Tensor Calculus Main references

Tensor Calculus

Tommaso Astarita

[email protected]

2 Tensor calculus T Astarita

Main references

Aris, Vectors, tensors, and the basic equations of fluid mechanics, 1989.

Borisenko and Tarapov, Vector and Tensor Analysis with Applications, 1979.

Fleisch, Student Guide to Vectors and Tensors, 2012.


Introduction - Plan

History.

Algebra of Cartesian vectors and tensors.

Calculus of Cartesian vectors and tensors.

General treatment of tensor algebra and calculus.

Conclusions.


History (http://en.wikipedia.org/wiki/Tensor#History)

The concepts of later tensor analysis arose from the work of Carl Friedrich Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic forms and invariants developed during the middle of the nineteenth century. The word "tensor" itself was introduced in 1846 by William Rowan Hamilton to describe something different from what is now meant by a tensor. The contemporary usage was introduced by Woldemar Voigt in 1898.

Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892. It was made accessible to many mathematicians by the publication of Ricci and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications).


Gregorio Ricci-Curbastro

Gregorio Ricci-Curbastro was an Italian mathematician born in Lugo di Romagna. He is most famous as the inventor of tensor calculus, but also published important works in other fields.

Born: January 12, 1853, Lugo, Italy

Died: August 6, 1925, Bologna, Italy

Mathematische Annalen

1900, Volume 54, Issue 1-2, pp 125-201

Méthodes de calcul différentiel

absolu et leurs applications

M. M. G. Ricci, T. Levi-Civita


Tullio Levi-Civita

Tullio Levi-Civita, was an Italian mathematician, most famous for his work on absolute differential calculus and its applications to the theory of relativity, but who also made significant contributions in other areas.

Born: March 29, 1873, Padua, Italy

Died: December 29, 1941, Rome, Italy

Mathematische Annalen

1900, Volume 54, Issue 1-2, pp 125-201

Méthodes de calcul différentiel

absolu et leurs applications

M. M. G. Ricci, T. Levi-Civita



In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer Marcel Grossmann. Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915 17, and was characterized by mutual respect:

I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.

Albert Einstein, The Italian Mathematicians of Relativity[18]



Tensors were also found to be useful in other fields such as continuum mechanics. Some well-known examples of tensors in differential geometry are quadratic forms such as metric tensors, and the Riemann curvature tensor. The exterior algebra of Hermann Grassmann, from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of differential forms, as naturally unified with tensor calculus.


Introduction - Scalars

Many physical quantities are correctly described by only a number (positive, negative or zero) and are called scalars.

By fixing a system of unit the magnitude of a scalar is independent of the reference system but can change by changing the point in space. No sense of direction is associated to scalars.

Scalars can be compared only if they have the same physical dimensions. Two scalars measured in the same system of units, are equal if they have the same magnitude and sign.

As an example, the Temperature T is a scalar and, in a fixed point in space, its value does not change. Clearly by changing the system of units its value changes e.g. 0 C=32 F.


Introduction - Vectors

Other physical quantities, that are correctly described, in each point in space, by a magnitude and a direction, are called vectors.

By fixing a system of unit the magnitude and direction of a vector are independent of the reference system but can change by changing the point in space.

Vectors can be compared only if they have the same physical dimensions. Two vectors measured in the same system of units, are equal if they have the same magnitude and direction.

The components of a vector (to be better defined in the following) depends on both the frame of reference and system of units.

As an example, the force f is a vector and, in a fixed point in space, its magnitude and direction does not change. Clearly by changing the system of units the magnitude does change.

In a more abstract way vectors are the elements of a vector space.


Introduction - Vectors

A vector space is a set of elements called vectors satisfying the following axioms.

To every pair, x and y of vectors in there corresponds a vector x + y, called the sum of x and y, such that:

1) x + y =y + x (addition is commutative);

2) (x + y) + z = x + (y + z) (addition is associative);

3) there exists in a unique vector zero 0, such that 0 + x = x, x ;

4) x in there corresponds a unique vector x such that x + ( x) = 0.

pair and x, where is a scalar real number and x is a vector in , there corresponds a vector , called the product of and x, such that:

1) ( ) = ( ) x (multiplication by scalars is associative),

2) 1x = x,

3) (x + y) = + (multiplication by scalars is distributive with respect to vector addition),

4) ( + ) x = + (multiplication by scalars is distributive with respect to scalar addition), , , x, y .


Introduction - Tensors

The sum and products (at least three of them are extremely relevant) of two vectors can be defined but (Aris):

"Although the quotient of two vectors cannot be defined satisfactorily, tensors arise physically in situations that make them look rather like this.

For example, a stress is a force per unit area. We have seen that a force is a vector and so is an element of area if we remember that we have to specify both its size and orientation, that is the direction of is normal.

If f denotes the vector of force and A the vector of magnitude equal to the area in the direction of its normal, the stress T might be thought as f/A.

However because division by a vector is undefined, it does not arise quite in this way. Rather we find that the stress system is such that given A we can find f by multiplying A by a new entity T which is like f/A only in the sense that f=A T."

The stress system is a tensor and it appears that (at least) two direction are associated to it.

Cartesian vectors and tensors


René Descartes:

René Descartes was a French philosopher, mathematician and writer who spent most of his life in the Dutch Republic.

Born: March 31, 1596, Descartes, Indre-et-Loire, France

Died: February 11, 1650, Stockholm, Sweden


Cartesian coordinates

In an Euclidean 3D the position of a point can be specified by the three Cartesian coordinates.

A frame of reference has to be fixed, as shown in the figure, we take a generic point O as the origin and draw three mutually perpendicular straight lines O1, O2 and O3 (with positive sense shown in figure).



The frame of reference is normally taken right handed.



In an Euclidean 3D the position of a point can be specified by the three Cartesian coordinates.

A frame of reference has to be fixed, as shown in the figure, we take a generic point O as the origin and draw three mutually perpendicular straight lines O1, O2 and O3 (with positive sense shown in figure).

The coordinates of P are the lengths of the projections of OP on the three axis O1, O2 and O3.

These lengths are indicated with x1, x2 and x3.



If the reference system is rigidly rotated the coordinates of P change.

The rotation can be specified by giving the direction cosines lij between Oi and . The new coordinates are related to the old by:

Conversely:

By introducing the Einstein notation (a repeated or dummy suffix imply a sum over the three values 1, 2 and 3; the other suffix, called free, can take any value):

332211 xlxlxlx jjjj

332211 xlxlxlx iiii

iijj xlx

jiji xlx


Albert Einstein

Theoretical Physicist

Albert Einstein was a German-born theoretical physicist and philosopher of science. He developed the general theory of relativity, one of the two pillars of modern physics.

Born: March 14, 1879, Ulm, Germany

Died: April 18, 1955, Princeton, New Jersey, United States


Vectors

The position vector is an example of vector and its components are the coordinates of P. We can therefore make the following definition:

A Cartesian vector, a, in three dimensions is a quantity with three components a1, a2 and a3, in the frame of reference O123, which, under rotation of the coordinate frame to become:

We will identity vectors by an underlined symbol, but often a bold letter is used.

iijj ala1


Vectors

If the position vector is a function of time:

Where:

Since the direction cosine lij are independent of time we have:

Therefore the derivatives, in particular the first and second one i.e. velocity and acceleration, of the position vector are vectors.

iijj xlx

txxtxx jjii

ni

n

ijn

jn

dt

xdl

dt

xd


Vectors

The position vector is an example of vector and its components are the coordinates of P. We can therefore introduce its length or magnitude:

If a=1 a is a unit vector and its components may be thought as the direction cosine. Thus

Is a unit vector and represent the direction of a. Clearly only two components are independent.

iiaaaa

aa


Scalar multiplication

If is a scalar the product of this scalar and the vector a is a vector with components ai. Clearly the direction of a is the same as that of a and:

aa


Addition of vectors

If a and b are two vectors with components ai and bi their sum is the vector with components ai +bi. Again we have:

Therefore the sum of two vectors is a vector. We have:

Subtraction may be defined by combining with a scalar multiplication:

iiijiijiijjj balblalba

ii bababa 1

cbacba

abba


Coplanar vectors

Any vector c which is in the same plane as a and b (with different direction i.e. ) can be represented as:

In component form:

By looking for a solution of this system of equation ( and are the unknown) we see that the following condition should hold:

bac

0

321

321

321

ccc

bbb

aaa

bbaa

iii cba


Kronecker Delta

The Kronecker Delta ij is defined as:

When ij appears in a formula with a repeated suffix it replace the dummy suffix with the other suffix of the Kronecker Delta:

ji,

ji,ij 0

1

iiiijij aaaaa 332211


Leopold Kronecker

Leopold Kronecker was a German mathematician who worked on number theory and algebra.

Born: December 7, 1823, Legnica, Poland

Died: December 29, 1891, Berlin, Germany


Unit vectors

The three unit vectors that have only one non-vanishing component are the natural basis of the frame of reference O123 and are mutually orthogonal:

Or:

Clearly we have:

1 0 0

0 1 0

0 0 1

3

2

1

,,e

,,e

,,e

ijjie

ii eaeaeaeaa 332211


Unit vectors

By considering a rotation given by an orthogonal matrix l the base vectors are transformed:

Therefore:

By comparing the second and third element we have:

That is the law of transformation of vector components.

iijjjjii elaeaeaa

iijj ele

ijji laa


Basis of non-coplanar vectors

The natural basis is not the only basis. By considering three non-coplanar vectors a, b and c (again they should not have the same direction) any vector d can be expressed as:

Where the constants can be determined by solving the following linear system of equations:

Since the vectors are non-coplanar the determinant does not vanish.

cbad

3333

2222

1111

dcba

dcba

dcba


Basis of non-coplanar vectors

If M is a non-singular matrix the vectors:

Are non-coplanar for:

Does not vanish if clearly the "barred" vectors form a new basis. But such a general transformation takes outside of the Cartesian vectors which are concerned with a basis of mutually orthogonal vectors.

cMc

bMb

aMa

333

222

111

333

222

111

cba

cba

cba

M

cba

cba

cba

0M


Scalar product:

The scalar (or dot) product of two vectors is defined as

(where is the angle between the two vectors a and b):

and read "a dot b". By recalling that the natural basis unit vectors are mutually orthogonal we have:

Thus:

The scalar product is invariant under rotation of axes as can be directly verified:

Where since l is an orthogonal matrix

cosabba

ijji ee

iiijjijijijjii babaeebaebeaba

ijij

ijij

lbb

laababababalllblababa iikiikkikjijkjkijijj

ikkjij ll


Scalar product:

If and are the unit vectors in the direction of a and b:

And

Is the projection of the vector a on the direction of the vector b. If the vectors are orthogonal then cos = 0 and the scalar product:

The scalar product is clearly commutative but also distributive with respect to addition:

0ba

cosaba

cosbacosabbaabbbaaba

cabaceeabeea

cabacabacba

cbaecbeacba

jjiijjii

ijjiijjiiiiiiii

ijjjijjjii


Vector product

The vector (or cross) product a b (read "a cross b") of two vectors is defined as the vector normal to the plane of a and b of magnitude ab sin directed in a way that a, b and (a b) form a right-handed system. Clearly the vector product is not commutative:

The magnitude of (a b) is the area of the parallelogram two of whose sides are the vectors a and b.

abba


Vector product

The vector product of the basis unit vectors are unit vectors and:

Therefore:

Or

0

213

132

321

ii ee

eee

eee

eee

312212211312332 ebabaebabaebabaebeaba jjii

321

321

321

bbb

aaa

eee

ba


Levi-Civita symbol

In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the sign of a permutation of the .

Born: March 29, 1873, Padua, Italy

Died: December 29, 1941, Rome, Italy


Vector product

A very valuable notation can be introduced with Levi-Civita (or permutation) symbol ijk.

Therefore:

It is easy to see that the Levi Civita symbol can be used to calculate the determinant of a matrix:

3 and 2 1, of npermutatio odd an is if1,-

3 and 2 1, of npermutatio even an is if1

same the are , , of twoany if0

ijk

ijk,

kji,

ijk

kjiijk ebaba

kjiijk cba

ccc

bbb

aaa

321

321

321

kijkji eee


Triple scalar product

The triple scalar product is defined as:

The vanishing of the triple product is the condition for co-planarity of the three vectors. The triple scalar product can be physically interpreted as the volume of the parallelepiped with sides a, b and c.

kjiijk cbabacacbcba


Triple vector product

The product a (b c) is called triple vector product and, since b c is a vector normal to the plane of both b and c and a (b c) is a vector normal to (b c), must be in the plane formed by b and c.

It can be evaluated starting from the identity:

For k=1 the first member is different from zero only if i j, l m and i 1, j 1, l 1, m 1. Therefore:

Hence:

jlimjmilklmkij

npermutatio jlim

jmil

ljmi

mjli

cbabcaecbaecba

ecbaecbaecba

ecbaecbaecbeacba

jjiiijij

ljimjlimljimjmilljimjlimjmil

ljimkijklmljimijkmklkjiijkmm


Vector identities

acb minus abc

Jacobi's identity

Pythagorean theorem

jlimjmilklmkij

ippjkijk 2

3ii

bacacbcba

cbabcacba0bacacbcbacbbadbcadcba

22 babbaaba

dcbacdba

adcbbdcadcba


Velocity due to a rigid body rotation

Suppose that a rigid body rotates about an axis through the origin with a direction given by . If is the angular velocity the rotation can be identified by the vector .

Let P be a point in the body at position x. Then x is a vector in the direction of PR of magnitude x sin .

In a short interval t the point P moves to R

and the vector PR= x is:

In the limit t 0, one finds:

Whenever the velocity of a point can be represented as a vector product of a constant vector with the position vector then the motion is due to pure rotation.

txx

xvt

xlimt 0


Second order tensors:

A vector was defined as: A Cartesian vector, a, in three dimensions is a quantity with three components a1, a2 and a3, in the frame of reference O123, which, under rotation of the coordinate frame to become:

Similarly we define a tensor A as an entity having 9 components Aij in the frame of reference O123, which, under rotation of the coordinate frame to become:

We will identify tensors with a double underlined symbol, but often a bold capital letter is used.

iijj ala1

ijjqippq AllA2



The components of a second order tensor can be written in matrix form:

The previous equation (2) in tensor form becomes:

Where l' is the transpose of l (i.e. ). A tensor A is called symmetric if . Clearly a symmetric tensor has only 6 distinct components.

A tensor A is called antisymmetric if . Clearly an antisymmetric tensor has only 3 distinct components.

333231

232221

131211

AAA

AAA

AAA

A

lAlA '

ji'ij ll

jiij AA

jiij AA



Care should be taken in dealing with non-symmetric tensors really the previous equation (2) may be misleading and should be correctly interpreted.

A more robust symbology (often called dyadic notation) is obtained if we do not drop the unit vectors from the equations:

In this case it is clear that lip is the transpose of l.

lAlA 'ijjqippq AllA2

qijjqippqpqp' eAlleeAelAlA


Examples of second order tensors

The Kronecker delta is a symmetric tensor:

Where the last equality is a consequence of the orthogonality of l. The components of are the same in all coordinate systems therefore is an isotropic tensor.

If a and b are two vectors their tensor product is a second order tensor:

a b is normally called a dyad.

Other important tensors are the inertia, stress and rate of strain tensors.

pqiqipijjqippq llll

jjii ebaeba

ijjqipjijqipjjqiipqppq AllballblalbaA


Scalar multiplication and addition

If is a scalar then the multiplication and addition of tensors are defined as:

By defining the symmetric and an antisymmetric part of A:

It is evident that any tensor can be represented as the sum of a symmetric and an antisymmetric part:

jiji eAeA

jijijijijijiji eBAeeBeeAeBA

jjiiji's eAAeAAA

2

1

2

1

jjiiji'a eAAeAAA

2

1

2

1

sa AAA


Tensor contraction and multiplication

The operation of summing the components of a tensor over two of its indices is called contraction:

For a second order tensor the contraction is a scalar and is called the trace of the tensor. Often the notation tr(A) is used to indicate the trace of the tensor A.

If A and B are second order tensor their tensor product A B is a fourth order tensor with components in the barred coordinate system:

That is the analogue of and, therefore will be used in the definition of higher order tensors.

The contractions of a fourth order tensor are second order tensors, e.g.

332211 AAAAii

kmijmskrjqipkmmskrijjqiprspq BAllllBllAllBA

ijjqippq AllA2

kjijjkjikiijjkij BABABABA


Tensor scalar product

Normally the scalar product is used instead of the contraction:

And for the other three:

Please note that e.g.:

because

kjkijikmkjmijikmkmjiji eBAeeBAeeBeeAeBA

kkiijjkkmmiijj'' eBAeeBeeAeBA

kjkjiikmkmjjii' eBAeeBeeAeBA

kkjijikkmmjiji' eBAeeBeeAeBA

kjijjkjikiijjkij BABABABA

ijkikiij ABBA

ABeABeeBAeBA jijkikkkiijj''


Tensor double scalar product

In general we have:

The double dot product (or double contraction) is defined as the scalar:

A cross product lowers the sum of the order of the tensor operands by a factor one while a single dot of two and a double dot by a factor 4. In another way: an arrow ( ) kills one (under)line and each dot ( ) kill two lines.

jiijkmkjmijikmkmjiji BAeBAeeBeeAeBA ::

''jjiikk

'kikjij

' ABeABeeBAeBA


The vector of an antisymmetric tensor:

An antisymmetric second order tensor has only three independent components so by introducing a vector one can write:

Since:

Where because both k and l should be different from i and j. Besides both even and odd permutations contribute to the sum.

The vector 2 of an antisymmetric second order tensor is therefore defined as:

One has: kijkijkkv ee22

0

0

0

12

13

23

aa

3

2

1

jkijkijiji eeee

lllijlkijklijlij

lqplqpjkijkilqplqpjiji

eee

eeeeeeeeee

2

::

klijlijk 2

aa


The vector of an antisymmetric tensor:

Really:

Where the condition of asymmetry has been used:

aeaeaea

eaeaeaa

iijjjijiiijj

mjmiljlimijlmlkmijkij

lmlkmkl

2

1

2

12

1

2

kkkijkij ee 2

jjijij ee


Canonical form of a symmetric tensor

For a particular vector a and tensor A it may occur that their scalar product has the same direction of a. In this case we have:

Or, in component notation:

That is a homogeneous system of three linear equations for the three unknown aj. A solution of this systems exists only if the determinant of the coefficient vanishes. Therefore, the following characteristic equation holds:

Where the quantities Ii are called the invariants of the tensor A.

aaA

0jijijjijijij aAaaaA

031

223 IIIA ijij

ij

ii

AI

QAAAAAAAAAAAAI

AtrI

3

3113211232231133221133222

1 )A(



We also have:

Clearly i are the characteristics values or eigenvalues and the corresponding ai the characteristics directions or eigenvectors.

It can be shown that when tr(A)=0:

311321123223113322113322

232313132112333322221111

332233112211333322221111

22222

2222

1

2222

12

1

2

1

2

1

AAAAAAAAAAAA

AAAAAAAAAAAA

AAAAAAAAAAAA

AAAAAtrAAtrAtrI jiijiijkijii

'ss

'aasa AAtrAAtrAAQI2



If the eigenvalues are all distinct then it can be proved that the eigenvector are mutually orthogonal. By normalising them and changing the symbol to l we have:

Thus lpi is an orthogonal matrix. By changing the reference system accordingly we have:

The tensor A in this new coordinate system has a diagonal form:

The characteristic directions are known as the tensor principal axes.

pqqp

ppp

ll

lAl

pqpqppqp lllAl

3

2

1

00

00

00

A


Vector and tensor identities

''AaaA

22 babbaaba

''' ABBA

aAAa '

''' ABBA'BBBA A:A:: '

AbaAba

bAabAa


Higher order tensors

In general we define a tensor A of order n (i.e. n times underlined) as an entity having 3n components Aij n provided that under rotation to a new coordinate frame they transform accordingly to:

If the interchange of two indices does not change the components the tensor it is said to be symmetric with respect to these indices. A similar definition for antisymmetry holds.

The algebra of higher order tensor remains practically unchanged:

n...ijntjqipt...pq Al...llA

ACABCBA

ABBABABA


The quotient rule

The quotient rule is used to prove that nine quantities Aij are the components of a tensor. If a and b are vectors and

then Aij are the components of a tensor. The usefulness of this is that A b=c may arise in many physical situation in which is known that both b and c are vectors therefore by the quotient rule we can say that the equation holds in any reference system. We should prove that:

By defining the first relation is satisfied and since both b and c are vectors:

That since A and b are independent proves the thesis.

The same proof may be easily extended, e.g

ijij cbA

pqpq cbA

0qijjqippqqjqijipjijipiippqpq bAllAblAlbAlclcbA

ijjqippq AllA

qppq bcA

qjqjjjqq blbblb iipp clc

ikjkij CBA


Isotropic tensors

An isotropic tensor is one whose components remain unchanged by any rotation of the frame of reference. Clearly scalar are isotropic. There are no isotropic vectors and the only isotropic second order tensor is the Kronecker delta.

First we take a permutation of the axes:

We have:

But for the isotropy we have:

and similar relation for the other components. By considering a permutation and a reflection of axes:

The off axis components should be zero Therefore:

001

100

010

l

ijjqippq AllA

1223

3311

AA

AA

122323

331111

AAA

AAA

100

001

010

l1212122112 AAAAA

ijijA


Isotropic tensors

An isotropic second order tensor is clearly connected to a sphere. Really since is the equation of a quadric surface that is invariant of the axes rotations.

The only isotropic third order tensor is ijk. See the Aris book for details.

The product of isotropic tensors is an isotropic tensor but it is quite intricate to find the general isotropic fourth order tensor again see the Aris book for details. A convenient representation of the general form is:

The second term is symmetric with respect to the first and second or third and fourth indices and the third term is antisymmetric with respect to them.

1jiij xxA

jpiqjqipjpiqjqippqijijpqT


Axial vectors

The only transformation that we have considered is a rotation of a right handed Cartesian system. A slight extension would allow also reflections i.e. a right handed coordinate frame is transformed in a left handed one. The matrix l is still orthogonal but with negative determinant.

Clearly, the vector product, that strictly depend on the choice of a right handed reference system, is not invariant with respect to a single reflection but changes the sign.

A vector which has this behaviour is normally called an axial vector or a pseudo vector.

It can be shown that the Levi Civita symbol is a pseudo tensor.

Documents

Tensor Calculus Main references