In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. It is used in differential geometry and the theory of manifolds, in algebraic geometry, in general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences and engineering. It is a generalisation of the idea of vector field, which can be thought of as a 'vector that varies from point to point'.

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.

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

v_m in V_m,

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 spaces) 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.

In the end, we 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.

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 functions — this makes little difference for an orientable manifold.)

One feature of the bundle of densities (again assuming orientablity) 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 operators 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.

See also jet bundle.