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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
8/13/2019 Lesson 2OK
88/88
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: