Everything about Tensor totally explained
A
tensor is an object which extends the notion of
scalar,
vector, and
matrix. The term has slightly different meanings in
mathematics and
physics. In the mathematical fields of
multilinear algebra and
differential geometry, a tensor is a
multilinear function. In
physics and
engineering, the same term usually means what a mathematician would call a
tensor field: an association of a different (mathematical) tensor with each point of a geometric space, varying continuously with position.
History
The word
tensor was introduced in 1846 by
William Rowan Hamilton to describe the
norm operation in a certain type of algebraic system (eventually known as a
Clifford algebra). The word was used in its current meaning by
Woldemar Voigt in 1899.
Tensor calculus was developed around 1890 by
Gregorio Ricci-Curbastro under the title
absolute differential calculus, and was made accessible to many mathematicians by the publication of
Tullio Levi-Civita's 1900 classic text of the same name (in Italian; translations followed). In the 20th century, the subject came to be known as
tensor analysis, and achieved broader acceptance with the introduction of
Einstein's theory of
general relativity, around 1915.
General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer
Marcel Grossmann, or perhaps from Levi-Civita himself. Tensors are used also in other fields such as
continuum mechanics.
Two usages of 'tensor'
Mathematical
In
mathematics, a
tensor is (in an informal sense) a generalized
linear 'quantity' or 'geometrical entity' that can be expressed as a multi-dimensional
array relative to a choice of
basis of the particular space on which it's defined. The intuition underlying the tensor concept is inherently geometrical: as an object in and of itself, a tensor is
independent of any chosen frame of reference. However, in the modern treatment, tensor theory is best regarded as a topic in
multilinear algebra. Engineering applications don't usually require the full, general theory, but
theoretical physics now does.
For example, the Euclidean
inner product (
dot product)—a real-valued function of two vectors that's linear in each—is a mathematical tensor. Similarly, on a smooth curved surface such as a
torus, the
metric tensor (field) essentially defines a different
inner product of
tangent vectors at each point of the surface. Just as a
linear transformation can be represented as a
matrix of numbers with respect to given vector
bases, so a tensor can be written as an organized collection of numbers. In physics, the numbers may be obtained as physical quantities that depend on a basis, and the collection is determined to be a tensor if the quantities transform appropriately under change of basis.
Physical - tensor fields
Many mathematical structures informally called 'tensors' are actually
tensor fields—a tensor valued function defined on a geometric or
topological space. This use of the term is analogous to
vector fields such as
electromagnetic fields, but with the 'tensor' defined so that it's invariant under a change of coordinates.
Differential equations posed in terms of tensor quantities are basic to modern
mathematical physics, so that tensor fields are usually defined on
differentiable manifolds.
Tensor rank
In mathematics, the term
rank of a tensor may mean either of two things, and it isn't always clear from the context which.
In the first definition, the rank of a tensor
T is the number of indices required to write down the components of
T. This is the sum of the number of
covariant and
contravariant indices. Expressed by means of the
tensor product of multilinear algebra, this is the number of factors of the tensor product needed to express
T.
In the second definition, the rank of a tensor is defined in a way that extends the definition of the
rank of a matrix given in linear algebra. A
tensor of rank 1 (also called a
simple tensor) is a tensor that can be written as a tensor product of the form
»
where
a,
b,...,
d are in
V or
V*. In indices, a tensor of rank 1 is a tensor of the form
»
Every tensor can be expressed as a
linear combination of rank 1 tensors. In general, the rank of
T is the minimum number of rank 1 tensors with which it's possible to express
T as a linear combination.
For example, a
matrix is a tensor with 2 indices, and so has rank 2 in the first definition. On the other hand, the rank of the tensor in the second definition is just the rank of the matrix. This latter meaning is possibly the intended one, whenever the array of components is two-dimensional.
To avoid this ambiguity, it's now preferred to use the terminology of
tensor order to denote the number of indices, and
tensor rank to designate the number of simple tensors necessary to decompose a tensor. Hence the definition of rank is now used in a way that's consistent with Linear Algebra.
Tensor valence
In physical applications, array indices are distinguished by being
contravariant (superscripts) or
covariant (subscripts), depending upon the type of transformation properties. The
valence of a particular tensor is the number and type of array indices; tensors with the same rank but different valence are not, in general, identical. However, any given
covariant index can be transformed into a contravariant one, and vice versa, by applying the
metric tensor. This operation is generally known as
raising or lowering indices.
Importance and applications
Tensors are important in
physics and
engineering. In the field of
diffusion tensor imaging, for instance, a tensor quantity that expresses the differential permeability of organs to water in varying directions is used to produce scans of the
brain; in this technique tensors are in effect made visible. Perhaps the most important engineering examples are the
stress tensor and
strain tensor, which are both
2nd rank tensors, and are related in a general linear elastic material by a fourth rank
elasticity tensor.
Specifically, a 2nd rank tensor quantifying stress in a 3-dimensional/solid object has components which can be conveniently represented as a 3x3 array. The three Cartesian faces of a cube-shaped infinitesimal volume segment of the solid are each subject to some given force. The force's vector components are also three in number (being in three-space). Thus, 3x3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment (which may now be treated as a point). Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Thus, the need for a 2nd order tensor is produced.
While tensors can be represented by multi-dimensional arrays of components, the point of having a tensor
theory is to explain further implications of saying that a quantity is a
tensor, beyond specifying that it requires a number of indexed components. In particular, tensors behave in specific ways under
coordinate transformations. The abstract theory of tensors is a branch of
linear algebra, now called
multilinear algebra.
The choice of approach
There are two ways of approaching the definition of tensors:
- The usual physics way of defining tensors, in terms of objects whose components transform according to certain rules, introducing the ideas of covariant or contravariant transformations.
The usual mathematics way, which involves defining certain vector spaces and not fixing any coordinate systems until bases are introduced when needed. Contravariant vectors, for instance, can also be described as one-forms, or as the elements of the dual space to the covariant vectors.
Physicists and engineers are among the first to recognise that vectors and tensors have a physical significance as entities, which goes beyond the (often arbitrary) co-ordinate system in which their components are enumerated. Similarly, mathematicians find there are some tensor relations which are more conveniently derived in a co-ordinate notation.
Examples
Physical examples
As a simple example, consider a ship in the water. We want to describe its response to an applied force. Force is a vector, and the ship will respond with an acceleration, which is also a vector. The relationship between force and acceleration is linear in classical mechanics. Such a relationship is described by a rank two tensor of type (1,1) (that is to say, here it transforms a plane vector into another such vector). The tensor can be represented as a matrix which when multiplied by a vector results in another vector. Just as the numbers which represent a vector will change if one changes the coordinate system, the numbers in the matrix that represents the tensor will also change when the coordinate system is changed.
In engineering, the stresses inside a solid body or fluid are also described by a tensor; the word "tensor" is Latin for something that stretches, for example, causes tension. If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force won't be orthogonal to the surface, but it'll depend on the orientation of the surface in a linear manner. This is described by a tensor of type (2,0), in linear elasticity, or more precisely by a tensor field of type (2,0) since the stresses may change from point to point.
Mathematical examples
Some well-known examples of tensors in differential geometry are quadratic forms, such as metric tensors, and the curvature tensor.
Formally speaking, a tensor has a particular type according to the construction with tensor products that give rise to it. For computational purposes, it may be expressed as the sequence of values represented by a function with a tuple-valued domain and a scalar valued range. Domain values are tuples of counting numbers, and these numbers are called indices. For example, a rank 3 tensor might have dimensions 2, 5, and 7. Here, the indices range from «1, 1, 1» through «2, 5, 7»; thus the tensor would have one value at «1, 1, 1», another at «1, 1, 2», and so on for a total of 70 values. As a special case, (finite-dimensional) vectors may be expressed as a sequence of values represented by a function with a scalar valued domain and a scalar valued range; the number of distinct indices is the dimension of the vector. Using this approach, the rank 3 tensor of dimension (2,5,7) can be represented as a 3-dimensional array of size 2 × 5 × 7. In this usage, the number of "dimensions" comprising the array is equivalent to the "rank" of the tensor, and the dimensions of the tensor are equivalent to the "size" of each array dimension.
A tensor field associates a tensor value with every point on a manifold. Thus, instead of simply having 70 values as indicated in the above example, for a rank 3 tensor field with dimensions «2, 5, 7»; every point in the space would have 70 values associated with it. In other words, a tensor field means there's some tensor-valued function which has, for example, Euclidean space as its domain.
Approaches, in detail
There are equivalent approaches to visualizing and working with tensors; that the content is actually the same may only become apparent with some familiarity with the material.
The classical approach
» The classical approach defines a tensor to a collection of multidimensional arrays, such that one array is associated to each possible coordinate system of any fixed vector space. This notion generalizes scalars, vectors, matrices, linear functionals, bilinear forms, etc. To represent a vector x as a tensor one can simply let the array associated to any basis B be the vector of coordinates of x with respect to B.
» However, to count as a tensor, the arrays need to obey a relation that precisely corresponds to how vectors, matrices, linear functionals, etc transform when one passes from one coordinate system to another.
The modern approach
» The modern (component-free) approach views tensors initially as abstract objects, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. This treatment has attempted to replace the component-based treatment for advanced study, in the way that the more modern component-free treatment of vectors replaces the traditional component-based treatment after the component-based treatment has been used to provide an elementary motivation for the concept of a vector. One could say that the slogan is 'tensors are elements of some tensor space'. Nevertheless, a component-free approach hasn't become fully popular, owing to the difficulties involved with giving a geometrical interpretation to higher-rank tensors.
The intermediate treatment of tensors attempts to bridge the two extremes, and to show their relationships.
In the end the same computational content is expressed. See glossary of tensor theory for a listing of technical terms.
Tensor densities
It is also possible for a tensor field to have a "density". A tensor with density r transforms as an ordinary tensor under coordinate transformations, except that it's also multiplied by the determinant of the Jacobian to the rth power. Invariantly, in the language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in the (1-dimensional) space of n-forms (where n is the dimension of the space), as opposed to taking their values in just R. Higher "weights" then just correspond to taking additional tensor products with this space in the range. In the language of vector bundles, the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles r times.
Further Information
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