Inmathematics andphysics, atensor field is afunction assigning atensor to each point of aregion of amathematical space (typically aEuclidean space ormanifold) or of thephysical space. Tensor fields are used indifferential geometry,algebraic geometry,general relativity, in the analysis ofstress andstrain in material object, and in numerous applications in thephysical sciences. As a tensor is a generalization of ascalar (a pure number representing a value, for example speed) and avector (a magnitude and a direction, like velocity), a tensor field is a generalization of ascalar field and avector field that assigns, respectively, a scalar or vector to each point of space. If a tensorA is defined on a vector fields setX(M) over a moduleM, we callA a tensor field onM.^[1]Many mathematical structures called "tensors" are also tensor fields. For example, theRiemann curvature tensor is a tensorfield as it associates a tensor to each point of aRiemannian manifold, which is atopological space.

LetM be amanifold, for instance theEuclidean planeRⁿ.

Definition. Atensor field of type (p,q) is a section

${\displaystyle T\in \mathrm{\Gamma }(M,V^{\otimes p}\otimes (V^{\ast }{)}^{\otimes q})}$

whereV is avector bundle onM,V^* is itsdual and ⊗ is thetensor product of vector bundles.

Equivalently, it is a collection of elementsT_x∈ V_x^⊗p ⊗ (V_x^*)^⊗q for all pointsx ∈ M, arranging into a smooth mapT : M → V^⊗p ⊗ (V^*)^⊗q. ElementsT_x are calledtensors.

Often we takeV = TM to be thetangent bundle ofM.

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 acurved space 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${\displaystyle g}$, such that given any two vectors${\displaystyle v,w}$ at point${\displaystyle x}$, their inner product is${\displaystyle g_x(v,w)}$. The field${\displaystyle g}$ 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 isTissot's indicatrix.

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.

FollowingSchouten (1951) andMcConnell (1957), the concept of a tensor relies on a concept of a reference frame (orcoordinate system), 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.^[2]

For example, coordinates belonging to then-dimensionalreal coordinate space${\displaystyle {\mathbb{R}}^n}$ may be subjected to arbitraryaffine transformations:

${\displaystyle x^k\mapsto A_j^k{x}^j+a^k}$

(withn-dimensional indices,summation implied). A covariant vector, or covector, is a system of functions${\displaystyle v_k}$ that transforms under this affine transformation by the rule

${\displaystyle v_k\mapsto v_i{A}_k^i.}$

The list of Cartesian coordinate basis vectors${\displaystyle {\mathbf{e}}_k}$ transforms as a covector, since under the affine transformation${\displaystyle {\mathbf{e}}_k\mapsto A_k^i{\mathbf{e}}_i}$. A contravariant vector is a system of functions${\displaystyle v^k}$ of the coordinates that, under such an affine transformation undergoes a transformation

${\displaystyle v^k\mapsto (A^{-1}{)}_j^k{v}^j.}$

This is precisely the requirement needed to ensure that the quantity${\displaystyle v^k{\mathbf{e}}_k}$ is an invariant object that does not depend on the coordinate system chosen. More generally, the coordinates of a tensor of valence (p,q) havep upper indices andq lower indices, with the transformation law being

${\displaystyle {T^{i_1\cdots i_p}}_{j_1\cdots j_q}\mapsto A_{i_1^{\prime }}^{i_1}\cdots A_{i_p^{\prime }}^{i_p}{T^{i_1^{\prime }\cdots i_p^{\prime }}}_{j_1^{\prime }\cdots j_q^{\prime }}(A^{-1}{)}_{j_1}^{j_1^{\prime }}\cdots (A^{-1}{)}_{j_q}^{j_q^{\prime }}.}$

The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to besmooth (ordifferentiable,analytic, etc.). A covector field is a function${\displaystyle v_k}$ of the coordinates that transforms by theJacobian of the transition functions (in the given class). Likewise, a contravariant vector field${\displaystyle v^k}$ transforms by the inverse Jacobian.

A tensor bundle is afiber bundle where the fiber is a tensor product of any number of copies of thetangent space and/orcotangent space of the base space, which is a manifold. As such, the fiber is avector space and the tensor bundle is a special kind ofvector bundle.

The vector bundle is a natural idea of "vector space depending continuously (or smoothly) on parameters" – the parameters being the points of a manifoldM. For example, avector space of one dimension depending on an angle could look like aMöbius strip or alternatively like acylinder. Given a vector bundleV overM, the corresponding field concept is called asection of the bundle: form varying overM, a choice of vector

v_m inV_m,

whereV_m is the vector space "at"m.

Since thetensor product concept is independent of any choice of basis, taking the tensor product of two vector bundles onM is routine. Starting with thetangent bundle (the bundle oftangent spaces) the whole apparatus explained atcomponent-free treatment of tensors carries over in a routine way – again independently of coordinates, as mentioned in the introduction.

We therefore can give a definition oftensor field, namely as asection of sometensor bundle. (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

${\displaystyle V\otimes \cdots \otimes V\otimes V^{\ast }\otimes \cdots \otimes V^{\ast },}$

whereV is thetangent space at that point andV^∗ is thecotangent space. See alsotangent bundle andcotangent bundle.

Given two tensor bundlesE →M andF →M, a linear mapA: Γ(E) → Γ(F) from the space of sections ofE to sections ofF can be considered itself as a tensor section of${\displaystyle E^{\ast }\otimes F}$ if and only if it satisfiesA(fs) =fA(s), for each sections in Γ(E) and each smooth functionf onM. Thus a tensor section is not only a linear map on the vector space of sections, but aC^∞(M)-linear map on themodule of sections. This property is used to check, for example, that even though theLie derivative andcovariant derivative are not tensors, thetorsion andcurvature tensors built from them are.

The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundleTM =T(M) might sometimes be written as

${\displaystyle T_0^1(M)=T(M)=TM}$

to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifoldM. This should not be confused with the very similar looking notation

${\displaystyle T_0^1(V)}$;

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 manifoldM.

Curly (script) letters are sometimes used to denote the set ofinfinitely-differentiable tensor fields onM. Thus,

${\displaystyle {\mathcal{T}}_n^m(M)}$

are the sections of the (m,n) tensor bundle onM that are infinitely-differentiable. A tensor field is an element of this set.

There is another more abstract (but often useful) way of characterizing tensor fields on a manifoldM, which makes tensor fields into honest tensors (i.e.single multilinear mappings), though of a different type (although this isnot usually why one often says "tensor" when one really means "tensor field"). First, we may consider the set of all smooth (C^∞) vector fields onM,${\displaystyle \mathfrak{X}(M):={\mathcal{T}}_0^1(M)}$ (see the section on notation above) as a single space — amodule over thering of smooth functions,C^∞(M), by pointwise scalar multiplication. The notions of multilinearity and tensor products extend easily to the case of modules over anycommutative ring.

As a motivating example, consider the space${\displaystyle {\mathrm{\Omega }}^1(M)={\mathcal{T}}_1^0(M)}$ 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 fieldω and a vector fieldX, we define

${\displaystyle \tilde{\omega }(X)(p):=\omega (p)(X(p)).}$

Because of the pointwise nature of everything involved, the action of${\displaystyle \tilde{\omega }}$ onX is aC^∞(M)-linear map, that is,

${\displaystyle \tilde{\omega }(fX)(p)=\omega (p)((fX)(p))=\omega (p)(f(p)X(p))=f(p)\omega (p)(X(p))=(f\omega )(p)(X(p))=(f\tilde{\omega })(X)(p)}$

for anyp inM and smooth functionf. 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!) onM as multilinear maps on vectors and covectors, we can regard general (k,l) tensor fields onM asC^∞(M)-multilinear maps defined onk copies of${\displaystyle \mathfrak{X}(M)}$ andl copies of${\displaystyle {\mathrm{\Omega }}^1(M)}$ intoC^∞(M).

Now, given any arbitrary mappingT from a product ofk copies of${\displaystyle \mathfrak{X}(M)}$ andl copies of${\displaystyle {\mathrm{\Omega }}^1(M)}$ intoC^∞(M), it turns out that it arises from a tensor field onM if and only if it is multilinear overC^∞(M). Namely${\displaystyle C^{\mathrm{\infty }}(M)}$-module of tensor fields of type${\displaystyle (k,l)}$ over M is canonically isomorphic to${\displaystyle C^{\mathrm{\infty }}(M)}$-module of${\displaystyle C^{\mathrm{\infty }}(M)}$-multilinear forms

${\displaystyle \underset{l\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}{\underbrace{{\mathrm{\Omega }}^1(M)\times \dots \times {\mathrm{\Omega }}^1(M)}}\times \underset{k\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s}}{\underbrace{\mathfrak{X}(M)\times \dots \times \mathfrak{X}(M)}}\to C^{\mathrm{\infty }}(M).}$^[3]

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 theLevi-Civita connection, which is a mapping of smooth vector fields${\displaystyle (X,Y)\mapsto {\mathrm{\nabla }}_{X}Y}$ taking a pair of vector fields to a vector field, does not define a tensor field onM. This is because it is only${\displaystyle \mathbb{R}}$-linear inY (in place of fullC^∞(M)-linearity, it satisfies theLeibniz rule,${\displaystyle {\mathrm{\nabla }}_X(fY)=(Xf)Y+f{\mathrm{\nabla }}_{X}Y}$)). 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.

The curvature tensor is discussed in differential geometry and thestress–energy tensor is important in physics, and these two tensors are related by Einstein's theory ofgeneral relativity.

Inelectromagnetism, the electric and magnetic fields are combined into anelectromagnetic tensor field.

Differential forms, used in defining integration on manifolds, are a type of tensor field.

Intheoretical physics and other fields,differential equations 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 todifferential calculus. Even to formulate such equations requires a fresh notion, thecovariant derivative. This handles the formulation of variation of a tensor fieldalong avector field. The originalabsolute differential calculus notion, which was later calledtensor calculus, led to the isolation of the geometric concept ofconnection.

An extension of the tensor field idea incorporates an extraline bundleL onM. IfW is the tensor product bundle ofV withL, thenW is a bundle of vector spaces of just the same dimension asV. This allows one to define the concept oftensor density, a 'twisted' type of tensor field. Atensor density is the special case whereL is the bundle ofdensities on a manifold, namely thedeterminant bundle of thecotangent bundle. (To be strictly accurate, one should also apply theabsolute value to thetransition functions – this makes little difference for anorientable manifold.) For a more traditional explanation see thetensor density article.

One feature of the bundle of densities (again assuming orientability)L is thatL^s is well-defined for real number values ofs; this can be read from the transition functions, which take strictly positive real values. This means for example that we can take ahalf-density, the case wheres =⁠1/2⁠. In general we can take sections ofW, the tensor product ofV withL^s, and considertensor density fields with weights.

Half-densities are applied in areas such as definingintegral operators on manifolds, andgeometric quantization.

WhenM is aEuclidean space and all the fields are taken to be invariant bytranslations by the vectors ofM, 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, itdoes 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.

As an advanced explanation of thetensor concept, one can interpret thechain rule 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 applyingfunctorial 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 thegeometric nature of tensorquantities; this kind ofdescent argument justifies abstractly the whole theory.

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 theJacobian of the inverse coordinate transformation to thewth power, is called a tensor density with weightw.^[4] Invariantly, in the language of multilinear algebra, one can think of tensor densities asmultilinear maps taking their values in adensity bundle such as the (1-dimensional) space ofn-forms (wheren is the dimension of the space), as opposed to taking their values in justR. 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 theEinstein–Hilbert action in general relativity. The most common example of a scalar 1-density is thevolume element, which in the presence of a metric tensorg is the square root of itsdeterminant in coordinates, denoted${\displaystyle \sqrt{detg}}$. The metric tensor is a covariant tensor of order 2, and so its determinant scales by the square of the coordinate transition:

${\displaystyle det(g^{\prime })={\left(det\frac{\mathrm{\partial }x}{\mathrm{\partial }{x}^{\prime }}\right)}^{2}det(g),}$

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 ofvector bundles, the determinant bundle of thetangent bundle is aline bundle that can be used to 'twist' other bundlesw 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 onorientable manifolds, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle ofn-forms are distinct. For more on the intrinsic meaning, seedensity on a manifold.

• Bitensor – Tensorial object depending on two points in a manifold
• Jet bundle – Construction in differential topology
• Ricci calculus – Tensor index notation for tensor-based calculations
• Spinor field – Geometric structurePages displaying short descriptions of redirect targets

• O'neill, Barrett (1983).Semi-Riemannian Geometry With Applications to Relativity. Elsevier Science.ISBN 9780080570570.
• Frankel, T. (2012),The Geometry of Physics (3rd edition), Cambridge University Press,ISBN 978-1-107-60260-1.
• Lambourne [Open University], R.J.A. (2010),Relativity, Gravitation, and Cosmology, Cambridge University Press,ISBN 978-0-521-13138-4.
• Lerner, R.G.; Trigg, G.L. (1991),Encyclopaedia of Physics (2nd Edition), VHC Publishers.
• McConnell, A. J. (1957),Applications of Tensor Analysis, Dover Publications,ISBN 9780486145020.
• McMahon, D. (2006),Relativity DeMystified, McGraw Hill (USA),ISBN 0-07-145545-0.
• C. Misner, K. S. Thorne, J. A. Wheeler (1973),Gravitation, W.H. Freeman & Co,ISBN 0-7167-0344-0{{citation}}: CS1 maint: multiple names: authors list (link).
• Parker, C.B. (1994),McGraw Hill Encyclopaedia of Physics (2nd Edition), McGraw Hill,ISBN 0-07-051400-3.
• Schouten, Jan Arnoldus (1951),Tensor Analysis for Physicists, Oxford University Press.
• Steenrod, Norman (5 April 1999).The Topology of Fibre Bundles. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press.ISBN 978-0-691-00548-5.OCLC 40734875.

• Multilinear algebra
• Differential geometry
• Differential topology
• Tensors
• Functions and mappings

• Articles with short description
• Short description is different from Wikidata
• Use American English from March 2019
• All Wikipedia articles written in American English
• Pages displaying short descriptions of redirect targets via Module:Annotated link
• CS1 maint: multiple names: authors list