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It should also be noted that many mathematical structures informally called 'tensors' are actually 'tensor fields', fields defined over a manifold which define a tensor at every point of the manifold. See the Tensor article for an elementary introduction to tensors.


GEOMETRIC INTRODUCTION


The geometric intuition for a vector field is of an 'arrow' attached to each point of a region, with variable length and direction. Our idea of a vector field on some curved space is supported by the example of a weather map showing horizontal wind velocity, at each point of the Earth's surface.

The general idea of tensor field combines the requirement of richer geometry — for example an Ellipsoid varying from point to point, in the case of a Metric Tensor — with the idea that we don't want our notion to depend on the particular method of mapping the surface. It should exist independently of latitude and longitude, or whatever particular 'cartographic projection' we are using to introduce numerical co-ordinates.


THE VECTOR BUNDLE EXPLANATION


The contemporary mathematical expression of the idea of tensor field breaks it down into a two-step concept.

There is the idea of Vector Bundle , which is a natural idea of ' Vector Space depending on parameters' — the parameters being in a manifold. For example a ''vector space of one dimension depending on an angle'' could look like a Möbius Band as well as a Cylinder . 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


the vector space 'at' ''m''.

Since the Tensor Product 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 (the bundle of Tangent Space s) the whole apparatus explained at Component-free Treatment Of Tensors carries over in a routine way — again independently of co-ordinates, as mentioned in the introduction.

We therefore can give a definition of tensor field, namely as a Section of some tensor bundle. 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

  • \otimes ... \otimes V^---



  • ⊗''F'' if and only if it satisfies ''A''(''fs'',...) = ''fA''(''s'',...) in each argument, where ''f'' is a smooth function on ''M''. Thus a tensor is not only a linear map on the vector space of sections, but a ''C''∞(''M'')-linear map on the module of sections. This property is used to check, for example, that even though the Lie Derivative and Covariant Derivative are not tensors, the Torsion and Curvature Tensor s 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 on the manifold ''M''. Do not confuse this 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{T}^m_n(M)
is the (''m'',''n'') tensor bundle on ''M'' of infinitely-differentiable tensor fields. A tensor field is an element of this set.


THE ''C''<SUP>&INFIN;</SUP>(''M'') MODULE EXPLANATION

There is another more abstract (but often useful) way of characterizing tensor fields on a manifold ''M'' which turns out to actually make tensor fields into honest tensors (i.e. ''single'' multilinear mappings), though of a different type (and 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{T}(M) (see the section on notation above) as a single space — a Module over the Ring 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.

  • (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''))

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).

  • (M) into ''C''∞(''M'').


  • (M) and ''l'' copies of \mathcal{T}(M) into ''C''∞(''M''), it turns out that it arises from a tensor field on ''M'' if and only if it is a 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
abla_{X} 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 is replaced by the Leibnitz rule,
abla_{X}(fY) = (Xf) Y +f
abla_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


For instance, the Curvature Tensor is discussed in differential geometry and the Stress-energy Tensor is important in physics and engineering. Both of these are related by Einstein's theory of General Relativity . In engineering, the underlying manifold will often be Euclidean 3-space .


TENSOR CALCULUS


In Theoretical Physics and other fields, Differential Equation 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 . 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'', 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 ''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. 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 of the Cotangent Bundle . (To be strictly accurate, one should also apply the Absolute Value to the Transition Function s — this makes little difference for an Orientable Manifold .)

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 s on manifolds, and Geometric Quantization .


THE FLAT CASE


Where ''M'' is a Euclidean Space and all the fields are taken to be invariant by Translations 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 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 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 argument justifies abstractly the whole theory.

See also Jet Bundle , Spinor Field .