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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a tensor field assigns a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
to each point of a mathematical space (typically a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
or
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
). Tensor fields are used in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
,
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
,
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, in the analysis of
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
and
strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...
in materials, and in numerous applications in the
physical sciences Physical science is a branch of natural science that studies non-living systems, in contrast to life science. It in turn has many branches, each referred to as a "physical science", together called the "physical sciences". Definition Phy ...
. As a tensor is a generalization of a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
(a pure number representing a value, for example speed) and a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
(a pure number plus a direction, like velocity), a tensor field is a generalization of a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
or vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor is defined on a vector fields set over a module , we call a tensor field on . Many mathematical structures called "tensors" are also tensor fields. For example, the
Riemann curvature tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
is a tensor ''field'' as it is defined on a manifold: it is named after
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
, and associates a tensor to each point of a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
, which is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
.


Geometric introduction

Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a
curved space Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
is a weather map showing horizontal wind velocity at each point of the Earth's surface. Now consider more complicated fields. For example, if the manifold is Riemannian, then it has a metric field g, such that given any two vectors v, w at point x, their inner product is g_x(v, w). The field g could be given in matrix form, but it depends on a choice of coordinates. It could instead be given as an ellipsoid of radius 1 at each point, which is coordinate-free. Applied to the Earth's surface, this is
Tissot's indicatrix In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 ...
. In general, we want to specify tensor fields in a coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates.


Via coordinate transitions

Following and , the concept of a tensor relies on a concept of a reference frame (or
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems. For example, coordinates belonging to the ''n''-dimensional real coordinate space \R^n may be subjected to arbitrary
affine transformation In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, ...
s: :x^k\mapsto A^k_jx^j + a^k (with ''n''-dimensional indices, summation implied). A covariant vector, or covector, is a system of functions v_k that transforms under this affine transformation by the rule :v_k\mapsto v_iA^i_k. The list of Cartesian coordinate basis vectors \mathbf e_k transforms as a covector, since under the affine transformation \mathbf e_k\mapsto A^i_k\mathbf e_i. A contravariant vector is a system of functions v^k of the coordinates that, under such an affine transformation undergoes a transformation :v^k\mapsto (A^)^k_jv^j. This is precisely the requirement needed to ensure that the quantity v^k\mathbf e_k is an invariant object that does not depend on the coordinate system chosen. More generally, a tensor of valence (''p'',''q'') has ''p'' downstairs indices and ''q'' upstairs indices, with the transformation law being :^\mapsto A^_\cdots A^_^(A^)^_\cdots (A^)^_. The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
(or
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, analytic, etc). A covector field is a function v_k of the coordinates that transforms by the Jacobian of the transition functions (in the given class). Likewise, a contravariant vector field v^k transforms by the inverse Jacobian.


Tensor bundles

A tensor bundle is a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
where the fiber is a tensor product of any number of copies of the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
and/or
cotangent space In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
of the base space, which is a manifold. As such, the fiber is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
and the tensor bundle is a special kind of
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
. The vector bundle is a natural idea of "vector space depending continuously (or smoothly) on parameters" – the parameters being the points of a manifold ''M''. For example, a ''vector space of one dimension depending on an angle'' could look like a Möbius strip or alternatively like a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
. Given a vector bundle ''V'' over ''M'', the corresponding field concept is called a ''section'' of the bundle: for ''m'' varying over ''M'', a choice of vector :''vm'' in ''Vm'', where ''Vm'' is the vector space "at" ''m''. Since the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
concept is independent of any choice of basis, taking the tensor product of two vector bundles on ''M'' is routine. Starting with the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
(the bundle of
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s) the whole apparatus explained at
component-free treatment of tensors In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or ...
carries over in a routine way – again independently of coordinates, as mentioned in the introduction. We therefore can give a definition of tensor field, namely as a
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of some
tensor bundle In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of t ...
. (There are vector bundles that are not tensor bundles: the Möbius band for instance.) This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space :V \otimes \cdots \otimes V \otimes V^* \otimes \cdots \otimes V^* , where ''V'' is the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at that point and ''V'' is the
cotangent space In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
. See also
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
and cotangent bundle. Given two tensor bundles ''E'' → ''M'' and ''F'' → ''M'', a linear map ''A'': Γ(''E'') → Γ(''F'') from the space of sections of ''E'' to sections of ''F'' can be considered itself as a tensor section of \scriptstyle E^*\otimes F if and only if it satisfies ''A''(''fs'') = ''fA''(''s''), for each section ''s'' in Γ(''E'') and each smooth function ''f'' on ''M''. Thus a tensor section is not only a linear map on the vector space of sections, but a ''C''(''M'')-linear map on the
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
of sections. This property is used to check, for example, that even though the
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
and covariant derivative are not tensors, the
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
and curvature tensors built from them are.


Notation

The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle ''TM'' = ''T''(''M'') might sometimes be written as :T_0^1(M)=T(M) =TM to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold ''M''. This should not be confused with the very similar looking notation :T_0^1(V); in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold ''M''. Curly (script) letters are sometimes used to denote the set of infinitely-differentiable tensor fields on ''M''. Thus, :\mathcal^m_n(M) are the sections of the (''m'',''n'') tensor bundle on ''M'' that are infinitely-differentiable. A tensor field is an element of this set.


The ''C''(''M'') module explanation

There is another more abstract (but often useful) way of characterizing tensor fields on a manifold ''M'', which makes tensor fields into honest tensors (i.e. ''single'' multilinear mappings), though of a different type (although this is ''not'' usually why one often says "tensor" when one really means "tensor field"). First, we may consider the set of all smooth (C) vector fields on ''M'', \mathcal(M) (see the section on notation above) as a single space — a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
over the
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
of smooth functions, ''C''(''M''), by pointwise scalar multiplication. The notions of multilinearity and tensor products extend easily to the case of modules over any
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
. As a motivating example, consider the space \mathcal^*(M) of smooth covector fields ( 1-forms), also a module over the smooth functions. These act on smooth vector fields to yield smooth functions by pointwise evaluation, namely, given a covector field ''ω'' and a vector field ''X'', we define :(''ω''(''X''))(''p'') = ''ω''(''p'')(''X''(''p'')). Because of the pointwise nature of everything involved, the action of ''ω'' on ''X'' is a ''C''(''M'')-linear map, that is, :(''ω''(''fX''))(''p'') = ''f''(''p'')''ω''(''p'')(''X''(''p'')) = (''fω'')(''p'')(''X''(''p'')) = (''fω''(''X''))(''p'') for any ''p'' in ''M'' and smooth function ''f''. Thus we can regard covector fields not just as sections of the cotangent bundle, but also linear mappings of vector fields into functions. By the double-dual construction, vector fields can similarly be expressed as mappings of covector fields into functions (namely, we could start "natively" with covector fields and work up from there). In a complete parallel to the construction of ordinary single tensors (not tensor fields!) on ''M'' as multilinear maps on vectors and covectors, we can regard general (''k'',''l'') tensor fields on ''M'' as ''C''(''M'')-multilinear maps defined on ''l'' copies of \mathcal(M) and ''k'' copies of \mathcal^*(M) into ''C''(''M''). Now, given any arbitrary mapping ''T'' from a product of ''k'' copies of \mathcal^*(M) and ''l'' copies of \mathcal(M) into ''C''(''M''), it turns out that it arises from a tensor field on ''M'' if and only if it is multilinear over ''C''(''M''). Thus this kind of multilinearity implicitly expresses the fact that we're really dealing with a pointwise-defined object, i.e. a tensor field, as opposed to a function which, even when evaluated at a single point, depends on all the values of vector fields and 1-forms simultaneously. A frequent example application of this general rule is showing that the Levi-Civita connection, which is a mapping of smooth vector fields (X,Y) \mapsto \nabla_ Y taking a pair of vector fields to a vector field, does not define a tensor field on ''M''. This is because it is only ''R''-linear in ''Y'' (in place of full ''C''(''M'')-linearity, it satisfies the ''Leibniz rule,'' \nabla_(fY) = (Xf) Y +f \nabla_X Y)). Nevertheless, it must be stressed that even though it is not a tensor field, it still qualifies as a geometric object with a component-free interpretation.


Applications

The curvature tensor is discussed in differential geometry and the
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
is important in physics, and these two tensors are related by Einstein's theory of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. In
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
, the electric and magnetic fields are combined into an electromagnetic tensor field. It is worth noting that
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s, used in defining integration on manifolds, are a type of tensor field.


Tensor calculus

In
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
and other fields,
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to
differential calculus In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
. Even to formulate such equations requires a fresh notion, the covariant derivative. This handles the formulation of variation of a tensor field ''along'' a vector field. The original ''absolute differential calculus'' notion, which was later called ''
tensor calculus In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
'', led to the isolation of the geometric concept of connection.


Twisting by a line bundle

An extension of the tensor field idea incorporates an extra
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
''L'' on ''M''. If ''W'' is the tensor product bundle of ''V'' with ''L'', then ''W'' is a bundle of vector spaces of just the same dimension as ''V''. This allows one to define the concept of tensor density, a 'twisted' type of tensor field. A ''tensor density'' is the special case where ''L'' is the bundle of ''densities on a manifold'', namely the
determinant bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisi ...
of the cotangent bundle. (To be strictly accurate, one should also apply the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
to the
transition functions In mathematics, a transition function may refer to: * a transition map between two charts of an atlas of a manifold or other topological space * the function that defines the transitions of a state transition system in computing, which may refer mo ...
– this makes little difference for an
orientable manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
.) For a more traditional explanation see the
tensor density In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is ...
article. One feature of the bundle of densities (again assuming orientability) ''L'' is that ''L''''s'' is well-defined for real number values of ''s''; this can be read from the transition functions, which take strictly positive real values. This means for example that we can take a ''half-density'', the case where ''s'' = ½. In general we can take sections of ''W'', the tensor product of ''V'' with ''L''''s'', and consider tensor density fields with weight ''s''. Half-densities are applied in areas such as defining
integral operator An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms invol ...
s on manifolds, and
geometric quantization In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a wa ...
.


The flat case

When ''M'' is a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
and all the fields are taken to be invariant by
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
by the vectors of ''M'', we get back to a situation where a tensor field is synonymous with a tensor 'sitting at the origin'. This does no great harm, and is often used in applications. As applied to tensor densities, it ''does'' make a difference. The bundle of densities cannot seriously be defined 'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion.


Cocycles and chain rules

As an advanced explanation of the ''tensor'' concept, one can interpret the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
in the multivariable case, as applied to coordinate changes, also as the requirement for self-consistent concepts of tensor giving rise to tensor fields. Abstractly, we can identify the chain rule as a 1- cocycle. It gives the consistency required to define the tangent bundle in an intrinsic way. The other vector bundles of tensors have comparable cocycles, which come from applying
functorial In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
properties of tensor constructions to the chain rule itself; this is why they also are intrinsic (read, 'natural') concepts. What is usually spoken of as the 'classical' approach to tensors tries to read this backwards – and is therefore a heuristic, ''post hoc'' approach rather than truly a foundational one. Implicit in defining tensors by how they transform under a coordinate change is the kind of self-consistency the cocycle expresses. The construction of tensor densities is a 'twisting' at the cocycle level. Geometers have not been in any doubt about the ''geometric'' nature of tensor ''quantities''; this kind of
descent Descent may refer to: As a noun Genealogy and inheritance * Common descent, concept in evolutionary biology * Kinship, one of the major concepts of cultural anthropology **Pedigree chart or family tree ** Ancestry ** Lineal descendant **Heritag ...
argument justifies abstractly the whole theory.


Generalizations


Tensor densities

The concept of a tensor field can be generalized by considering objects that transform differently. An object that transforms as an ordinary tensor field under coordinate transformations, except that it is also multiplied by the determinant of the Jacobian of the inverse coordinate transformation to the ''w''th power, is called a tensor density with weight ''w''. Invariantly, in the language of multilinear algebra, one can think of tensor densities as
multilinear map In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W are ...
s taking their values in a
density bundle In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called the d ...
such as 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. A special case are the scalar densities. Scalar 1-densities are especially important because it makes sense to define their integral over a manifold. They appear, for instance, in the
Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...
in general relativity. The most common example of a scalar 1-density is the
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
, which in the presence of a metric tensor ''g'' is the square root of its
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
in coordinates, denoted \sqrt. The metric tensor is a covariant tensor of order 2, and so its determinant scales by the square of the coordinate transition: :\det(g') = \left(\det\frac\right)^2\det(g), which is the transformation law for a scalar density of weight +2. More generally, any tensor density is the product of an ordinary tensor with a scalar density of the appropriate weight. In the language of
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s, the determinant bundle of the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
is a
line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
that can be used to 'twist' other bundles ''w'' times. While locally the more general transformation law can indeed be used to recognise these tensors, there is a global question that arises, reflecting that in the transformation law one may write either the Jacobian determinant, or its absolute value. Non-integral powers of the (positive) transition functions of the bundle of densities make sense, so that the weight of a density, in that sense, is not restricted to integer values. Restricting to changes of coordinates with positive Jacobian determinant is possible on
orientable manifold In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
s, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of ''n''-forms are distinct. For more on the intrinsic meaning, see
density on a manifold In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a section of a certain line bundle, called t ...
.


See also

* * *


Notes


References

* * . * . * . * . * . * . * . * . * {{Manifolds Multilinear algebra Differential geometry Differential topology Tensors