Clasificación

The tensor term indicates either a tensor itself, or a field of tensors. In Multilinear algebra and differential Geometry, fields of mathematics, a tensor indicates a multilinear function . In physics and engineerings, the same term usually indicates what the mathematicians call field of tensor: an application which associates with each point of a geometrical space a different tensor, a tensor which varies continuously with the position.

History

The tensor word is resulting from English of Latin origin tensor , word introduced in 1846 by William Rowan Hamilton to describe the Norme in an algebraic system (finally named Algèbre of Clifford). The word was used with its current direction by Woldemar Voigt in 1899.

Tensorial differential calculus was developed towards 1890 under the name of absolute differential calculus , and was made available to many mathematicians by the publication by Tullio Levi-Civita 1900 traditional text of the same name (in Italian, follow-up of translations). At the 20th century, the subject becomes known under the name of analyzes tensorial , and acquired a broader recognition with the introduction of the theory of relativity of Albert Einstein, around 1915.

General relativity is completely formulated in the language of the tensors. Einstein learned how to use them, with great difficulty, of the geometrician Marcel Grossmann or perhaps of Levi-Civita itself. One also uses the tensors in other fields, such as for example the Mécanique of the continuous mediums.

Definitions

Mathematics

In differential Geometry, a tensor is an object defined on the variety S authorizing in speaking about fields of Endomorphisme S, of fields of multilinear applications as well as the fields of vectors. They generalize the corresponding tools of Linear algebra. The tensors are also necessary tools to carry out analysis on the varieties. Among the important tensors in mathematics, let us quote the metric riemanniennes or the tensor of curve.

There exist several approaches to define a tensor. The formal approach, of use in mathematics, consists in defining the tensors as total fiber sections vectorial obtained by tensorial product, external algebra and symmetrical algebra starting from the tangent Espace and of the space cotangent. The second approach consists in introducing matrices of functions corresponding to the form of the tensor into local charts, checking invariances or contravariances by board swappings. This approach is systematic in Physique, and in particular in General relativity, general Mécanique and Mécanique of the continuous mediums: the objects are not posed a priori as fiber sections but are essential a posteriori like such by coherence in calculations or the theory.

Concept of tensor

Examples with order 0,1 and 2

When one has a bases of a vector Space E on a body $\backslash \; mathbb\; \{K\}$, any vector of this space can be described by its coordinates in this base. In the same way, a linear application between two vector spaces, when one has a base of each one of these spaces, can be described by a matrix.

Thus, in a base $(\backslash \; vec\; e\_\; \{1\},\; \backslash \; vec\; e\_\; \{2\},\; \backslash \; vec\; e\_\; \{3\})$ given, the vector $\backslash \; vec\; u$ will be described by its components ( U ₁, U ₂, U ₃). If one changes basic, the components (the numbers U ₁, U ₂ and U ₃) change, but the vector $\backslash \; vec\; u$ remains the same one. The tensor represents the whole of the representations of $\backslash \; vec\; u$ in all the bases. A vector is a tensor known as “of order 1”.

A Linear application F of a space E towards a space F is described by a matrix M whose coefficients depend on the base of E and of that of F . The tensor represents the whole of the representations of F in all the bases. A matrix is a tensor known as “of order 2”.

A scalar is a simple number, which does not depend on any base. It is said that the scalar is a “tensor of order 0”.

Generalization: tensor of order N

Another manner of seeing is the following one: a matrix M can be noted by its coefficients ( M_ij ), or rather ( M_i^j ), to further see -   that is to say two indices  -, a vector $\backslash \; vec\; u$ by its components ( u_i ) -   that is to say a indice  -, and a scalar has by itself simply -   that is to say zero index. One can consider objects defined with three, four, N indices ( A_ijk… ). an object defined by N indices is a tensor of order N . Attention, the object must moreover check the formulas of basic change (cf distinction between vectors and pseudovecteurs).

On a vector space of finished size m , each index can take the values of 1 with m . A tensor of order N on this vector space thus has m ^N coefficients. If the tensor “connects” N vector spaces of different size m ₁, m ₂,… m_n , then the tensor contains? _{I = 1,… N} m_i coefficients.

Such a tensor of order N represents a multilinear application ( N-linear form ) of $E\; \backslash \; times\; E\; \backslash \; times\dots \; \backslash \; times\; E$ in $\backslash \; mathbb\; \{K\}$:

$T\; (\backslash \; vec\; a~,\; \backslash \; vec\; b~,\dots ,\; \backslash \; vec\; l~)\; =\; a^i\; b^j\dots \; \backslash \; l^u\; \backslash \; T\; (e\_i,\; e\_j,\dots ,\; e\_u)$

One finds the coefficients of the tensor T by identifying $T\_\; \{ij\dots \; U\}\; =\; T\; (e\_i,\; e\_j,\dots ,\; e\_u).\; \backslash$

Physics - field of tensors

Many mathematical structures called informellement “tensor” are in fact of the Champs of tensors, i.e. of a tensor associated with a point with the space, and which thus varies from one point to another. The modern Mathematical physics rests on differential equations posed in terms of tensorial quantities. Thus, the methods of differential Calculus also apply to the tensors.

In physics, the first tensors were introduced to represent the state of constraint and deformation of a volume subjected to force S, from where them name (tensions).

Notations

In the notations, $T\_\; \{ijk\dots \}$ represents the component of the tensor T of coordinates $(I,\; J,\; K,\dots )$. When one wants to indicate a tensor in his globality while indicating the order of this tensor, one can underline the name of the tensor of as much of feature than the order of the tensor. Thus, with this notation, a vector will be noted $\backslash \; underline\; \{U\}$ rather than $\backslash \; vec\; u$, and a tensor of mechanical constraints (of order 2) will be noted $\backslash \; underline\; \{\backslash \; underline\; \{\backslash \; sigma\}\}$. This is particularly useful when one handles tensors of different natures, which is the case in Elastic strain, for which one characterizes the behavior of deformation of materials by a tensor $\backslash \; underline\; \{\backslash \; underline\; \{\backslash \; underline\; \{\backslash \; underline\; \{M\}\}\}\}$ of order 4, and the deformations $\backslash \; underline\; \{\backslash \; underline\; \{\backslash \; forced\; epsilon\}\}$ and $\backslash \; underline\; \{\backslash \; underline\; \{\backslash \; sigma\}\}$ by tensors of order 2.

Order

The order of a tensor is the number of matric indices necessary to describe such a quantity. For example in traditional mechanics Mass, Temperature, and other scalar quantities are of the tensors of order 0, but Force, Déplacement and other vector quantities is of the tensors of order 1. The theory of the tensors offers new aspects starting from order 2 and superiors.

Valence

In the physical applications, one distinguishes the matric indices, according to whether they are Contravariant S (by putting them while exposing) or Covariant S (by putting them in index), according to the behavior from the tensorial size considered vis-a-vis linear transformations of space. The valence of a tensor is the number of the matric indices associated with the type of each one of them; of the same tensors order but of different valences do not behave in the same way during change of the frame of reference. In addition, an index Covariant can be changed into index Contravariant by tensorial product contracted with the metric Tenseur. One invites this operation to raise or lower indices.

One notes the valence by saying that the tensor is of type (N, m) where N is the number of indices contravariants and m the number of indices covariants. The valence does not note the order of the indices. The valence is also used when one notes the tensor by a letter, an index in top means whereas the tensor is contravariant for this index, an index in bottom means that the tensor is covariant for this index. One will thus note the vectors with a high index, and the forms linear with a low index.

Examples:

The Vecteur S are of the tensors of order 1 contravariants. they are thus tensor of valence (1,0)

The linear forms are of the tensors of order 1 covariants. they are of valence (0,1)

The interest of such a notation, it is that in the event of basic change, it directly gives the number of multiplications by the matrix of basic change to carry out: N, and by its reverse: Mr.

For the basic change of a tensor (1,1), there will be a multiplication by the matrix of basic change, and a multiplication by his reverse, exactly as for the matrices in linear algebra.

Importance and applications

The tensors are important in Physique and Engineerings. For example, in medical imagery, one uses a tensorial quantity expressing the differential Perméabilité bodies to water according to the direction to produce cerebral images scanner. The tensor of effort and the tensor of compression are the most important uses in engineerings. They are both of the tensors of order 2, and they are dependant, in a material with linear behavior, by a tensor of elasticity, order 4.

A tensor of order 2 which quantifies compression in a three-dimensional object (a solid) has components that one can easily represent by a table 3x3, where 9 components describe the compression of this element of volume. The interest of a theory of the tensors is to explain the mathematical properties and physics which rises owing to the fact that a quantity is tensorial. In particular, the tensors behave in a special way when one makes a Transformation of coordinates. The abstract theory of the tensors is a branch of the Linear algebra, now called Multilinear algebra.

Examples

In Physics

In physics, a simple example: let us consider a floating boat on water. One wants to describe the effect of the application of a force on the displacement of the center of the boat in the horizontal plane. The force applied can be modelled by a vector, and the acceleration which the boat by another vector will undergo. These two vectors are horizontal. But their directions, which should be identical for an object of round form, are not it any more for one boat, which is lengthened more in a direction than in the other. The relation between the two vectors, which is thus not a relation of proportionality, is however a linear relation, at least if a small force is considered. Such a relation can be described by using a tensor of the type (1,1) (1 time contravariant, 1 time covariant) (i.e. that here it transforms a vector of the plan into another vector of the plan). This tensor can be represented by a matrix (= table of numbers), which, when one multiplies it by a vector, gives another vector. Same manner as the numbers which represent a vector change when one changes frame of reference, the numbers which represent the tensor in the matrix change when the frame of reference changes.

In engineerings, one can also describe the tensions, the interior forces undergone by a solid or a fluid by a tensor. The tensor word comes indeed from the tender verb, which means to subject to a tension. Let us consider an element of surface inside material; the parts of material located on a side of surface exert a force on the other side of surface (and reciprocally). In general, this force is not orthogonal on the surface, but will depend linearly on the orientation of surface. We can describe it by a tensor of linear, tensor elasticity of type (2,0) (2 times contravariant, 0 times covariant), or more precisely, by a field of tensors of the type (2,0), since the forces of tension vary the point-to-point one.

In mathematics

The bilinear forms the such metric tensor or the tensor of curve are well-known examples of tensors in differential geometry.

Formally, the type of tensor depends on the way in which it is defined in term of tensorial product. For example, a tensor of order 3 could have dimensions 2,5,7. Here the indices go from 1,1,1 up to 2,5,7; thus the tensor will have a value with 1,1,1, another to 1,1,2 and so on for a total of 70 values. One can write this tensor like a succession of numbers arranged in a three-dimensional matrix of size 2*5*7. The number of dimensions of the matrix is then equivalent to the order of the tensor.

A field of tensor associates a tensor with each point of a variety. Thus, instead of simply having 70 values, as in the example above, for a tensor of row 3, and dimensions 2,5,7; each point of space would be associated with 70 values. In other words, a field of tensor a function with tensorial value which has as a field, for example, Euclidean space.

Operations on the tensors

The tensors are mathematical objects, more complex than the numbers, but with which one can carry out mathematical operations.

• Multiplication by a scalar. The result is a of the same tensor order and of the same valence. Thus, $(\backslash \; underline\; \{\backslash \; underline\; \{E\}\}\; has)\; \_\; \{ij\}\; =\; has\; \backslash \; underline\; \{\backslash \; underline\; \{E\}\}\; \_\; \{ij\}$.

• Addition of of the same tensors order and same valences. The result is a of the same tensor order and of the same valence than the two starting tensors. In this case, $(\backslash \; underline\; \{\backslash \; underline\; \{has\}\}\; +\; \backslash \; underline\; \{\backslash \; underline\; \{B\}\})\; \_\; \{ij\}$ = $\backslash \; underline\; \{\backslash \; underline\; \{has\}\}\; \_\; \{ij\}\; +\; \backslash \; underline\; \{\backslash \; underline\; \{B\}\}\; \_\; \{ij\}$.

• There exist finally operations specific to the tensors: the tensorial product and the product (tensorial) contracted, which can be carried out between tensors of different natures, and whose result is a tensor still of another kind:

The Produces tensorial between $A$ order N, and $B$ of order p produces a tensor of order (n+p). The valences of the respective indices are unchanged.

As for the produces tensorial contracted between $A$ of order N, and $B$ of order p, it produces a tensor of order (n+p-2). The contraction of the product consists in fact, compared to the tensorial product, to reduce the order of the result of 2, by an equivalent of scalar product between the last component of $A$ and the first of $B$. This analogy with the scalar product is obvious when one applies it to tensors of order 1 (i.e. vectors): in this case, the result is a tensor of order zero (i.e. a scalar) whose value is precisely the scalar product of the 2 vectors. A generalization of this contracted product is the contracted double-product (whose result is a tensor of order n+p-4), the contracted triple-product (whose result is a tensor of order n+p-6), etc In a general way, the contracted product defines a scalar product for the vector space of the tensors of order p. It contracted double-product in particular is very much used to describe the elastic strain of materials.

• lowering of index: a high index can be changed into a low index by multiplication with the metric tensor, g^ab: T^a_c=g^abT_bc

The result is a of the same tensor order but of different valence: a contravariant index became covariant in the tensor result.

• rise in index: a low index can be changed into high index by multiplication with the metric tensor reverses g_ab: T_ac=g_abT^b_c

The result is a tensor of the same order but of different valence: a covariant index became contravariant in the tensor result.

• contraction: equalization of a covariant index and a contravariant index: T^a = T^ac_c (One uses the Convention of Einstein, the sign summons on the index C is implied)

The result is a tensor of order N2 where N is the order of the starting tensor. The valence of the other indices is unchanged.

Operations on the fields of tensors

• Gradient: the gradient of a tensor of order N is a tensor of order n+1. N first indices have the same valence as the starting tensor. The additional index is covariant. It is a kind of space derivative.

Basic changes

Vectors of a space with 3 dimensions

In the $B\; bases\; (\backslash \; vec\; \{E\}\; \_1,\; \backslash \; vec\; \{E\}\; \_2,\; \backslash \; vec\; \{E\}\; \_3)$, the components of the vector $\backslash \; vec\; u$ are ( U ₁, U ₂, U ₃). In the base $B\text{'}\; (\backslash \; vec\; \{e\text{'}\}\; \_1,\; \backslash \; vec\; \{e\text{'}\}\; \_2,\; \backslash \; vec\; \{e\text{'}\}\; \_3)$, they are ( u' ₁, u' ₂, u' ₃). One seeks how to pass from the one to the other of the representations.

In the base B , the vectors of the base B' are written:

$\backslash \; vec\; \{e\text{'}\}\; \_\; \{I\}\; =\; e\_\; \{1i\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; \_\; \{1\}\; +\; e\_\; \{2i\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; \_\; \{2\}\; +\; e\_\; \{3i\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; \_\; \{3\}$

By definition of a base, each vector $\backslash \; vec\; \{E\}\; \_\; \{I\}$ breaks up according to a single linear combination of the vectors of B' . One can thus define the matrix of basic change P of B towards B' :

the columns of the matrix of basic change are the coordinates of the vectors of the old base in the news. There is then

$(u\_\; \{1\},\; u\_\; \{2\},\; u\_\; \{3\})\; =\; P\; \backslash \; cdot\; (u\text{'}\_\; \{1\},\; u\text{'}\_\; \{2\},\; u\text{'}\_\; \{3\})$ and

$(u\text{'}\_\; \{1\},\; u\text{'}\_\; \{2\},\; u\text{'}\_\; \{3\})\; =\; P^\; \{-\; 1\}\; \backslash \; cdot\; (u\_\; \{1\},\; u\_\; \{2\},\; u\_\; \{3\})$.

When the two bases B and B' are orthonormées, P checks in in addition to

$P^\; \{-\; 1\}\; =\; ^\; \{\backslash \; rm\; T\}\; P$.

The basic change is done by multiplication of only one matrix of basic change, the tensor is known as of order 1.

Matrices and linear applications

A matrix M represents a linear application ƒ of a space towards another for a base given in each space. One can thus change basic in the starting space and the space of arrival. One can thus define two matrices, P ₁ and P ₂ for each space. The matrix Me representative ƒ for the two new bases is thus calculated by making

$M\text{'}\; =\; ^\; \{\backslash \; rm\; T\}\; P\_\; \{1\}\; \backslash \; cdot\; M\; \backslash \; cdot\; P\_\; \{2\}$

The basic change is done by multiplication of two matrices of basic change, the tensor is known as of order 2.

Components covariantes and contravariantes

Convention of Einstein

A tensor can have components covariantes and contravariantes, which explains that certain indices are noted in top and others in bottom, for example T_l^mn .

One often adopts the convention of notation of Einstein which consists in summoning when an index is in top and bottom in a product, for example

$\backslash \; sum\_\; \{J\}\; p^\; \{J\}\; \_\; \{I\}\; \backslash \; cdot\; u\_\; \{J\}$ and $\backslash \; sum\_\; \{J\}\; p^\; \{I\}\; \_\; \{J\}\; \backslash \; cdot\; f^\; \{\backslash \; J\}$

note respectively

$p^\; \{J\}\; \_\; \{I\}\; \backslash \; cdot\; u\_\; \{J\}$ and $p^\; \{I\}\; \_\; \{J\}\; \backslash \; cdot\; f^\; \{\backslash \; J\}$

Linear forms and basic change

Let us consider a space with three dimensions provided with a nonorthogonal base (one will suppose it normalized to simplify the presentation). Indeed, there are many examples in nature where there are “natural” axes which are not orthogonal, for examples the axes of some crystals. In fact, when a phenomenon is anisotropic, one can often find axes known as “principal” for which calculations are simplified, and these axes are not always orthogonal.

Let us consider a linear Forme ƒ on this space, which with a vector $\backslash \; vec\; u$ associates a scalar

$f\; (\backslash \; vec\; U)\; =\; f^\; \{1\}\; \backslash \; cdot\; u\_\; \{1\}\; +\; f^\; \{2\}\; \backslash \; cdot\; u\_\; \{2\}\; +\; f^\; \{3\}\; \backslash \; cdot\; u\_\; \{3\}$

(the indices relating to the linear form are noted in top to make it possible to distinguish them). Let us consider the base $(\backslash \; vec\; \{E\}\; ^\; \{\backslash \; *i\})$, known as “dual base”, defined by

$\backslash \; vec\; \{E\}\; ^\; \{\backslash \; *1\}\; =\; \backslash \; vec\; \{E\}\; \_\; \{2\}\; \backslash \; wedge\; \backslash \; vec\; \{E\}\; \_\; \{3\}$

$\backslash \; vec\; \{E\}\; ^\; \{\backslash \; *2\}\; =\; \backslash \; vec\; \{E\}\; \_\; \{3\}\; \backslash \; wedge\; \backslash \; vec\; \{E\}\; \_\; \{1\}$

$\backslash \; vec\; \{E\}\; ^\; \{\backslash \; *3\}\; =\; \backslash \; vec\; \{E\}\; \_\; \{1\}\; \backslash \; wedge\; \backslash vec\; \{E\}\; \_\; \{2\}$

there is then

$\backslash \; vec\; \{E\}\; \_\; \{I\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; ^\; \{\backslash \; *j\}\; =\; \backslash \; delta\_\; \{I\}\; ^j$ (Symbole of Kronecker)

that is to say

$\backslash \; vec\; \{E\}\; \_\; \{I\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; ^\; \{\backslash \; *j\}\; =\; 1$ if $i\; =\; j$

$\backslash \; vec\; \{E\}\; \_\; \{I\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; ^\; \{\backslash \; *j\}\; =\; 0$ if not

If one defines the vector

$\backslash \; vec\; \{F\}\; =\; f^\; \{1\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; ^\; \{\backslash \; *1\}\; +\; f^\; \{2\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; ^\; \{\backslash \; *2\}\; +\; f^\; \{3\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; ^\; \{\backslash \; *3\}$

one can then write

$f\; (\backslash \; vec\; U)\; =\; \backslash \; vec\; \{F\}\; \backslash \; cdot\; \backslash \; vec\; u$

The base of the functions g ⁱ “produces scalar by $\backslash \; vec\; \{E\}\; ^\; \{\backslash \; *i\}$”

$g^i\; \backslash :\; E\; \backslash \; rightarrow\; \backslash \; mathbb\; \{R\}$

$\backslash \; vec\; U\; \backslash \; mapsto\; \backslash \; vec\; \{E\}\; ^\; \{\backslash \; *i\}\; \backslash \; cdot\; \backslash \; vec\; u$

is a base of the linear forms of space; one often identifies this base of functions ( g ^I ) with the base of vectors $(\backslash \; vec\; \{E\}\; ^\; \{\backslash \; *i\})$ itself. The vector space formed by the linear forms is called “dual Espace” or “spaces reciprocal”.

If one makes a basic change of direct space, then the components of the vector $\backslash \; vec\; u$ change according to

$u\_\; \{I\}\; =\; \backslash \; sum\_\; \{J\}\; p^\; \{J\}\; \_\; \{I\}\; \backslash \; cdot\; u\_\; \{J\}$

where p ^j_i is the coefficient of the matrix of basic change (noted e_ji in the preceding paragraph). On the other hand, the components of $\backslash \; vec\; \{F\}$ change according to

$f^\; \{I\}\; =\; \backslash \; sum\_\; \{J\}\; p^\; \{I\}\; \_\; \{J\}\; \backslash \; cdot\; f^\; \{J\}$

it is seen that in the case of the change of the base of linear forms, one multiplies by the matrix of basic change, whereas in the case of the change of the base of vectors, one multiplies by its transposed.

It is thus seen that one has two types of components. On the one hand components of the type “vector”, noted with an index in bottom (for example U _I ), obtained by projection of the vector on the axes parallel to the other axes, and changing during a basic change by the product of transposed of the matrix of basic change ( P ). These components are known as contravariantes .

In addition components of the type “forms linear”, noted with an index in top (for example ƒ ^I ), obtained by projection on the axes perpendicular to the axes ($\backslash \; vec\; \{E\}\; ^\; \{\backslash \; *2\}$ and $\backslash \; vec\; \{E\}\; ^\; \{\backslash \; *3\}$ is perpendicular to $\backslash \; vec\; \{E\}\; \_\; \{1\}$), and changing during a basic change by the product of the “direct” matrix of basic change ( P ). These components are known as covariantes .

According to the formula of basic change of the matrices, one once sees that those are once covariantes, contravariantes, one should thus note M_i ^j . However, one uses only seldom this tensorial notation for the matrices.

Symmetry

In the case of order 2, a tensor can be symmetrical or antisymmetric. For a symmetrical tensor, one with the T_ab relation = T_ba. For an antisymmetric tensor, one with the T_ab relation = - T_ba. In general, a tensor is neither symmetrical, nor antisymmetric. An unspecified tensor can be broken up into a symmetrical part S and an antisymmetric part has, with the relations:

• S_ab = 1/2 (T_ab + T_ba)
• A_ab = 1/2 (T_ab - T_ba)

The symmetrical and antisymmetric parts joined together gather as much information than the original tensor.

This rule can be wide with the tensors of an unspecified nature. It will be said whereas the tensor is symmetrical for a pair of index , if it is invariant by exchange of the two indices, and that it is antisymmetric for a pair of index if it is transformed into its opposite by exchange of the two indices.

The indices of the pair considered must have even valence.

In the particular case of a vector space of dimension 3, an antisymmetric tensor of order 2 bears the name of Pseudovecteur.

See too

Internal bonds

• tensorial Champ

• Produces tensorial
• Tenseur constraints
• Tenseur of the deformations
• Tenseur of the rates of deformations
• metric Tenseur
• dual Espace

External bonds

• http://jgarrigues.perso.egim-mrs.fr/tenseurs.pdf

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