tensor

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Tensor theory is a branch of mathematics and has important applications in mechanics. The term tensor originated from mechanics. It was originally used to express the stress state of each point in the elastic medium. Later, tensor theory developed into a powerful mathematical tool in mechanics and physics. Tensor is important because it can satisfy the property that all physical laws must be independent of the choice of coordinate system. The concept of tensor is the generalization of the concept of vector, and vector is a first-order tensor. Tensor is a multilinear function that can be used to express the linear relationship between some vectors, scalars and other tensors.

Chinese name: tensor
Foreign name: Tensor
Presenter: William Ron Hamilton

Proposed time: 1846
Applicable fields: Continuum mechanics
Applied discipline: Mechanics, Mathematics^[2]

catalog

ten Tensor density
eleven Tensor correlation

Physical name

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tensor (Tensor) is defined in some vector spaces and Dual space The coordinates of a multilinear mapping on the Cartesian product of are| n |In dimensional space, there are| n |Pieces weight A quantity of, in which each component is coordinate function These components are also made according to certain rules during coordinate transformation linear transformation 。 R is called the Rank or rank (and Rank of matrix And order).

stay isomorphism The zero order tensor (r=0) is scalar (Scalar), the first order tensor (r=1) is vector (Vector), the second order tensor (r=2) becomes matrix （Matrix）。 For example, for 3 Dimensional space The tensor when r=1 is a vector: (x, y, z). Due to different transformation methods, tensors are divided into three categories: covariant tensor (the indicator is the lower), contravariant tensor (the indicator is the upper), and mixed tensor (the indicator is the upper and the indicator is the lower).

stay mathematics Tensor is a geometric entity, or "quantity" in a broad sense. The concept of tensor includes scalar, vector and linear operator. Tensors can be expressed in coordinate systems and recorded as scalar arrays, but they are defined as "independent of the selection of reference systems". Tensors are important in physics and engineering. For example, in diffusion tensor imaging, the tensor of the differential permeability of the expression organ to water in all directions can be used to generate brain scans. Perhaps the most important engineering examples are stress tensors and strain tensors, which are second-order tensors. For general linear materials, the relationship between them is determined by a fourth-order elastic tensor.

Although tensors can be represented by multidimensional arrays of components, the significance of the existence of tensor theory lies in further explaining the meaning of calling a quantity a tensor, not just that it needs a certain number of components with index indexes. In particular, in coordinate transformation The component value of the tensor obeys certain transformation rules. The abstract theory of tensors is linear algebra Branches, now called Multilinear algebra 。

Tensor is the basic language of continuum mechanics, and is the necessary basis for establishing the concept system of mechanics. Tensor is a multilinear functional that inputs several vectors and outputs a number, and the assignment is linear with respect to each vector.^[4]

background knowledge

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The word "tensor" was originally used by William Ron Hamilton stay 1846 Introduced, but he used the word to refer to what is now called model Object of. The modern meaning of the word is Waldmar Vogt stay 1899 Started using.

This concept is defined by Gregorio Ricci Courbastro stay 1890 On《 Absolute differential geometry 》Developed under the title of 1900 Levy Chiveta 's classic article《 Absolute differential 》(Italian, later published in other translations), which became known to many mathematicians. along with 1915 about Einstein Of General relativity The introduction of tensor calculus has gained wider recognition. General relativity is completely expressed by tensor language. Einstein learned a lot of tensor language from Levi Chiveta himself (actually Marcel Grossman, who was Einstein's Zurich Federal Institute of Technology His classmate, a geometer, was also Einstein's mentor and helpful friend in tensor language - see Abraham Pais's Subtle is the Lord, and he learned very hard. But tensors are also used in other fields, such as Continuum mechanics , such as Strain tensor (See Linear elasticity ）。

Note that the word "tensor" is often used Tensor field And the tensor field is right manifold Each point of is given a tensor value. To better understand the tensor field, we must first understand the basic ideas of tensors.

regulations

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1. Agreement on summation

It means that in a given item, if there are two indicators that are the same, it means that the indicator is from 1 to the spatial dimension N Sum. For example, in 3D space,

2. Tensor index

Including dummy index and free index. Dumb indicators refer to the same indicators in pairs of the top and bottom indicators. For example, the indicator in the above formula i It is a dummy index. Free index refers to the index that appears only once in all terms of the equation.^[1]

definition

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There are two ways to define tensors:

1. Defined according to transformation law

If a coordinate system

Quantity

With another coordinate system

Quantity

Meet the exchange rule between

be called r Order inversion and s The components of the order covariant mixed tensor. if s =0, then

be called r The component of the order inverse tensor. if r =0, then

be called s The components of the order covariant tensor. The above tensor notation is called component notation.

2. Defined by invariance

Any quantity that can be written in the following invariant form in any coordinate system is defined as r + s Order tensor:

Where

and

Coordinate system

and

The covariant (inverse) basis vector in. The above tensor notation is called invariance notation or dyadic notation.^[1]

Basic operation

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1. Addition and subtraction

The sum (difference) of two or more isomorphic tensors of the same order is still the same as their isomorphic tensors.

2. Merging

The union product of two tensors is a new tensor whose order is equal to the sum of the original two tensor orders.

3. Collapse

The operation of making a superscript and a subscript of a tensor the same, the result is a new tensor of order 2 lower than the original tensor.

4. Dot product

The joint operation of union product and contraction union between two tensors. For example, in the polar decomposition theorem, three second-order tensors R 、 U and V Intermediate primary dot product R·U and V·R The result of is a second order tensor F 。

5. Symmetrization and antisymmetry

For tensored n Index n 1 Different permutations n The operation of the arithmetic mean of a new tensor is called symmetrization. The operation of calculating the arithmetic mean after taking the inverse sign of the new tensor of the index through odd permutation is called inverse symmetry.

6. Additive decomposition

Any second order tensor can be uniquely decomposed into the sum of the symmetric part and the antisymmetric part. For example, velocity gradient

Can be decomposed into

, where

and

Are

The symmetric and antisymmetric parts of the

and

。

1. Law of Quotient

The law affirming the tensority of certain quantities.^[1]

Special tensor

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There are four kinds of special tensors:

① The result of the dot product of two base vectors of a metric tensor.

and

They are called covariant and inverse metric tensors respectively, while mixed metric tensors

, here

(or written as

）For Kronecker Symbol , which is defined as:

② A staggered tensor or an Eddington tensor can be defined as

, here

Representation element

Is determinant, but the replacement symbol

express

（

Is an even permutation of (1,2,3), - 1（

Is odd permutation of (1,2,3), 0 (other cases)

③ Transposed tensor versus arbitrary second order tensor

The result of component index replacement of is recorded as

。

④ Orthogonal tensor A second order tensor that preserves the length of the image.

Christophel symbol

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Category I and II Christophel symbol They are defined as:

and

。^[1]

Covariant derivative and operator

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1. Covariant derivative

Covariant vector

And inverter vector

about

The covariant derivatives of are defined as:

and

。 The above results can be extended to the covariant derivatives of higher-order tensors.

2. Invariant differential operator

The concept of vector analysis is extended to any tensor field T There are four kinds of invariant differential operators, namely gradient ▽ T , divergence ▽· T , curl ▽× T And Laplacian ▽ two T 。

In the rectangular coordinate system, the difference between covariant and contravariant disappears, so it can be specified that all indicators are written as subscripts. In addition, since the Christophel symbol is zero, the covariant derivative becomes a common partial derivative.^[1]

example

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A tensor can be expressed as a sequence of values Define Fields And a scalar value. these ones here Define Fields The vector in is Natural number And these numbers are called index 。 For example, a third-order tensor can have dimensions 2, 5, and 7. Here, indicators range from<1,1,1,>to<2,5,7>. Tensors can have one value in the indicator<1,1,1>, another value in the indicator<1,1,2>, and a total of 70 values. (Similarly, a vector can be expressed as a sequence of values Define Fields And a scalar value. The number in the definition field is Natural number , called indicators, and the number of different indicators is sometimes called vector dimension 。）

A tensor site Is on Euclidean space Each point in is given a tensor value. This is not like the simple example above, which has 70 values. For a third-order tensor, the dimension is<2,5,7>, and every point in the space has 70 values related to it. In other words, a tensor field represents a tensor valued function whose domain of definition is Euclidean space. Not all functions work -- see for more details on these requirements Tensor field 。

Not all relationships in nature are linear, but many are Differentiable So it can be used locally Multilinear mapping To local approximation. In this way, most quantities in physics can be expressed in terms of tensors.

As a simple example, consider a boat in the water. If you want to describe its response to the force, the force is a vector, while the ship's response is an acceleration, which is also a vector. Generally, the acceleration is not in the same direction as the force, because of the specific shape of the hull. However, the relationship between this force and acceleration is actually linear Of. Such a relationship can be represented by a tensor of type (1,1) (that is, it turns one vector into another vector). This tensor can be used matrix That is, when it multiplies one vector, it will get another as the result. When the coordinate system changes, the number representing a vector will change, as will the number representing the tensor matrix.

In engineering, rigid body or fluid Internal stress It is also expressed by a tensor; The Latin word for "tensor" means a certain stretch that causes tension. If a specific surface element in the material is selected, the material on one side of the surface will exert a force on the other side. Normally, this force is not orthogonal to the surface, but it will be linearly dependent on the orientation of the surface. This can be accurately described with a tensor of type (2,0), or more precisely, with a tensor of type (2,0) site Because tensors may be different in each.

Other famous examples of tensors in geometry are Quadratic form , and Curvature tensor 。 Examples of physical tensors are Dynamic tensor ， inertia and Polarization tensor 。

geometry And physical quantities can be considered freedom To classify. Scalars are those that can be represented by a number—— rate ， quality ， temperature , etc. There are some vector type quantities, such as power , it needs a list of numbers to express. Finally, a quantity such as a quadratic form needs to be represented by a multidimensional array. These latter quantities can only be regarded as tensors.

In fact, the concept of tensor is quite extensive and can be used in all the above examples; Scalars and vectors are special cases of tensors. The distinguishing feature between scalars and vectors and between these two and more general tensors is the number of indicators representing their arrays. This number is called tensor rank 。 Thus, a scalar is a tensor of order 0 (no index is required), and a vector is a tensor of order 1.

Another example of tensors is General relativity In Riemannian curvature tensor , which is a fourth order tensor with dimensions<4,4,4>(3 spatial dimensions+time dimensions=4 dimensions). It can be regarded as a matrix (or vector) of 256 components (256=4 × 4 × 4 × 4), which is actually 4 dimension Group). The fact that only 20 components are independent of each other can greatly simplify its actual expression.

Tensor density

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Tensor field There can also be a Density. Density is r The tensor of is the same coordinate transformation as the ordinary tensor, but it has to be multiplied by Jacobian matrix Of determinant Value r Power. The best explanation for this may be to use Vector bundle : where, Cutterplex The determinant bundle of is a Linear plexus , can be used to 'twist' other clumps r Times.

Tensor correlation

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1. The theoretical source of tensor.

Arthur Cayley's research Invariant theory (variant theory) Matrix theory The establishment of determinant The algebraic expression of, which becomes Projective geometry Important tools. Kelley's Invariant theory Britain, which emerged in the first half of the 19th century, focused on Algebra And the application of algebra in geometry. Matrix theory introduces the algebraic definition of vector into the study of linear transformation, which is the precursor of the concept of tensor.

On the other hand, Georg Friedrich Bernhard Riemann proposed N-dimensional manifold The concept of Algebra Formal subject. Riemannian geometry Thought is expanding geometry While improving Algebra The degree of abstraction in representing geometric objects. After Riemann Christophel With the efforts of Ricci, Levi Chevita and others, the tensor analysis Mathematical method ， Riemannian geometry It has also been established.

2. Definition, nature and application value of tensor

Algebraically speaking, it is the generalization of vector. As you know, vector Can be regarded as one-dimensional“ form ”(that is, the components are arranged in a row in order), matrix yes two-dimensional The n-order tensor is the so-called n-dimensional "table". The strict definition of tensor is to use Linear mapping To describe. Similar to vector, the definition is defined by several Coordinate system The set of ordinal numbers satisfying certain coordinate transformation relations when changing is a tensor.

Geometrically speaking, it is a real Geometric quantity In other words, it is a Frame of reference Of coordinate transformation And change things. vector It also has this characteristic.

Scalar can be regarded as a tensor of order 0, vector It can be regarded as a tensor of order 1. There are many special forms in tensor, such as symmetric tensor, antisymmetric tensor and so on.

Sometimes, people directly Coordinate system The tensor is represented by several numbers (called components), and certain transformation rules should be met between components in different coordinate systems (see covariant law, Contravariant Laws), such as matrices, multivariable linear forms, etc. some physical quantity For example, the stress and strain of elastic body and the energy , momentum, etc. need to be expressed in terms of tensors. stay differential geometry In the development of C.F Gaussian 、B. Riemann 、E.B. Christophel The concept of tensor was introduced by G Richie And his student T. Levitzivita developed into tensor analysis, A Einstein In its General relativity Tensors are widely used in.

Riemannian Geometry As a kind of non Euclidean geometry, it is related to Lobachevsky geometry Compared with, there are substantial differences. Roche geometry The main work is to establish a set of Euclid Of《 Geometric primitives 》Logical system of; and Riemannian Geometry The core problem of differential geometry Based on curve Differential method in coordinate system.

Roche geometry It is the first non Euclidean geometry proposed. Its basic viewpoints are:

First, Fifth public establishment Cannot be proved;

Second, a series of reasoning can be carried out in the new axiom system to obtain a series of new theorem To form a new theory.

Roche geometry The difference between the axiom system of Euclidean geometry and Euclidean geometry is that Axiom of parallelism Change to: From a point outside the line, you can make at least two lines parallel to this line. Riemannian Geometry And Roche geometric Axiom of parallelism Contrary: When passing through a point outside the straight line, the straight line cannot be parallel to the known straight line. in other words, Riemannian Geometry It is stipulated that any two straight lines in the same plane have common points, Riemannian geometry Non recognition of existence Parallel line 。 Naturally, there is another common assumption: a straight line can be extended to any length, but the length is limited, which can be compared to a sphere 。 Riemannian Geometry Yes differential geometry So it is fundamentally different from Roche geometry.

Riemannian geometry The axiom system of Riemann introduced a curved geometric space (which can be described by the curvilinear coordinate system introduced by Lame), and Riemann tried to establish a corresponding algebraic structure 。 Riemann himself failed to achieve this goal, but along the road he opened up, Christophel and Ritchie completed the new geometry Construction of. In other words, tensor analysis constitutes Riemannian geometry The core content of.

This is manifested in several aspects:

one Riemannian space The curvature in is a tensor, and its related operations need to be absolute Differential method ；

2. The metric of Riemannian space is expressed by metric tensor;

3. Parallelism in Riemannian space is defined as Scalar product Remains unchanged (i.e included angle Remains unchanged), relying on the Christophel symbol;

4. Lines in Riemannian space（ Geodesic ）The establishment of equation depends on Covariant differential 。

Because of the tool of tensor analysis, Riemannian Geometry Just got something like Calculus The same computing function, thus getting rid of the constraints on the level of logical structure differential geometry It has realized inheritance and the progress of differential geometry from linear coordinate system to curve coordinate system, making geometry And Algebra Connect more closely.

In a word, tensor analysis is a generalization of vector analysis on the one hand, and differential geometry Development. Tensor Analysis and Riemannian Geometry Develop and promote each other in interweaving.

The input of a color image is a three-dimensional tensor whose size is composed of the width, height and depth of the image^[2] 。

From the perspective of data storage form, data can be divided into vector representation, matrix representation and tensor representation^[3] 。

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