algebra of Vectors


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

δ⋅ = ⋅ = ⋅ = =

= + +

u v e e e e

v ⋅u v

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


Algebra of Vectors


{ }ie

( ) ( ) ( ) ( )

( )

1 2 1 21 2 1 2

1 22 2 21 2 3

i i j j i j ij i iu u u u u u

u u u

δ= ⋅ = ⋅ = =

= + +

u u u e e

u u

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


Assignment 1.1


( )( )

( )( )( )( )

v

= ⋅

= ⋅

= ⋅ ⋅

= ⋅ ⋅

v a b c

v a b c

a b c d

v a b c c d

Cross Product The cross (or vector) product of 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, .


Algebra of Vectors


u v×u v

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

e0, 0⋅ = ⋅ =u e v e

Norm of the Cross Product of two Vectors The norm (or magnitude or 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,


Algebra of Vectors


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

u v ×u v

×u v

×u v

u

v

Permutation Symbol The permutation (or alternating or Levi-Civita) symbol is defined as,


Algebra of Vectors


ijkε

1 for even permutations of ( , , ), i.e. 123, 231, 3121 for odd permutations of ( , , ), i.e. 213,132, 321 0 if there is repeated index

ijk

i j ki j kε

+= −

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


Algebra of Vectors


ijkε

2 2 6, ,ijp klp ik jl il jk ikl jkl ij ijk ijk iiε ε δ δ δ δ ε ε δ ε ε δ= − = = =

1 2 3

1 2 3

1 2 3

det , deti i i ip iq ir

ijk j j j ijk pqr jp jq jr

k k k kp kq kr

δ δ δ δ δ δε δ δ δ ε ε δ δ δ

δ δ δ δ δ δ

= =

( )( )( )2ijk

i j j k k iε

− − −=

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


Algebra of Vectors


jeie

i j ijk kε× =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 ee e e e e e e e e

e e e e e e

Taking the right-handed orthonormal basis , the cross (or vector) product of two basis vectors and , denoted as can defined as,

{ }ie,i j×e e

1e

2e

3e

1,x

2 ,x

3,x

O

Cross Product Taking the basis , the cross (or vector) product of two vectors and , denoted as , is another vector, perpendicular to the plane defined by the two vectors, and can be written in index form as,


Algebra of Vectors


( ) ( ) ( )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 ku v u v u v u v

u v u v u v u v u v u v

u u uv v v

ε ε× = × = × = =

= − + − + −

=

u v e e e e e e

e e e

e e e

{ }ieu v ×u v

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


Algebra of Vectors


( )

( ) ( ) ( )( )

, , ||α α α

× = − ×

× = ≠ ≠ ⇔

× = × = ×

× + = × + ×

u v v uu v 0 u 0 v 0 u v

u v u v u v

u v w u v u w

Assignment 1.2 Using the expression, obtain the following relationships between the permutation symbol and the Kronecker delta,


Assignment 1.2


2 2 6, ,ijp klp ik jl il jk ipq jpq ij ijk ijk iiε ε δ δ δ δ ε ε δ ε ε δ= − = = =

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

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


Algebra of Vectors


( ) ( )ijk i j k i j kε = ⋅ × = × ⋅e e e e e e

, ,i j ke e e ( )i j k⋅ ×e e e

Box Product Taking the orthonormal basis , the box (or triple scalar) product of three vectors , denoted as , is a scalar quantity and using index notation can be written as,


Algebra of Vectors


( ) ( ) ( )

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 w

u v w

u u uv v vw w w

ε

⋅ × = ⋅ × = ⋅ ×

=

=

u v w e e e e e e

{ }ie, ,u v w ( )⋅ ×u v w

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


Algebra of Vectors


( )V = ⋅ ×u v w

, ,u v w

( )V = ⋅ ×u v w

v

w

u

×v w

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


Algebra of Vectors


( ) ( ) ( )( ) ( )( ) ( ) 0

⋅ × = ⋅ × = ⋅ ×

⋅ × = − ⋅ ×

⋅ × = ⋅ × =

u v w w u v v w u

u v w u w v

u u v v u v

Triple Vector Product Taking the orthonormal basis , the triple vector product of three vectors , denoted as , is a vector and using index notation can be written as,


Algebra of Vectors


( ) ( )

( )

( ) ( )

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

ε ε

ε ε

δ δ δ δ

× × =

=

= −

= −

= ⋅ − ⋅

u v w e

e

e

e eu w v u v w

{ }ie, ,u v w ( )× ×u v w

Triple Vector Product Taking the orthonormal basis , the triple vector product of three vectors , denoted as , is a vector and using index notation can be written as,


Algebra of Vectors


( ) ( )

( )

( ) ( )

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

ε ε

ε ε

δ δ δ δ

× × =

=

= −

= −

= ⋅ − ⋅

u v w e

e

e

e e

u w v v w u

{ }ie, ,u v w ( )× ×u v w

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


Algebra of Vectors


( ) ( ) ( )( ) ( ) ( )

( ) ( )

× × = ⋅ − ⋅

× × = ⋅ − ⋅

× × ≠ × ×

u v w u w v u v w

u v w u w v v w u

u v w u v w

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 transformation that assigns a vector to a vector , The second-order unit tensor, denoted as , satisfies the following identity,

Introduction to Vectors and Tensors > Algebra of Tensors

Algebra of Tensors


Au v

=v Au

v u

1

=u 1u

Second-order Tensors As linear operators defining linear transformations, second-order tensors have the following properties,


Algebra of Tensors


( )( ) ( )

( )

α α

α α

+ = +

=

± = ±

A u v Au Av

A u Au

A B u Au Bu

Tensor Product The tensor product (or dyad) of two vectors and , denoted as , is a second-order tensor which linearly transforms a vector into a vector with the direction of following the rule, A dyadic is a linear combination of dyads.


Algebra of Tensors


⊗u vu v

u

( ) ( ) ( )⊗ = ⋅ = ⋅u v w u v w v w u

w

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

Generally, the tensor product (or dyad) is not commutative, i.e.


Algebra of Tensors


( )( ) ( ) ( )( )

( ) ( ) ( )( )( ) ( ) ( )

( ) ( )

α α

α β α β

⊗ + = ⊗ + ⊗

+ ⊗ = ⊗ + ⊗

⊗ = ⋅ = ⋅

⊗ ⊗ = ⋅ ⊗ = ⊗ ⋅

⊗ = ⊗

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 uu v w x w x u v

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 operator such that,


Algebra of Tensors


i j⊗e eie

je

( ) ( )i j k j k i jk iδ⊗ = ⋅ =e e e e e e e

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 the uniquely 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 tensor may be written as,


Algebra of Tensors


A

ij i jA= ⊗A e e

ijAA

{ }ie

i j⊗e e{ }ie

A

ij i j i iδ= ⊗ = ⊗1 e e e e

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


Algebra of Tensors


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

δ

δ

⋅ = ⋅ ⊗ = ⋅

= ⋅ = ⋅ = =

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 ijδ⋅ = ⋅ =e 1e e e

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


A[ ]A

[ ]11 12 13

21 22 23

31 32 33

A A AA A AA A A

=

A

ij i jA= ⊗A e e

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


[ ]1 0 00 1 00 0 1

=

1

1[ ]1


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


i j i ju v⊗ = ⊗u v e e

i ju v

{ }ie

i j⊗e e{ }ie

⊗u v

u v⊗u v

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


[ ]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 vu v u v u v

⊗ =

u v

⊗u v[ ]⊗u v

i j i ju v⊗ = ⊗u v e e

Second-order Tensors Using index notation, the linear transformation can be written as, yielding


Algebra of Tensors


=v Au

( )( ) ( )i i ij i j k k ij k i j k ij k jk i

i i ij j i

v A u A u A u

v A u

δ

=

= ⊗ = ⊗ =

=

v Au

e e e e e e e e

e e

[ ] [ ][ ], ,i ij jv A u= = =v Au v A u

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


Algebra of Tensors


A

0⋅ > ∀ ≠u Au u 0

Positive Definite Second-order Tensor A second-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


Algebra of Tensors


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

A

,,

T

Ti 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 components of the transpose of a second-order tensor satisfy, Using index notation the transpose of may be written as,

( ):T T Tij i j j i jiij

A A= = ⋅ = ⋅ =A e A e e Ae

A

AT

ji i jA= ⊗A e e


Algebra of Tensors


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

TT

A A AA A AA A A

= =

A A

A

ij i j

Tji i j

A

A

= ⊗

= ⊗

A e e

A e e


Algebra of Tensors


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

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

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

A

T=A A

A

,Tij i j ji i j ij jiA A A A= ⊗ = ⊗ = =A e e e e A

A

[ ] [ ]TT = = A A A


Algebra of Tensors


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

A B

( ) ( )= ∀AB u A Bu u

AB


Algebra of Tensors


Dot Product The components of the dot product along an orthonormal basis read,

( ) ( ) ( )( )

ij i j i jij

i kj k i k kj

ik kj

C

B B

A B

= = ⋅ = ⋅

= ⋅ = ⋅

=

AB e AB e e A Be

e A e e Ae

AB{ }ie

, ij ik kjC A B= =C AB


Algebra of Tensors


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

( ) ( )2

= =

=≠

AB C A BC ABC

A AAAB BA


Algebra of Tensors


Trace The trace of a dyad , denoted as , is a scalar quantity given by,

⊗u v

( )tr i iu v⊗ = ⋅ =u v u v

( )tr ⊗u v

The trace of a second-order tensor , denoted as , is a scalar quantity 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

AA A A

δ

= ⊗ = ⊗

= ⋅ =

=

= + + =

A e e e e

e e

A

A tr A


Algebra of Tensors


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

( ) ( )( )( ) ( )

tr trtr tr

tr tr tr

tr tr

T

α α

=

=

+ = +

=

A AAB BA

A B A B

A A


Algebra of Tensors


Double Dot Product The double dot product of two second-order tensors and , denoted as , is a scalar quantity 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


Algebra of Tensors


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

( ) ( ) ( )( ) ( )

( ) ( ) ( )( )( ) ( ) ( )( )

: : tr: :

: : :

: :

:

:

T T

i j k l i k j l ik jlδ δ

= ==

= =

⊗ = ⋅ = ⊗

⊗ ⊗ = ⋅ ⋅

⊗ ⊗ = ⋅ ⋅ =

A 1 1 A AA 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


Algebra of Tensors


Euclidean Norm The euclidean 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


Algebra of Tensors


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

16ijk i j k ijk pqr pi qj rk

A A AA A AA A A

A A A A A Aε ε ε

=

=

= =

A A

det A


Algebra of Tensors


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

( )( )

3

det detdet det

det det det

T

α α

=

=

=

A AA A

AB A B


Algebra of Tensors


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


Algebra of Tensors


Inverse of a Second-order Tensor For a non-singular second-order tensor , i.e. , there exist a unique non-singular inverse second-order tensor, denoted as , satisfying, yielding,

A

( ) ( )1 1− −= = = ∀AA u A A u 1u u u

det 0≠A

1−A

1 1 1 1, ik kj ik kj ijA A A A δ− − − −= = = =AA A A 1


Algebra of Tensors


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

α α

−−

− − −

− − −

− − −

− − −

−−

=

= =

=

=

=

=

A A

A A A

A A

AB B A

A A A

A A


Algebra of Tensors


Orthogonal Second Order Tensor An orthogonal second-order tensor is a linear operator that preserves the norms and the angles between 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


Algebra of Tensors


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 between 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


Algebra of Tensors


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

A

, ij ij ijA S W= + = +A S W

( ) ( )

( ) ( )

1 1

2 21 1

2 2

,

,

T Tij ij ji ji

T Tij ij ji ji

S A A S

W A A W

= + = = + =

= − = − = − = −

S A A S

W A A W

SW


Algebra of Tensors


Symmetric/Skew-symmetric Second-order Tensors The matrices of components of a symmetric second-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 WS S S W WS S S W W

= = − − −

S W

S


Algebra of Tensors


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

S

12 12 13 13 23 23 12 12 13 13 23 23

:

0

ij ijS WS W S W S W S W S W S W

=

= + + − − −=

S W


Algebra of Tensors


Skew-symmetric Second-order Tensor A skew-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,

ω

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

ε ω ε ω

ω ε ω ε

= ⊗ = − ⊗ = −

= = − = −

W e e e e

e eω

2

= × ∀

=

Wu u u

W

ω

ω


Algebra of Tensors


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

[ ]

[ ] [ ] [ ]

12 13 3 2

12 23 3 1

13 23 2 1

1 2 3 23 13 12

0 00 0

0 0

, , , ,T T

W WW WW W

W W W

ω ωω ωω ω

ω ω ω

− = − = − − − −

= = − −

W

ω

( )2 2 2 2 2 212 13 23 1 3 32 2 2W W W ω ω ω= + + = + + =W ω


Algebra of Tensors


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

and has the following properties,

A

[ ]0 0

0 00 0

αα

α

=

A

, ij ijAα αδ= =A 1

tr tr 3 , 3ii iiAα α αδ α= = = =A 1


Algebra of Tensors


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

A

tr 0, 0iiA= =A


Algebra of Tensors


Spherical/Deviatoric Additive Split A second-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 31 1

3 3

dev , dev

tr ,

dev tr , dev

esf esfij ij ij

esf esfij kk ij

ij kk ijij

A A A

A A

A A A

δ

δ

= + = +

= =

= − = −

A A A

A A 1

A A A 1

esfAdev A

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),


Algebra of Tensors


Ain

( ),i i i i iλ λ= − =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 λ λ= − =A 1

iλ−A 1

iλ ∈

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 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.


Algebra of Tensors


Aiλ ∈ in

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


Algebra of Tensors


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,

( ) ( ) 3 21 2 3det 0p I I Iλ λ λ λ λ= − = − + − + =A 1

A

( )

( ) ( )( )

1 1 2 3

2 22 1 2 2 3 3 1

3 1 2 3

1

2

tr

tr tr

det

I

I

I

λ λ λ

λ λ λ λ λ λ

λ λ λ

= = + +

= − = + +

= =

A A

A A A

A A

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),


Algebra of Tensors


dev A iλ′∈ i′n

( )dev , devi i i i iλ λ′ ′ ′ ′ ′= − =An n A 1 n 0In 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 λ λ′ ′= − =A 1

dev iλ′−A 1

Characteristic Polynomial and Principal Invariants The characteristic polynomial of the deviatoric part of a second-order tensor takes the form,


Algebra of Tensors


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,

( ) ( ) 3 21 2 3det dev 0p I I Iλ λ λ λ λ′ ′ ′ ′ ′ ′ ′ ′= − = − + − + =A 1

( )

( ) ( ) ( )

( )

1

22 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

λ λ λ λ λ λ

λ λ λ

′ = =

′ ′ ′ ′ ′ ′ ′= − = − + +

′ ′ ′= =

A A

A A

A A

dev A

Eigenvalues/Eigenvectors The eigenvalues/eigenvectors problem for a symmetric second-order tensor and its the deviatoric part satisfy the following relations (Einstein notation does not applies here),


Algebra of Tensors


dev A

( )dev , devi i i i iλ λ′ ′ ′ ′ ′= − =An n A 1 n 0

( ) ( ) ( )1

3dev tri i i i i iλ λ λ ′ ′ ′ ′− = − + = − =

A 1 n A A 1 n A 1 n 0

( ),i i i i iλ λ= − =An n A 1 n 0

A

Then,

( )1

3tr ,i i i iλ λ′ ′= + =A n n

yielding,

For a second-order unit tensor, all three eigenvalues are equal to one, , 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 takes the form,


Algebra of Tensors



1 2 3 1λ λ λ= = =

1,3 i i i ii== ⊗ = ⊗∑1 n n n n

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


Algebra of Tensors


A

1,3 i i iiλ

== ⊗∑A n n

If there are two equal eigenvalues , the spectral decomposition takes the form,

1 2λ λ λ= =

( )3 3 3 3 3λ λ= − ⊗ + ⊗A 1 n n n n

1 2 3λ λ λ λ= = =

1,3 i i i iiλ λ λ

== = ⊗ = ⊗∑A 1 n n n n

If the three eigenvalues are equal , the spectral decomposition takes the form,

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 , such that,

A third-order tensor has components.

Introduction to Vectors and Tensors > Higher-order Tensors

Third-order Tensors


3 3 3

1 1 1 ijk i j k ijk i j ki j k= = == ⊗ ⊗ = ⊗ ⊗∑ ∑ ∑ e e e e e e

i j k⊗ ⊗e e e

33 27=

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 , such that,

A fourth-order tensor has components.


Fourth-order Tensors


3 3 3 3

1 1 1 1 ijkl i j k l ijkl i j k li j k l= = = == ⊗ ⊗ ⊗ = ⊗ ⊗ ⊗∑ ∑ ∑ ∑ e e e e e e e e

i j k l⊗ ⊗ ⊗e e e e

43 81=

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




: ,ijkl kl i j ij i j ij ijkl klA B B A= = ⊗ = ⊗ =B A e e e e

A B

Transpose of a Fourth-order Tensor The transpose of a fourth-order tensor is uniquely defined as a fourth-order tensor such that for arbitrary second-order tensors and the following relationship holds,




BAT

: : : : , ,T T Tij ijkl kl kl klij ij ijkl klijA B B A= = =A B B A

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

( )TT =

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




( ),ij kl i j k l ijkl i j k l

ijkl ij klijkl

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

( )

( ) ( ),

TTij kl i j k l

Tijkl i j k l

TTijkl ij klijkl ijkl

B A

B A

= ⊗ = ⊗ = ⊗ ⊗ ⊗

= ⊗ ⊗ ⊗

= ⊗ = ⊗ =

A B B A e e e e

e e e e

A B B A

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




( ) ( )1 1ˆ2 2

ik jl i j k l

il jk i j k l

ik jl il jk i j k l

δ δ

δ δ

δ δ δ δ

= ⊗ ⊗ ⊗

= ⊗ ⊗ ⊗

= + = + ⊗ ⊗ ⊗

e e e e

e e e e

e e e e

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




: ,

:ik jl kl i j ij i j

ij ik jl k l kl k l

A AA Aδ δ

δ δ

= ⊗ = ⊗ =

= ⊗ = ⊗ =

A e e e e AA e e e e A

: : : : : :: : : :

TT

= = = ⇒ ==

A B A B B A B AA B B A

The fourth-order identity tensor is symmetric, satisfying,





:

:

Til jk kl i j ji i j

Tij il jk kl k l lk k l

A A

A A A

δ δ

δ δ

= ⊗ = ⊗ =

= ⊗ = ⊗ =

A e e e e A

A e e e e A

: : : : : :: : : :

T TT

T

= = =⇒ =

=

A B A B B A B AA B B A






( ) ( )

( ) ( )

1 1ˆ : :2 21 1ˆ: :2 2

T

T

= + = +

= + = +

A A A A

A A A A

( ) ( )1 1ˆ ˆ: : : : : : ˆ ˆ2 2ˆ ˆ: : : :

T TT

T

= + = + = ⇒ ==

A B A B B B A A B A

A B B A


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




dev

( )1 1

3 3,dev dev ik jl ij klijkl

δ δ δ δ= − ⊗ = −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 Aδ δ δ δ δ

= = − ⊗ = −

= = − = −

A A 1 1 A A A 1

A


Algebra of Tensors


Algebra of Tensors

( )

,,

,

,

,

,

: ,

: ,

i i i i

ijk j k i i ijk j k

ijk i j k ijk i j k

i j i j ij i j

ij j i i ij j

ik kj i j ij ik kj

ij ij ij ij

ijkl kl i j ij ijkl k

a u v a u vu v w u v

a u v w a u v wu v A u vA u v A u

A B C A Ba A B a A B

A B A

ε ε

ε ε

= ⋅ = == × = =

= ⋅ × = =

= ⊗ = ⊗ =

= = =

= = ⊗ =

= = =

= = ⊗ =

u vw u v e

u v wA u v e e

v Au eC AB e e

A BB A e e

,l

ij kl i j k l ijkl ij klA B A B= ⊗ = ⊗ ⊗ ⊗ =A B e e e e

Nabla The nabla vector differential operator, denoted as , is defined as,

Introduction to Vectors and Tensors > Differential Operators

Differential Operators


∇

iix

∂∇ =

∂e

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

∆

2

2ix

∂∆ =

∂

Hessian The hessian symmetric second-order tensor differential operator, denoted as , is defined as,




2

i ji jx x∂

∇⊗∇ = ⊗∂ ∂

e e

∇⊗∇

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




( ) ( ) ( )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

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




( ) ( ) ( ) ( )2

2 divgradix

∂∆ = ∇⋅∇ = =

∂

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

( ) ( ) ( )2

grad gradi ji jx x

∂∇⊗∇ = ⊗ =

∂ ∂e e

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




The gradient of a scalar field is a vector field defined as,

( ) ( ) ( ) ( ) ( )grad i ii ix x

∂ ∂= ∇⊗ = ⊗ = ∇ =

∂ ∂e e

,grad i i iixφφ φ φ∂

= ∇ = =∂

e e

Laplacian of a Scalar Field The laplacian differential operator is defined as,




The laplacian of a scalar field is a scalar field defined as,

2

,2divgrad iiixφφ φ φ φ∂

∆ = = ∇⋅∇ = =∂

( )∆

( ) ( ) ( ) ( )2

2divgradix

∂∆ = = ∇⋅∇ =

∂

Hessian of a Scalar Field The hessian differential operator is defined as,




The hessian of a scalar field is a symmetric second-order tensor field defined as,

2

,i j ij i ji jx xφφ φ∂

∇⊗∇ = ⊗ = ⊗∂ ∂

e e e e

( )∇⊗∇

( ) ( )2

i ji jx x

∂∇⊗∇ = ⊗

∂ ∂e e

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




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

( ) ( ) ( ) ( ) ( )grad i ii ix x

∂ ∂= ∇⊗ = ⊗ = ∇ =

∂ ∂e e

,grad ij i j i j i j

j j

u ux x

∂∂= ∇⊗ = ⊗ = ⊗ = ⊗

∂ ∂uu u e e e e e

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




The curl of a vector field is a vector field defined as,

,curl k kj j k ijk i ijk k j i

j j j

u u ux x x

ε ε∂ ∂∂= ∇× = × = × = =

∂ ∂ ∂uu u e e e e e

( ) ( ) ( )curl iix

∂= ∇× = ×

∂e

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 0 | gradφ φ= ⇒ ∃ =u u

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




The gradient of a scalar field is a vector field defined as,

,grad i i iixφφ φ φ∂

= ∇ = =∂

e e

( ) ( ) ( ) ( ) ( ) ( )curl , gradi ii ix x

∂ ∂= ∇× = × = ∇⊗ = ⊗

∂ ∂e e

The curl of the gradient of a vector field is a null vector,

,curlgrad ijk kj iφ φ ε φ= ∇×∇ = =e 0

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




The divergence of a vector field is a scalar field defined as,

,div i i ij i j ij i i

j j j i

u u u ux x x x

δ∂ ∂ ∂∂= ∇ ⋅ = ⋅ = ⋅ = = =

∂ ∂ ∂ ∂uu u e e e

( ) ( ) ( )div iix

∂= ∇ ⋅ = ⋅

∂e

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 0 | curl= ⇒ ∃ =u v u v

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




The curl of a vector field is a vector field defined as,

,curl k kj j k ijk i ijk k j i

j j j

u u ux x x

ε ε∂ ∂∂= ∇× = × = × = =

∂ ∂ ∂uu u e e e e e

( ) ( ) ( ) ( ) ( ) ( )div , curli ii ix x

∂ ∂= ∇ ⋅ = ⋅ = ∇× = ×

∂ ∂e e

The divergence of a curl of a vector field is a null scalar,

( ) ,divcurl 0ijk k jiuε= ∇ ⋅ ∇× = =u u

Laplacian of a Vector Field The laplacian differential operator is defined as,




The laplacian of a vector field is a vector field defined as, 2

,2i

i i jj ij

u ux

∂∆ = ∇⋅∇ = =

∂u u e e

( )∆

( ) ( ) ( )2

2ix

∂∆ = ∇⋅∇ =

∂

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




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

,

div

ik ikj i k j kj i

j j j

iji ij j i

j

A Ax x xA

Ax

δ

= ∇ ⋅∂ ∂∂

= ⋅ = ⊗ ⋅ =∂ ∂ ∂

∂= =∂

A AA e e e e e

e e

( ) ( ) ( )div iix

∂= ∇ ⋅ = ⋅

∂e

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




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

,

curl

kjl l k j

l l

kjlki i j lki kj l i j

l

Ax xA

Ax

ε ε

= ∇×∂∂

= × = × ⊗∂ ∂∂

= ⊗ = ⊗∂

A AAe e e e

e e e e

( ) ( ) ( )curl iix

∂= ∇× = ×

∂e

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




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

( ) ( ) ( ) ( ) ( )grad i ii ix x

∂ ∂= ∇⊗ = ⊗ = ∇ =

∂ ∂e e

,

grad

kk

iji j k ij k i j k

k

xA

Ax

= ∇⊗∂

= ⊗∂∂

= ⊗ ⊗ = ⊗ ⊗∂

A AA e

e e e e e e





, , ,

,

,

,

,

,

,

,

grad , ,

divgrad ,

curl

div

grad

curl

i i ii ij i j

i i

i j i j

ijk k j i

i jj i

ij j i

ij k i j k

lki kj l i j

uu

uuA

AA

φ φ φ φ φ φ φ φ

ε

ε

= ∇ = ∆ = ∇⋅∇ = ∇⊗∇ = ⊗

= ∇⋅ =

= ∇⊗ = ⊗

= ∇× =

∆ = ∇⋅∇ =

= ∇⋅ =

= ∇⊗ = ⊗ ⊗

=∇× = ⊗

e e eu u

u u e eu u eu u e

A A eA A e e eA A e e

Assignment 1.3 Establish the following identities involving a smooth scalar field and a smooth vector field ,


Assignments


( )( )

(1)

(2)

div div grad

grad grad grad

φ φ φ

φ φ φ

= + ⋅

= ⊗ +

v v v

v v v

φv

Assignment 1.4 [Classwork] Establish the following identities involving the smooth scalar fields and , smooth vector fields and , and a smooth second order tensor field ,


Assignments


φ u

( )( ) ( )( )( ) ( )( ) ( ) ( )

(1)

(2)

(3)

(4)

(5)

div div grad

div div : grad

div curl curl

div grad div

grad grad grad

T

φ φ φ

φψ φ ψ φ ψ

= +

= ⋅ +

× = ⋅ − ⋅

⊗ = +

= +

A A A

A v A v A v

u v v u u v

u v u v u v

ψ vA

Assignment 1.5 [Homework] Establish the following identities involving the smooth scalar field , and smooth vector fields and ,


Assignments


φ 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

φ φ φ

⋅ = +

= × +

× = − + −

∆ = −

∆ ⋅ = ∆ ⋅ + + ⋅∆

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

Assignment 1.6 [Classwork] Given the vector determine


Assignments


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

Introduction to Vectors and Tensors > Integral Theorems

Integral Theorems


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

uV∂ n

divV V V

dV dV dS∂

= ∇ ⋅ = ⋅∫ ∫ ∫u u u n

V∂ nA

divV V V

dV dV dS∂

= ∇ ⋅ =∫ ∫ ∫A A An

Introduction to Vectors and Tensors > Integral Theorems

Integral Theorems


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

u S∂n

( ) ( )curlS S S

dS dS d∂

⋅ = ∇× ⋅ = ⋅∫ ∫ ∫u n u n u r