Tensor field
From Wikipedia
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'. Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), Nature) is the science of Nature in the broadest sense. ... Engineering is the application of science to the needs of humanity. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ... Stress has different meanings in different fields: Stress in physics, see also pressure. ... The strain tensor [ε] is a symmetric tensor used to quantify the strain of an object undergoing a 3-dimensional deformation: the diagonal coefficients εii are the relative change in length in the direction of the i direction (along the xi-axis) ; the other terms εij (i ≠ j) are the... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ...
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. This is the current mathematics collaboration of the week! Please help improve it to featured article standard. ...
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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. Definition In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ...
In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...
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 In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...
The fundamental concept in linear algebra is that of a vector space or linear space. ...
The word cylinder has several meanings. ...
- vm in 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 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 mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...
In mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ...
In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...
Note: This is a fairly abstract mathematical approach to tensors. ...
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. Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is...
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. A tensor field assigns to any given point of the manifold a tensor in the space In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... The stress-energy tensor is a tensor quantity in relativity. ... General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
where V is the tangent space at that point and V* is the cotangent space. See also tangent bundle and cotangent bundle. In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...
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
to emphasize that the tangent bundle is a tensor field of (1,0) tensors on the manifold M. Do not confuse this with the very similar looking notation
- ;
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, In mathematics, a smooth function is one that is infinitely differentiable, i. ...
is the (m,n) tensor bundle on M of infinitely-differentiable tensor fields. A tensor field is an element of this set.
Tensor calculus
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. Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, predicting physical phenomena through a physical theory. There are three types of theories in physics; mainstream theories, proposed theories and fringe theories. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ...
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.) In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... Graph of absolute value function In mathematics, the absolute value (or modulus) of a number is the difference between that number and 0. ... This article or section should be merged with Orientability. ...
One feature of the bundle of densities (again assuming orientablity) 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. In mathematical physics, geometric quantization is a mathematical approach to define a quantum theory corresponding to a given classical theory in such a way that certain analogies between the classical theory and the quantum theory remain manifest, for example the similarity between the Heisenberg equation in the Heisenberg picture of...
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. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...
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. In calculus, the chain rule is a formula for the derivative of the composition of two functions. ...
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. For functors in computer science, see the function object article. ...
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 tranform 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. In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in topology. ...
See also jet bundle, spinor field. In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. ...
