lecture 3 tensor

8/2/2019 Lecture 3 Tensor

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EARTH AND ATMOSPHERIC SCIENCE DEPARTMENT

COLLEGE OF NATURAL SCIENCES AND MATHEMATICS

UNIVERSITY OF HOUSTON

Evgeny M. CHESNOKOV

Groundwork of Vector and TensorAnalysis

Lecture

- HOUSTON, UH, 2012 -

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CONTENTS

1.1. Concept of scalar and vectors

1.2. Transformation of systems of coordinates. General conceptof tensor

1.3. Division theorem

1.4. Tensors in Cartesian system of coordinates

1.5. Specific tensors

1.6. Eigen values and Eigen vectors of second rank tensors

1.7. Curvilinear integrals. Stocks theorem

1.8. Matrices and Determinants

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Concept of Scalars and Vectors

Parameters determining the fundamental physical characteristics - force, velocity, displacement,tenseness of electric field, etc. are determined by directions and values. These characteristics can berepresented by directed segment in a three dimensional space and named - vectors or tensors offirst rank .

Lets characterize vectors in 3-D space:

1. vectors are equal if their directions are identical and lengths are equal; 2. the unit vector - vector with unit length;

3. the negative vector - vector with the same length as an initial one, but with opposite direction

4. the zero vector - vector with zero length and indefinite direction.

In order to continue the consideration of other properties of first rank tensors lets introduce the zerorank tensors (scalars):

Physical characteristics, which determined by quantities only named tensors of zero rank orscalars.

As vectors scalars are fundamental characteristics of objects in physics and mathematics. In physicsfor example, scalars characterize energy, mass, volume, period, frequency, temperature and etc.These values do not depend on directions, but determine the properties of studied subjects as a

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Mathematical properties of first rank tensors

1. the addition of vectors:

( 1 )

Or in a general case:

2. the multiplication of a vector by a scalar:

( 2 )

fabqqcba

......

kbf

X

Y

Cjbaibaba yyxx

a

b

C

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2. The vector product of a vector and a scalar:

kbf

f

k

2b

X

Y

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3. the scalar product of two vectors:

( 3 )

10/9/2008 2

Definition

Scalar Multiplication

a

b

cos( ) a b b a ab

Scalar Product

zzyyxxzyxzyx fqfqfqkfjfifkqjqiqfq )()()(

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8. the vector product of two vectors:

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Definition

Vector

Multiplication

sin( )V a b b a ab e

b

a

e

sin( )V a b b a ab e

V

Vector product

kfqfqjfqfqifqfq

fff

qqq

kji

fq xyyxxzzxyzzy

zyx

zyx

)()()(][

It is easily to show from (9) that: 0][ qq

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Another forms of previous expressions

Components of any rank tensors (except zero one) can be represented visuallyand shortly over indexes denotes. In this case system of coordinate OXYZchanges to the nameless OXi and components of vectors can be written in aform: ai , bj, c

k, dl.

Z

O X

Y

1

X

3X

2X

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Nonrecurring indexes named free. For example, in an

expression:

one index is free only.

Recurring indexes named mute. In previous expressions we

met from one up to two mute indexes.

The rank of the tensor is determined by the number of free

indexes. For example, a second rank tensor can be written

with two free indexes:

or in more general form:

a b c Ri j j j npp qpp

, , , ,,

i j k j k a b

,,ij

ijDD ij

ij DD ,,

kkij

ij

jkijip VUBA

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In 3-D case, where both free indexes change from one up to

three, the expression for represents nine components of a

second rank tensor:

Index written form is very convenient. For example, a systemof linear algebraic equation:

can be represented in an index notation in a form:

3,2,1,,

333231

232221

131211

ji

DDD

DDD

DDD

Dij

3332321313

3232221212

3132121111

zczczcX

zczczcX

zczczcX

jiji zcX 10


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Using this expression we can write for second rank tensor:

or in an expanded form for one component:

Up to now we were discussed just a convenience ofindex notations, but did not introduce the conception oftensors.

knjnikijAbbA

)()(

)(

3113131132231312

2112121113213122121121111

3

1,

11

AAbbAAbb

AAbbAbAbAbAbbA knnk

nk

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TRANSFORMATION OF SYSTEM OF COORDINATES.GENERAL CONCEPT OF TENSOR.

12

TRANSFORMATIONOFSYSTEMOFCOORDINATES GENERALCONCEPTOFTENSOR


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TRANSFORMATION OF SYSTEM OF COORDINATES. GENERAL CONCEPT OF TENSOR.

Lets consider an arbitrary 3-D Cartesian system of

coordinate (old) )and any other Cartesian system

of coordinate (new) (i=1,2,3).

Formulas of transformation of coordinate:

determine a new set of coordinate for any point

of an old system .

Xi

i

),,(321

XXXii

i

321 ,, XXX Xi

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TRANSFORMATION OF SYSTEM OF COORDINATES. GENERAL CONCEPT OF TENSOR

Determinant:

is the Jacobian of transformation. Here or is

arbitrary curvilinear or orthogonal system of coordinates.

3

3

2

3

1

3

3

2

2

2

1

2

3

1

2

1

1

1

XXX

XXX

XXX

J

Xi i

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Lets define a component of differential from

::

This expression determines class of tensors named as conravariantfirst rank tensors or vectors. In general case mathematical objects

named contravariant first rank tensors if they obey to the low oftransformation:

There is an existenceso called covariant tensors, components of

which are follow of rule:

In general theory for visualization of covariant tensors usually usethe low indexes.

TRANSFORMATION OF SYSTEM OF COORDINATES. GENERAL CONCEPT OF TENSOR.

),,(321

XXXii

j

j

ii

dXX

d

rs

s

j

r

iij

BXX

B

'

rsj

s

i

r

ij DXX

D

'

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For illustration of a concept of contra and co-variant first order tensors (vectors),

lets consider an example.

It is well known that any vector can be represented on a plane OX1X2 in a form:

fig 1

( * )

where: are unit vectors of expanded basis; are the rectangularcoordinates of vector (fig 1).


22

11

eaeaa

2

2

1

1eaeaa

1a

2aa

1x

2x

2,1e 2,1aa

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In a case when the coordinate axes are not orthogonal each other,

then vector can be represented in a two different ways (fig 2.).

fig 2

2,1x

Different Vector ComponentsDifferent Vector Components

a

a1

a2

a1

a2

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If we want to determine the vector via values , then we have to use

the expression (*) and named as contravariant components of vector .

In the another form of a definition of vector through gives the

expression:

The quantities named as covarint components of a vector .


a 2,1

a

a

2,1a

a

2,1a

|||sin||||

|||cos||||

2222

1111

eaeaa

eaeaa

x

x

2,1aa

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Tensors in the Cartesian System of

Coordinates

19

TensorsintheCartesianSystemofCoordinates.


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Tensors in the Cartesian System of Coordinates.

Transformation one system of homogeneous coordinate to the

another one are named orthogonal.

We will consider the Cartesian tensors . For this type of tensors noany difference between contravariant and covariant components

and this is a reason why we will use the low indexes only. As can be

shown below the partial derivations in (18) and (19) can be changed

by constants. In fact, lets consider two orthogonal systems of

coordinate (green) and (red).

In 2-D :

OX X X1 2 3 OX X X 1 2 3' ' '

20

TensorsintheCartesianSystemofCoordinates.


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In this case an orientation of any axes of one system relatively

another one can be determined by table:

or by tensor of transformation aij. In this case components of anarbitrary vector Ui in initial (without primes) system of coordinatesare connected with those coordinates in prime system of coordinateby expression:

(**)


a11

a12

a13

a21 a22 a23

a31 a32 a33

'

1X'

2X'

3X

1X

2X

3X

jiji UaU

'

21

Tensorsinthe CartesianSystemofCoordinates.


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This equality determines the low of transformation of the first rank

Cartesian tensors (vectors). The inverse relationship is valid as well:

(***)

Combination of (**) and (***) leads to the expression:

( # )

Generalization of (#) leads to low of transformation of second rank

Cartesian tensors:


ijij UaU'

kjkkjkkikjij UUaUaaU

knjnikij FaaF '

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Properties of Tensors

1. Tensor is called symmetric when:

And antisymmetric if:

Rank of tensor is defined by the number of free indexes

jiij

AA

jiij AA

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Algebra of Tensors 1. Addition

Addition is valid for the same rank tensors only. In accordance of the

rule:

2. Multiplication

There are two sort of multiplication - internal and external:

ijkijkijkijk TDBA ..........

Internal ExternalaI bj = Tij ai Eik = fk

Dij Tkm = Fijkm Eij Ejm = Gim

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lecture 3 tensor

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