Lesson 2OK

8/13/2019 Lesson 2OK

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Lesson 2 :An introduction to tensors


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OUTLINE

1. Introduction2. Vector algebra (recalls)

3. Tensor algebra

4. Scalar, vector, tensor functions

AppendixBibliography


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1. INTRODUCTION

In (nonlinear) continuum mechanics, physical quantities can be described by :

oscalars (or real numbers) denoted by italic lightface letters like t, l, T, W, ...

measure of the quantity (eventually with negative sign) associated to a unity

Ex : time (s), length (m), temperature (C, eergy J, ass desity kg/,


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1. INTRODUCTION

o

vectors often designated by lowercase bold-face Latin letters such asx , v , f

Elsewhere, the notation is also employed

model physical quantities having both direction and length (or intensity)

represented by triplets of real numbers associated to basis vectors (of unit length)

Ex : position vectors (m), displacement (m), velocities /s , fores N,


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o

second-order tensors represented by uppercase bold-face Latin letters like F, E , T ,

also denoted

generalize scalars and vectors that can be interpreted as 0th-order and 1st-order tensors

may be thought as linear operators acting on vectors

represented by matrices associated to basis tensors (of unit length)higher order tensors (3rd and 4th order) will also be considered

Ex : deformation (-, stress easures N/, elastiity of a aterial ,

stress tensor

Unit normal nStress vector t t(n)=T . n linear relationship


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IN SUMMARY

oTensors of different orders (up to 4) are sufficient to describe all continuum mechanical

quantities

oThey can either be defined as global variables for a whole body or local variables in every

point of the body :

Mass, resulting force acting on the body gloal

density, velocity, acceleration loal

oContinuum mechanics brings into play :

Scalar-valued , vector-valued and tensor-valued functions

of scalar , vector and tensor variables

vector and tensor algebra

tensor analysis (gradient, derivation, integration)


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2. Vector algebra (recalls)

Three-dimensional Euclidian space considered

Fixed set of three basis vectors called a Cartesian basis

such that

orthonormal system

Any vector is re presented uniquely by a linear combination of :

or With the three Cartesian components

dot product)


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2.1 Dot product :

The dot (scalar or inner) product is a positive-defined bilinear form

bilinear means linear in both arguments

with the geometrical meaning,

angle between u and v

norm (length) of u : (also denoted | u|)

uand vare perpendicular if

eis a unit vector if


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2.2 Direct, index, explicit notations :

Direct (intrinsic) notation so far employs bold-face letters without referring to any basis :

convenient and concise to manipulate vectors and tensors

Index notation that only retains the generic component of a vector or tensor, previously

decomposed on a basis (of or ) :

useful for some (complicated) calculations

Explicit notation that enumerates all the components of a vector or tensor:

(2 or 3 components)

(4 or 9 components)


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2.3 Summation convention :

If an index is repeated in the same term, a summation is implied over the range of this index

(unless stated otherwise)

(1) means

(2)

(3) or

The summation symbol is left out !

This convention is adopted in the subsequent


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An index that is summed over is called a dummy index and can be replaced by any other letter

without changing the value of the expression

In (1) and (2), iand mare dummy ranging from 1 to 3

and for ex, ambmand arbrhave the same meaning

An index that is not summed over is a freeindex. In the same expression, an index is eitherdummy or free

In (3), iis free and enumerates the (two) components of y

The indexjis dummy ranging from 1 to 2


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2.4 Delta operator :

Kronecker delta symbol

it orrespods to the opoets of the d-order identity tensor I

For the (Cartesian) basis (orthonormal conditions)

Some related properties:

(summation convention !)

replacement operation of a free indexji

evolving the dot product,

componentj of the vectorx (projection)

and the norm


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1. Meaning of the following expressions ?


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2.5 Cross product and alternating symbol :

The vector product is bilinear and anticommutative (or skew)

If holds, uis parallel to v(they are linearly dependent)

Geometrical interpretation :

For the right-handed orthonormal basis,


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How to express the cross product in terms of components ?

Perutatio alternating) symbol

even permutations of the indices

odd permutations of the indices

if there are repeated indices

thus, and,

opoets of the d-order alternating tensor

The ie relatios for the asis etors a e re-written as

by instance,


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Coordinate expression for

explicitly,

also,

detis the determinant of the matrix


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2. Check that

determinant of the matrix [A]

3. Starting from the classical (Lagrange) identity

show that

and deduce the relation

2 6 T i l l d


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2.6 Triple scalar product :

iff : if and only if

for allThis product is trilinear

The vectors are linearly dependent iff their

triple scalar product vanishes !

Note that,

and

3 T l b


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3. Tensor algebra

First definition of a second-order tensor :

A second order tensorA is a linear operator that associates a (given) vectorx with a vector y

such as : y=Ax

linear means

All 2nd-order tensors form a linear space of dimension 9, denoted Lin, with the operations

Ex:

2nd-order zero tensor O:

2nd-order identity tensor:

maps the vectorx to the zero vector o

mapsx to itself


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Second (equivalent) definition :

A can be viewed equivalently as the representative factor of a bilinear formb

bis characterized by

linear in both arguments (bilinear)

More appropriate to introduce the deformation tensor (in solid mechanics)

Useful to extent the definition to higher order tensors by considering multilinear forms

3 1 T d t Gibb (1881)


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3.1 Tensor product Gibbs (1881)

The projection of any vectorx on a unit vector n is the simplest (non trivial) linearoperation

direct notation

index notation

explicit notation

This operation of projection can be represented by a second order tensor denoted

The product is called tensor or dyadic product of two vectors : considered as the

generating operation of tensors


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The tensor or dyadic product of the vectors a and b denoted is a second-order

tensor that assigns to any third vectorxthe following vector

It is viewed as the scalar product ofx and b along the direction of a

Not to be confused with the dot or cross product !

As a tensor, it should satisfy the characteristic linearity property

Generally,


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Ex: Replacing a , bby the basis vectors e1, e2

clearly here,

Any second order tensorA may be expressed as a linear combination of the of nine dyads

formed by the Cartesian basis

form a basis of

also called a tensor space denoted

Aimare the nine Cartesian components ofA


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Matrix [A]in the basis

In particular ,

With aibjthe components of the tensor (or dyad)

Explicitly,


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The tensor product satisfies

5. Prove the remaining relations .


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Index and explicit notation of the vector transformation also denoted y=A.x

Thus, dummy indexj

Explicitly,

4. Deduce that the components

Calculate

3 2 Composition of two second order tensors


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3.2 Composition of two second-order tensors

Given two linear transformations

The composition of g byf is a linear functiong o f defined by

the circle o is often omitted

Composition of two tensorAB is another tensor C such that

In matrix form

classical ultipliatio of atries

Ex: Generally ,

3 3 Some particular tensors


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3.3 Some particular tensors

Identity tensor I :

Because of one has called spectral decomposition of I

In matrix form,

Clearly,


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Transpose of a tensor :

The unique transpose ofA denoted byAT is such that


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with the properties,

Symmetric tensor :

A tensor S is symmetric iff


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Skew (antisymmetric) tensor :

A tensor W isskew iff

Any tensor T can be uniquely decomposed into a a symmetric and skew tensor :

also denoted


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Dual ector of a ske tensor :

A unique vectora can always be associated to a skew tensor denoted Waas


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Ex : in particular ifx = e1anda = e3

in this case,

and,

rotation of vector e1at right angle

3 4 Trace and scalar product


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3.4 Trace and scalar product

The trace of a tensor T is a scalar defined by

with the important properties,

homogeneity of degree 1

additivitytr ( ) linear operator

6. Show that Tr I = 3


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The salar produt : or doule otratio of to tesorsA andB is given by

sum over i andj !

Ex:

The norm of a tensorA is the nonnegative real number :

As vectors,

U andV are orthogonalif

U is a unittensor if


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Useful properties of the double contraction :

Deviatoric tensor :

Every tensor Acan be decomposed into its spherical part and its deviatoric part

= definition of the deviator devA

by construction the trace of devA is always zero :


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7. Prove that

and deduce that basis tensors are mutually orthogonal andunit tensors

8. Given the tensor

Show that

3.5 Determinant


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3.5 Determinant

The determinant of a tensor T is defined by the determinant of the matrix [T]

also denoted IIIT

this scalar (real number) satisfies,

homogeneity of degree 3

recall,

3.6 Inverse, orthogonal tensors


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3.6 Inverse, orthogonal tensors

If there exists a unique inverse ofA, denotedA -1 such as

The tensorA is said to be invertible

set of all invertible tensors

With the properties


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A tensor Q isorthogonal if

or

This implies,

The norm of the two vectors and their relative

orientation are preservedbecause

= rotation (Q is said to be proper)

= reflection (Q is said to be improper)

3.7 Change of basis


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3.7 Change of basis

Let represent the old basis

The vector v can be decomposed such as

Relationship between the components vi and i?

By introducing the (proper) orthogonal tensor such as

it rotates the old asis to e oe

Thus

with the geometrical meaning,

represent the new basis


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In matrix form,

Ex: rotation of angle arounde3

The components of v are related by,

and similarly,

with


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For the components of a tensor T

in the old basis (of Lin)

in the new basis (of Lin)

Applying the relations between the basis vectors ej andej ,

By identifying,

In matrix form,


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Application : rotation of angle arounde1

The relation is given explicitly

Here,

After lengthy (!) calculations,


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Consequently, vectors and tensors can also be characterized by the laws of change of their

components

Vector v :

Second-order tensor T :(can be generalized to higher-

order tesors

Scalars are invariant

Ex:

3.8 Eigenvalues and eigenvectors of tensors


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g g

For a given tensor T, a vector v is said to be aneigenvectorwith an associated eigenvalue if

Tv=.

From linear algebra, the eigenvalues are the roots of the third-degree polynomial equation :

or explicitly,

(characteristic polynomial for T)

where ,TII and TIIIare the first andthird principalinvariant of T (the trace and the determinant

respectively)

is the second principal invariant of T

(also denoted sec (T) with the property : sec(T=2sec(T) )


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Every second-order tensorA satisfies its own characteristic equation

(Cayley-Hamilton theorem)

Proof :

Application : useful to derive certain intrinsic relations

Applying the linear tr(.) operator in both sides with the definition of TII

simplifying,Clearly, det(.) is a nonlinear operator and

det(T=3det(T)


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For a symmetric tensorS , it is possible to prove that there exits three real roots of the

characteristic equation 1, 2, 3

The three corresponding (real) eigenvectors n1, n2, n3are moreover mutually orthogonal

i = 1, 2, 3 no summation here !

and

can be used as an alternative Cartesian basis

In solids mechanics, symmetric tensors are often encountered such as stress or deformation

tesors


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A symmetric tensor S is positive definite if

Its eigenvalues are then all positive

Its principal invariants are also positive


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The symmetric tensor S can be expressed in the form

= spectral decomposition of S

In matrix form,

where and

The three principal invariants are simply


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9. Consider the symmetric tensor

Show that its eigenvalues and associated (unit) eigenvectors are

Express [P] and check the relation

3.9 Higher order tensors (notions)


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Generalization of the relation y=x

where the second order tensorA (with 32 components) maps the vectorxinto the vector y

A third order tensor

can be used to linearly map

- a tensor A into the vectorx as

- a vectorx into the tensorA as

(also denoted )

may be expressed as

where Tijkare the 33 = 27 components

Linear mappings can be written as

although other index contractions could be defined as


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The triadic product of the vectors a, b, c is a particular third order tensor denoted

satisfying

Ex : Alternating tensor

with,

In explicit form, thus

one obtains where

and the interpretation,


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A fourth order tensor

is used in practice to map a second tensorA into a second order B

(also denoted )

may be expressed as

where Tijklare the 34 = 81 components

In term of components, ones writes

The product of the vectors a, b, c, d is a particular fourth order tensor denoted

satisfying


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The tensor product gives a fourth order tensor by

Three fourth orderunit

tensors can be introduced

with the important properties,

Ex : projection tensor

4. Scalar, vector, tensor functions


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Functions of one scalar variable, such as time t :

scalar-valued functions yare

assigns to each element t of its domain Duniquely one element y of its range (or image)

Ex:

by extension, vector-valued and tensor valued functions are

such as

4.1 Derivatives


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Their first derivative with respect to t , or rate of change, is given by

Because and the rules below

Usual rules of differentiation,

The derivation is a linear operation

4.2 Vector / scalar fields


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Vector-valued function y of one vector variablex (point of R3)

with scalar-valued functions

In particular,

linear transformation

affine transformation

Scalar field

4.3 Other useful functions


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Scalar-valued function of one tensor variableA

Ex : principal invariants ofA :

linear function

nonlinear functions

Tensor-valued function B ofone tensor variableA

Ex : linear function

fourth order (constant) tensor

4.4 Gradients or related operators


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Gradient of a (smooth) scalar field F(x)

Gradient also denoted grad F or = vector !

Writing

is the derivative along the direction u = variation of F along u(directional derivative)

Denoting,

Thus,


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Ex :

Variation of Fat (1,1,1) along the unit vector


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10. Considering the quadratic form

show that


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Gradient of a (smooth) vector field v(x)

Gradient also denoted = second-order tensor !

with

along e1,

For the component 11 of the tensor

Thus, and,


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Ex:

(1) :

In index notation

(2) :

By extension, the gradient of a (smooth) tensor fieldA(x) is the third order tensor

with 27 components


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11. Find the gradient of the vector field

h, g are constant vectors


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Gradient of a (smooth) scalar-valued function (A), A second order tensor

Gradient of atA = second order tensor!

In index notation,

= scalar product

In particular

or


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Gradient of a tensor-valued function B(A)

Gradient of B atA fourth order tensor

In index notation

In particular used to derived incremental relations (laws)

between stress and strain tensors in mechanics

(nonlinear behavior)

or


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Ex:

12. Prove that

Using the previous results and the expression of det in terms tr,

One finds after some calculations


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Divergence of a vector v is the scalar

Rotational of a vector v is the vector

with

Explicitly,


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Divergence of a second order tensor T is a vector

Explicitly,

Serves to establish the equilibrium law in mechanics

Some properties for sooth salar, etor, tesor fields , u, v, A, B

4.5 Integral theorems


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Given scalar, vector and tensor fields denoted

respectively , v andS

or

or

(Gauss) divergence theorem

By setting S=I and

or

4.6 Some rules for differential operators


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4.7 Vector fields


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When the components of a vector or tensor depend on thecoordinates we have a vector or tensor field. Es: a flow with velocityv=v(x,y,,t!

, where s is a parameter alon" the tra#ectory (for instance, the arc len"th!.

$n a physical vector field, the operator s%ch as the diver"ence,the c%rl and the "radient have a partic%lar meanin" connectedfl%xes and so%rces of some physical &%antities

'ssociated with any vector field a(x! are its tra#ectories, which arethe family of c%rves everywhere tan"ent to the local vector a

4.8 Gradient of a scalar


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so:

$f is a scalar f%nction of the position(for example the temperat%re in a vol%me!and is a small displacement in the direction n, then

is the derivative in the ndirection. sin" )aylor*s theorem:

)h%s the "radient of can +e viewed as the rate of chan"e ofs%ch a scalar &%antity in the directions ni

ince it represent the rate of chan"e in all the directions, the"radient of a scalar is a vector (the "radient of a vectoris a tensor, etc..!.

4.9 Divergence of a vector field


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-or any differentia+le varia+le a(x,x/,x0!, we can write the diver"ence as:

We ta1e now a parallelepiped with one corner 2at x,x/,x0and the dia"onally opposite one 3 at(x4dx, x/4dx/, x04dx0!.

)he o%tward %nit normal to the face thro%"h 3 parallelto xis e, whereas the one thro%"h 2 is 5e.6n the first face:

6n the second face a=a(x,2,3!.)h%s, denotin" nthe o%tward normal and d the areadx/,dx0 of thesefaces, the &%antity avaries from face to face, contri+%tin" to the s%rfaces

inte"ral and which denotes the fl%x over the s%rface d

We eval%ate the contri+%tion to the s%rface inte"ral as:


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imilar terms can +e o+tained for the contri+%tion of the other twofaces (with a/x/, a0x0! so that for the whole parallelepiped of

vol%me dV=dxdx/dx0we have:

$f ais tho%"ht of as a fl%x, then andis the net flux out of the volume.A vector field whose divergence is zero is called SO!"O#DA

$f the fl%x field of a certain property is solenoidal, there is no "eneration of

that property within the field

which +ecome, for avol%me shrin1in" to ero:

4.$% a&lacian


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$f a is a the "radient of a scalar f%nction, , its diver"ence is calledthe a&lacianof

)he followin" is the 8aplace*s e&%ation :

' f%nction which satisfies this e&%ation is called a&otential function

)he 8aplacian represents the fl%x density of the "radient flow of af%nction.

Es: $n electrostatics, the 8aplacian of the electrostatic potentialassociated to a char"e distri+%tion is the char"e distri+%tion itselfEs/: the 8aplacian of the "ravitational potential is the mass

distri+%tion.

' f%nction which satisfies this e&%ation is called a&otential function

4.$$ 'url of a vector field


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)he cross prod%ct of the na+la operator with a vector field a is thec%rl of the vector field

$t is connected to therotation of the field,as we will demonstrate

We ta1e an elementary rectan"le in theplane normal to x with one one corner2 at x,x/,x0and the dia"onally oppositeone 3 at (x, x/4dx/, x04dx0!.

We want to calc%late the line inte"ral aro%ndthis elementary circ%it , a t ds , where tisthe tan"ent to the circ%it (the pro#ection of atan"ent to the circ%it!.

)he line thro%"h 2 parallel to x0has tan"ent5e0, and the parallel side thro%"h 3 has

tan"ent e0, and each has len"th dx0

)h%s the fl%x on these faces contri+%ites to the inte"ral a t ds of:


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)h%s, the fl%x on these faces contri+%ites to the inte"ral a t ds of:

imilarly, on the two other sides, there is a contri+%tion :

)h%s, writin" d'=dx0dx/, we have

9: the s%ffix indicate that the line inte"ral has

+een comp%ted only on a plane parallel to x

imilarly, we can comp%te the other two components for line inte"ral


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y p p "aro%nd rectan"les on a plane normal to x

;oreover, we can reach the same res%lt +y considerin" an infinitesimaltrian"le 23


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$f we consider now a fo%rth point S at (xdx,x/,x0! with 3


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$t is also possi+le to show that if any small c%rve in the plane withnormal nshrin1s on the point x, the limit of a tds divided +y the areais the pro#ection of c%rl aon the normal n

is called#((O)A)#O"A, since the circ%lation aro%nd anyinfinitesimal c%rve vanishes

We now define thecirculationof a vector &%antity aaro%nd aclosed c%rve >, as the inte"ral of a aro%nd > (where is thetan"ent to >!:

)h%s, the c%rl correspond to the circ%lation of aaro%nd aninfinitesimal c%rve. ;oreover, a vector field afor which:

9: if the inte"ral aro%nd any simple closed c%rve vanishes, the val%e ofthe inte"ral from ' to is independent of the path. $n fact, followin" twodifferent paths >, >/ from ' to to form the closed c%rve >,

the total inte"ral vanishes +y hypothesis,meanin" that the two inte"rals alon"different paths are e&%al

4.$* Green+s theorem


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Green's theorem relates a certain volume integral to an integral over the

bounding surface:If we think of aas the flux of some physical property, the integral of an

over the whole surface is the total flux out of a closed volume, which isthus equal to the integral of ain the enclosed volume.

%ppose that V is a vol%me with a closed s%rface and a any vectorfield defined in V and . ?efinin" the diver"ence, we havedemonstrated that, for infinitesimal vol%mes,

$f we s%m all of the infinitesimal vol%mes which constit%te V, we "et avol%me inte"ral on the left5hand5side.

't the ri"ht.5hand side, the contri+%tion of an d , from the to%chin"faces of two ad#acent elements of vol%me are e&%al in ma"nit%de +%topposide in si"n (the o%tward normal point in opposite directiion!

)h%s, s%mmin" %p the terms, only the terms on the o%ter s%rface s%rvive, "ivin":

4.$* Stoes+ theorem


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to1es* theorem relates the s%rface inte"ral of a &%antity awiththe line inte"ral aro%nd the +o%ndin" c%rve of the s%rface.

$f is the s%rface +o%nded +y >, we can divide this s%rface into a lar"e

n%m+er of small trian"les for each of which the e&. a+ove is tr%e %mmin" the ri"ht5hand sides we have the inte"ral over the whole s%rface %mmin" the left5hand sides, the contri+%tions from ad#acent sides of

trian"les will cancel (since they are traversed in opposite directions!,leavin" only the contri+%tions from the +o%ndin" c%rve >, o+tainin":

We have demonstrate that for an infinitesimal trian"%lar area the lineinte"ral of an is e&%al to the pro#ection of the c%rl on the normal n

to1es* theorem says that the total circ%lation of aalon" the +order of as%rface is e&%al to the c%rl of aover the normal of the s%rface

9: 6ne conse&%ence of the form%la is that the field lines of an irrotationalvector field cannot +e closed conto%rs.

4.$- )he classification of vector fields


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)he vector fields can +e cate"oried with respect to their properties

$


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$f c%rl a=,there exists a scalar f%nction s%ch that:which is called the potential of a.

ince c%rl a= , to1es* theorem says that the circ%lation inte"ral

aro%nd any closed c%rve vanishes :

)h%s, the line inte"ral from the ori"in to 2 is independent of the path.$f we define:

, we are th%s s%re that it is a definite scalar f%nction which depends onlyon the position 2. $f we ta1e a near+y point, 3 (x4dx,x/,x0! we have:

$f we choose a line parallel to eto "o from 2 to 3, we have t=e, and so

with A


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$f we repeat the same proced%re with 23 parallel to the other two axesand esta+lishes that:

)h%s, an irrotational field is characteried +y one of the followin" properties:

>%rl a=

-

9: ince is in the direction of the normal to a family of s%rfaces

(x,x/,x0! =constant, the irrotational of the vector field implies that

there is a family of s%rfaces everywhere normal to the tra#ectory of thevector field

4.$ Solenoidal vector fields


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Solenoidal vector fields satisfies

;oreover, it is possi+le todemonstrate that it can +erepresented dependin" on two scalarf%nctions, and,in partic%lar:

Which can +e restated as:

Bere is a vector f%nction of position, which is not %ni&%e +%t m%st+e irrotational

which implies, for the Creen theorem :

$t may also +e demonstrated that, for any finite, contin%o%s vectorfield which vanishes at infinty one can always find 0 scalar

f%nctions, , and(or a solenoidalvector field ! s%ch that:

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