Тензорлық өрістер: математикалық кеңістікте үздіріссіз өзгеру

Assignment of a tensor continuously varying across a mathematical space

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. If a tensor A is defined on a vector fields set X(M) over a module M, we call A a tensor field on M.

Many mathematical structures called "tensors" are also tensor fields. For example, the Riemann curvature tensor is a tensor field as it associates a tensor to each point of a Riemannian manifold, which is a topological space.

Let M be a manifold, for instance the Euclidean plane Rn. Equivalently, it is a collection of elements Tx ∈ Vx⊗p ⊗ (Vx*)⊗q for all points x ∈ M, arranging into a smooth map T : M → V⊗p ⊗ (V*)⊗q. Elements Tx are called tensors. Often we take V = TM to be the tangent bundle of M.

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 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 , such that given any two vectors at point , their inner product is The field 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 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.

The curvature tensor is discussed in differential geometry and the stress–energy tensor is important in physics, and these two tensors are related by Einstein's theory of general relativity. In electromagnetism, the electric and magnetic fields are combined into an electromagnetic tensor field. It is worth noting that differential forms, used in defining integration on manifolds, are a type of tensor field.

In theoretical 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 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.

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 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 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.) For a more traditional explanation see the tensor density article. One feature of the bundle of densities (again assuming orientability) L is that Ls 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 Ls, and consider tensor density fields with weight s.

Half densities are applied in areas such as defining integral operators on manifolds, and geometric quantization.

When 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.

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.

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 wth power, is called a tensor density with weight w. Invariantly, in the language of multilinear algebra, one can think of tensor densities as multilinear maps taking their values in a density bundle 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 in general relativity. The most common example of a scalar 1 density is the volume element, which in the presence of a metric tensor g is the square root of its determinant in coordinates, denoted The metric tensor is a covariant tensor of order 2, and so its determinant scales by the square of the coordinate transition:

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 bundles, the determinant bundle of the tangent bundle is a line bundle 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 manifolds, 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.